From 0b7dc7680496b19c48b8c8ebdd7c36b20a3f292e Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Mon, 6 May 2024 15:57:58 +0000 Subject: [PATCH] build based on bc0818c --- dev/.documenter-siteinfo.json | 1 + dev/acceleration/index.html | 6 +- dev/api/index.html | 8 +- dev/assets/documenter.js | 913 ++++++++++++++++-- dev/assets/search.js | 267 ----- dev/assets/themes/documenter-dark.css | 4 +- dev/assets/themes/documenter-light.css | 4 +- dev/assets/themeswap.js | 82 +- dev/assets/warner.js | 87 +- dev/citing/index.html | 6 +- dev/contributing/index.html | 9 +- dev/decomposition/index.html | 8 +- .../closest_correlation_matrix/index.html | 10 +- dev/examples/index.html | 6 +- dev/examples/logistic_regression/10279931.svg | 349 +++++++ dev/examples/logistic_regression/f5fc5625.svg | 341 +++++++ dev/examples/logistic_regression/index.html | 708 +------------- dev/examples/lovasz_petersen/abe5edc1.svg | 95 ++ dev/examples/lovasz_petersen/index.html | 105 +- dev/examples/lp/index.html | 6 +- dev/examples/maxcut/b98ec54e.svg | 47 + dev/examples/maxcut/index.html | 57 +- .../portfolio_optimisation/11509310.svg | 47 + .../portfolio_optimisation/index.html | 69 +- dev/examples/qp/index.html | 12 +- dev/examples/sum_abs_k_eigenvalues/index.html | 48 +- dev/examples/svm_primal/2f3f2fa2.svg | 144 +++ dev/examples/svm_primal/ba6f4b3f.svg | 148 +++ dev/examples/svm_primal/index.html | 306 +----- dev/examples/two_way_partitioning/index.html | 10 +- dev/getting_started/index.html | 8 +- dev/index.html | 8 +- dev/jump/index.html | 6 +- dev/lin_solver/index.html | 6 +- .../build/arbitrary_precision/index.html | 10 +- dev/literate/build/custom_cone/index.html | 12 +- .../portfolio_model_updates/ab8679a2.svg | 47 + .../build/portfolio_model_updates/index.html | 53 +- dev/logistic_regression/index.html | 4 +- dev/method/index.html | 6 +- dev/objects.inv | Bin 0 -> 2488 bytes dev/performance/index.html | 4 +- dev/search/index.html | 6 - dev/search_index.js | 2 +- 44 files changed, 2291 insertions(+), 1784 deletions(-) create mode 100644 dev/.documenter-siteinfo.json delete mode 100644 dev/assets/search.js create mode 100644 dev/examples/logistic_regression/10279931.svg create mode 100644 dev/examples/logistic_regression/f5fc5625.svg create mode 100644 dev/examples/lovasz_petersen/abe5edc1.svg create mode 100644 dev/examples/maxcut/b98ec54e.svg create mode 100644 dev/examples/portfolio_optimisation/11509310.svg create mode 100644 dev/examples/svm_primal/2f3f2fa2.svg create mode 100644 dev/examples/svm_primal/ba6f4b3f.svg create mode 100644 dev/literate/build/portfolio_model_updates/ab8679a2.svg create mode 100644 dev/objects.inv delete mode 100644 dev/search/index.html diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json new file mode 100644 index 00000000..23049a6b --- /dev/null +++ b/dev/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.10.3","generation_timestamp":"2024-05-06T15:57:53","documenter_version":"1.4.1"}} \ No newline at end of file diff --git a/dev/acceleration/index.html b/dev/acceleration/index.html index 8b4d96ee..9d504f4a 100644 --- a/dev/acceleration/index.html +++ b/dev/acceleration/index.html @@ -1,7 +1,7 @@ -Acceleration · COSMO.jl
Edit on GitHub

Acceleration

By default COSMO's ADMM algorithm is wrapped in a safeguarded acceleration method to achieve faster convergence to higher precision. COSMO uses accelerators from the COSMOAccelerators.jl package.

By default, the solver uses the accelerator type AndersonAccelerator{T, Type2{QRDecomp}, RestartedMemory, NoRegularizer}. This is the classic type-II Anderson acceleration method where the least squares subproblem is solved using an updated QR method. Moreover, the method is restarted, i.e. the history of iterates is deleted, after mem steps and no regularisation for the least-squares method is used.

In addition, the method is safeguarded (safeguard = true), i.e. the residual-norm of the accelerated point can not deviate too much from the current point. Otherwise, the point is discarded and the ADMM algorithm performs a normal step instead.

The acceleration method can be altered as usual via the solver settings and the accelerator keyword. To deactivate acceleration pass an EmptyAccelerator:

settings = COSMO.Settings(accelerator = EmptyAccelerator)

To use the default accelerator but with a different memory size (number of stored iterates) use:

settings = COSMO.Settings(accelerator = with_options(AndersonAccelerator, mem = 15))

To turn the safeguarding off use:

settings = COSMO.Settings(safeguard = false)

To use an Anderson Accelerator of Type-I with a rolling-memory (oldest iterate replaced by newest) approach, use:

settings = COSMO.Settings(accelerator = AndersonAccelerator{Float64, Type1, RollingMemory, NoRegularizer})

For more fine-grained control look at the implementation of the accelerator here.

When JuMP is used, the accelerator settings can be passed in the usual way:

model = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, "accelerator" => with_options(AndersonAccelerator, mem = 15)))

Or using the set_optimizer_attribute() method:

model = JuMP.Model(COSMO.Optimizer);
-set_optimizer_attribute(model, "accelerator", with_options(AndersonAccelerator, mem = 15))
+
GitHub

Acceleration

By default COSMO's ADMM algorithm is wrapped in a safeguarded acceleration method to achieve faster convergence to higher precision. COSMO uses accelerators from the COSMOAccelerators.jl package.

By default, the solver uses the accelerator type AndersonAccelerator{T, Type2{QRDecomp}, RestartedMemory, NoRegularizer}. This is the classic type-II Anderson acceleration method where the least squares subproblem is solved using an updated QR method. Moreover, the method is restarted, i.e. the history of iterates is deleted, after mem steps and no regularisation for the least-squares method is used.

In addition, the method is safeguarded (safeguard = true), i.e. the residual-norm of the accelerated point can not deviate too much from the current point. Otherwise, the point is discarded and the ADMM algorithm performs a normal step instead.

The acceleration method can be altered as usual via the solver settings and the accelerator keyword. To deactivate acceleration pass an EmptyAccelerator:

settings = COSMO.Settings(accelerator = EmptyAccelerator)

To use the default accelerator but with a different memory size (number of stored iterates) use:

settings = COSMO.Settings(accelerator = with_options(AndersonAccelerator, mem = 15))

To turn the safeguarding off use:

settings = COSMO.Settings(safeguard = false)

To use an Anderson Accelerator of Type-I with a rolling-memory (oldest iterate replaced by newest) approach, use:

settings = COSMO.Settings(accelerator = AndersonAccelerator{Float64, Type1, RollingMemory, NoRegularizer})

For more fine-grained control look at the implementation of the accelerator here.

When JuMP is used, the accelerator settings can be passed in the usual way:

model = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, "accelerator" => with_options(AndersonAccelerator, mem = 15)))

Or using the set_optimizer_attribute() method:

model = JuMP.Model(COSMO.Optimizer);
+set_optimizer_attribute(model, "accelerator", with_options(AndersonAccelerator, mem = 15))
diff --git a/dev/api/index.html b/dev/api/index.html index 2de51ecc..944afc62 100644 --- a/dev/api/index.html +++ b/dev/api/index.html @@ -1,12 +1,12 @@ -API Reference · COSMO.jl
Edit on GitHub

API Reference

Model

COSMO.Model — Type
Model{T <: AbstractFloat}()

Initializes an empty COSMO model that can be filled with problem data using assemble!(model, P, q,constraints; [settings, x0, s0, y0]).

source
COSMO.assemble! — Function
assemble!(model, P, q, constraint(s); [settings, x0, y0, s0])

Assembles a COSMO.Model with a cost function defind by P and q, and a number of constraints.

The positive semidefinite matrix P and vector q are used to specify the cost function of the optimization problem:

min   1/2 x'Px + q'x
+
GitHub
API Reference
Model
COSMO.Model — Type
Model{T <: AbstractFloat}()
Initializes an empty COSMO model that can be filled with problem data using assemble!(model, P, q,constraints; [settings, x0, s0, y0]).
source
COSMO.assemble! — Function
assemble!(model, P, q, constraint(s); [settings, x0, y0, s0])
Assembles a COSMO.Model with a cost function defind by P and q, and a number of constraints.
The positive semidefinite matrix P and vector q are used to specify the cost function of the optimization problem:
min   1/2 x'Px + q'x
 s.t.  Ax + b ∈ C
constraints is a COSMO.Constraint or an array of COSMO.Constraint objects that are used to describe the constraints on x.
The optional keyword argument settings can be used to pass custom solver settings:
custom_settings = COSMO.Settings(verbose = true);
 assemble!(model, P, q, constraints, settings = custom_settings)
The optional keyword arguments x0 and y0 can be used to provide the solver with warm starting values for the primal variable x and the dual variable y.
x_0 = [1.0; 5.0; 3.0]
-COSMO.assemble!(model, P, q, constraints, x0 = x_0)
source
COSMO.set! — Function
set!(model, P, q, A, b, convex_sets, [settings])
Sets model data directly based on provided fields.
source
COSMO.empty_model! — Function
empty_model!(model)
Resets all the fields of model to that of a model created with COSMO.Model() (apart from the settings).
source
COSMO.warm_start_primal! — Function
warm_start_primal!(model, x0, [ind])
Provides the COSMO.Model with warm starting values for the primal variable x. ind can be used to warm start certain components of x.
source
COSMO.warm_start_slack! — Function
warm_start_slack!(model, s0, [ind])
Provides the COSMO.Model with warm starting values for the primal slack variable s. ind can be used to warm start certain components of s.
source
COSMO.warm_start_dual! — Function
warm_start_dual!(model, y0, [ind])
Provides the COSMO.Model with warm starting values for the dual variable y. ind can be used to warm start certain components of y.
source
Constraints
COSMO.Constraint — Type
Constraint{T <: AbstractFloat}(A, b, convex_set_type, dim = 0, indices = 0:0)
Creates a COSMO constraint: Ax + b ∈ convex_set.
By default the following convex set types are supported: ZeroSet, Nonnegatives, SecondOrderCone, PsdCone, PsdConeTriangle.
Examples
julia> COSMO.Constraint([1 0;0 1], zeros(2), COSMO.Nonnegatives)
+COSMO.assemble!(model, P, q, constraints, x0 = x_0)
source
COSMO.set! — Function
set!(model, P, q, A, b, convex_sets, [settings])
Sets model data directly based on provided fields.
source
COSMO.empty_model! — Function
empty_model!(model)
Resets all the fields of model to that of a model created with COSMO.Model() (apart from the settings).
source
COSMO.warm_start_primal! — Function
warm_start_primal!(model, x0, [ind])
Provides the COSMO.Model with warm starting values for the primal variable x. ind can be used to warm start certain components of x.
source
COSMO.warm_start_slack! — Function
warm_start_slack!(model, s0, [ind])
Provides the COSMO.Model with warm starting values for the primal slack variable s. ind can be used to warm start certain components of s.
source
COSMO.warm_start_dual! — Function
warm_start_dual!(model, y0, [ind])
Provides the COSMO.Model with warm starting values for the dual variable y. ind can be used to warm start certain components of y.
source
Constraints
COSMO.Constraint — Type
Constraint{T <: AbstractFloat}(A, b, convex_set_type, dim = 0, indices = 0:0)
Creates a COSMO constraint: Ax + b ∈ convex_set.
By default the following convex set types are supported: ZeroSet, Nonnegatives, SecondOrderCone, PsdCone, PsdConeTriangle.
Examples
julia> COSMO.Constraint([1 0;0 1], zeros(2), COSMO.Nonnegatives)
 Constraint
 Size of A: (2, 2)
 ConvexSet: COSMO.Nonnegatives{Float64}
For convex sets that require their own data, it is possible to pass the pass the instantiated object directly rather than the type name.
Examples
julia> COSMO.Constraint([1 0;0 1], zeros(2), COSMO.Box([-1.;-1.],[1.;1.]))
@@ -18,4 +18,4 @@
 ConvexSet: COSMO.ZeroSet{Float64}
Notice that extra columns of A have been added automatically.
julia>Matrix(c.A)
 2×4 Array{Float64,2}:
 0.0  1.0  0.0  0.0
-0.0  0.0  1.0  0.0
source
COSMO.ZeroSet — Type
ZeroSet(dim)
Creates the zero set $\{ 0 \}^{dim}$ of dimension dim. If x ∈ ZeroSet then all entries of x are zero.
source
COSMO.Nonnegatives — Type
Nonnegatives(dim)
Creates the nonnegative orthant $\{ x \in \mathbb{R}^{dim} : x \ge 0 \}$  of dimension dim.
source
COSMO.Box — Type
Box(l, u)
Creates a box or intervall with lower boundary vector $l \in  \mathbb{R}^m \cup \{-\infty\}^m$ and upper boundary vector$u \in \mathbb{R}^m\cup \{+\infty\}^m$.
source
COSMO.SecondOrderCone — Type
SecondOrderCone(dim)
Creates the second-order cone (or Lorenz cone) $\{ (t,x) \in \mathrm{R}^{dim} : || x ||_2  \leq t \}$.
source
COSMO.PsdCone — Type
PsdCone(dim)
Creates the cone of symmetric positive semidefinite matrices $\mathcal{S}_+^{dim}$. The entries of the matrix X are stored column-by-column in the vector x of dimension dim. Accordingly  $X \in \mathbb{S}_+ \Rightarrow x \in \mathcal{S}_+^{dim}$, where $X = \text{mat}(x)$.
source
COSMO.PsdConeTriangle — Type
PsdConeTriangle(dim)
Creates the cone of symmetric positive semidefinite matrices. The entries of the upper-triangular part of matrix X are stored in the vector x of dimension dim. A $r \times r$ matrix has $r(r+1)/2$ upper triangular elements and results in a vector of $\mathrm{dim} = r(r+1)/2$.
Examples
The matrix
\[\begin{bmatrix} x_1 & x_2 & x_4\\ x_2 & x_3 & x_5\\ x_4 & x_5 & x_6 \end{bmatrix}\]
is transformed to the vector $[x_1, x_2, x_3, x_4, x_5, x_6]^\top$ with corresponding constraint  PsdConeTriangle(6).
source
COSMO.ExponentialCone — Type
ExponentialCone(MAX_ITERS = 100, EXP_TOL = 1e-8)
Creates the exponential cone $\mathcal{K}_{exp} = \{(x, y, z) \mid y \geq 0 ye^{x/y} ≤ z\} \cup \{ (x,y,z) \mid   x \leq 0, y = 0, z \geq 0 \}$
source
COSMO.DualExponentialCone — Type
DualExponentialCone(MAX_ITERS::Int = 100, EXP_TOL = 1e-8)
Creates the dual exponential cone $\mathcal{K}^*_{exp} = \{(x, y, z) \mid x < 0,  -xe^{y/x} \leq e^1 z \} \cup \{ (0,y,z) \mid   y \geq 0, z \geq 0 \}$
source
COSMO.PowerCone — Type
PowerCone(alpha::Float64, MAX_ITERS::Int = 20, POW_TOL = 1e-8)
Creates the 3-d power cone $\mathcal{K}_{pow} = \{(x, y, z) \mid x^\alpha y^{(1-\alpha)} \geq  \|z\|, x \geq 0, y \geq 0 \}$ with $0 < \alpha < 1$
source
COSMO.DualPowerCone — Type
DualPowerCone(alpha::Float64, MAX_ITERS::Int = 20, POW_TOL = 1e-8)
Creates the 3-d dual power cone $\mathcal{K}^*_{pow} = \{(u, v, w) \mid \left( \frac{u}{\alpha}\right)^\alpha \left( \frac{v}{1-\alpha}\right)^{(1-\alpha)} \geq  \|w\|, u \geq 0, v \geq 0 \}$ with $0 < \alpha < 1$
source

+0.0  0.0  1.0  0.0
source
COSMO.ZeroSet — Type
ZeroSet(dim)

Creates the zero set $\{ 0 \}^{dim}$ of dimension dim. If x ∈ ZeroSet then all entries of x are zero.

source
COSMO.Nonnegatives — Type
Nonnegatives(dim)

Creates the nonnegative orthant $\{ x \in \mathbb{R}^{dim} : x \ge 0 \}$ of dimension dim.

source
COSMO.Box — Type
Box(l, u)

Creates a box or intervall with lower boundary vector $l \in \mathbb{R}^m \cup \{-\infty\}^m$ and upper boundary vector$u \in \mathbb{R}^m\cup \{+\infty\}^m$.

source
COSMO.SecondOrderCone — Type
SecondOrderCone(dim)

Creates the second-order cone (or Lorenz cone) $\{ (t,x) \in \mathrm{R}^{dim} : || x ||_2 \leq t \}$.

source
COSMO.PsdCone — Type
PsdCone(dim)

Creates the cone of symmetric positive semidefinite matrices $\mathcal{S}_+^{dim}$. The entries of the matrix X are stored column-by-column in the vector x of dimension dim. Accordingly $X \in \mathbb{S}_+ \Rightarrow x \in \mathcal{S}_+^{dim}$, where $X = \text{mat}(x)$.

source
COSMO.PsdConeTriangle — Type
PsdConeTriangle(dim)

Creates the cone of symmetric positive semidefinite matrices. The entries of the upper-triangular part of matrix X are stored in the vector x of dimension dim. A $r \times r$ matrix has $r(r+1)/2$ upper triangular elements and results in a vector of $\mathrm{dim} = r(r+1)/2$.

Examples

The matrix

\[\begin{bmatrix} x_1 & x_2 & x_4\\ x_2 & x_3 & x_5\\ x_4 & x_5 & x_6 \end{bmatrix}\]

is transformed to the vector $[x_1, x_2, x_3, x_4, x_5, x_6]^\top$ with corresponding constraint PsdConeTriangle(6).

source
COSMO.ExponentialCone — Type
ExponentialCone(MAX_ITERS = 100, EXP_TOL = 1e-8)

Creates the exponential cone $\mathcal{K}_{exp} = \{(x, y, z) \mid y \geq 0 ye^{x/y} ≤ z\} \cup \{ (x,y,z) \mid x \leq 0, y = 0, z \geq 0 \}$

source
COSMO.DualExponentialCone — Type
DualExponentialCone(MAX_ITERS::Int = 100, EXP_TOL = 1e-8)

Creates the dual exponential cone $\mathcal{K}^*_{exp} = \{(x, y, z) \mid x < 0, -xe^{y/x} \leq e^1 z \} \cup \{ (0,y,z) \mid y \geq 0, z \geq 0 \}$

source
COSMO.PowerCone — Type
PowerCone(alpha::Float64, MAX_ITERS::Int = 20, POW_TOL = 1e-8)

Creates the 3-d power cone $\mathcal{K}_{pow} = \{(x, y, z) \mid x^\alpha y^{(1-\alpha)} \geq \|z\|, x \geq 0, y \geq 0 \}$ with $0 < \alpha < 1$

source
COSMO.DualPowerCone — Type
DualPowerCone(alpha::Float64, MAX_ITERS::Int = 20, POW_TOL = 1e-8)

Creates the 3-d dual power cone $\mathcal{K}^*_{pow} = \{(u, v, w) \mid \left( \frac{u}{\alpha}\right)^\alpha \left( \frac{v}{1-\alpha}\right)^{(1-\alpha)} \geq \|w\|, u \geq 0, v \geq 0 \}$ with $0 < \alpha < 1$

source
diff --git a/dev/assets/documenter.js b/dev/assets/documenter.js index 6adfbbbf..c6562b55 100644 --- a/dev/assets/documenter.js +++ b/dev/assets/documenter.js @@ -1,15 +1,15 @@ // Generated by Documenter.jl requirejs.config({ paths: { - 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/languages/julia.min', + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', - 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.12.1/jquery-ui.min', - 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/contrib/auto-render.min', - 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.6.0/jquery.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', - 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min', - 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/highlight.min', - 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/languages/julia-repl.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', }, shim: { "highlight-julia": { @@ -70,13 +70,91 @@ $(document).ready(function() { hljs.highlightAll(); }) +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let timer = 0; +var isExpanded = true; + +$(document).on("click", ".docstring header", function () { + let articleToggleTitle = "Expand docstring"; + + debounce(() => { + if ($(this).siblings("section").is(":visible")) { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + $(this) + .find(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + $(this).siblings("section").slideToggle(); + }); +}); + +$(document).on("click", ".docs-article-toggle-button", function (event) { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + let animationSpeed = event.noToggleAnimation ? 0 : 400; + + debounce(() => { + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(animationSpeed); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(animationSpeed); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); + }); +}); + +function debounce(callback, timeout = 300) { + if (Date.now() - timer > timeout) { + callback(); + } + + clearTimeout(timer); + + timer = Date.now(); +} + }) //////////////////////////////////////////////////////////////////////////////// require([], function() { function addCopyButtonCallbacks() { for (const el of document.getElementsByTagName("pre")) { const button = document.createElement("button"); - button.classList.add("copy-button", "fas", "fa-copy"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + el.appendChild(button); const success = function () { @@ -85,7 +163,7 @@ function addCopyButtonCallbacks() { }; const failure = function () { - button.classList.add("error", "fa-times"); + button.classList.add("error", "fa-xmark"); button.classList.remove("fa-copy"); }; @@ -94,7 +172,7 @@ function addCopyButtonCallbacks() { setTimeout(function () { button.classList.add("fa-copy"); - button.classList.remove("success", "fa-check", "fa-times"); + button.classList.remove("success", "fa-check", "fa-xmark"); }, 5000); }); } @@ -138,30 +216,682 @@ require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { // Manages the top navigation bar (hides it when the user starts scrolling down on the // mobile). window.Headroom = Headroom; // work around buggy module loading? -$(document).ready(function() { - $('#documenter .docs-navbar').headroom({ - "tolerance": {"up": 10, "down": 10}, +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, }); +}); + }) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); + + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +/* +To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +PSEUDOCODE: + +Searching happens automatically as the user types or adjusts the selected filters. +To preserve responsiveness, as much as possible of the slow parts of the search are done +in a web worker. Searching and result generation are done in the worker, and filtering and +DOM updates are done in the main thread. The filters are in the main thread as they should +be very quick to apply. This lets filters be changed without re-searching with minisearch +(which is possible even if filtering is on the worker thread) and also lets filters be +changed _while_ the worker is searching and without message passing (neither of which are +possible if filtering is on the worker thread) + +SEARCH WORKER: + +Import minisearch + +Build index + +On message from main thread + run search + find the first 200 unique results from each category, and compute their divs for display + note that this is necessary and sufficient information for the main thread to find the + first 200 unique results from any given filter set + post results to main thread + +MAIN: + +Launch worker + +Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) + +On text update + if worker is not running, launch_search() + +launch_search + set worker_is_running to true, set last_search_text to the search text + post the search query to worker + +on message from worker + if last_search_text is not the same as the text in the search field, + the latest search result is not reflective of the latest search query, so update again + launch_search() + otherwise + set worker_is_running to false + + regardless, display the new search results to the user + save the unfiltered_results as a global + update_search() + +on filter click + adjust the filter selection + update_search() + +update_search + apply search filters by looping through the unfiltered_results and finding the first 200 + unique results that match the filters + + Update the DOM +*/ + +/////// SEARCH WORKER /////// + +function worker_function(documenterSearchIndex, documenterBaseURL, filters) { + importScripts( + "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" + ); + + let data = documenterSearchIndex.map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; + }); + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", + ]); + + let index = new MiniSearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + + word = word.toLowerCase(); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); + + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } + + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`${querystring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(querystring)}`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${escape(result.title)}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; + } + + self.onmessage = function (e) { + let query = e.data; + let results = index.search(query, { + filter: (result) => { + // Only return relevant results + return result.score >= 1; + }, + }); + + // Pre-filter to deduplicate and limit to 200 per category to the extent + // possible without knowing what the filters are. + let filtered_results = []; + let counts = {}; + for (let filter of filters) { + counts[filter] = 0; + } + let present = {}; + + for (let result of results) { + cat = result.category; + cnt = counts[cat]; + if (cnt < 200) { + id = cat + "---" + result.location; + if (present[id]) { + continue; + } + present[id] = true; + filtered_results.push({ + location: result.location, + category: cat, + div: make_search_result(result, query), + }); + } + } + + postMessage(filtered_results); + }; +} + +// `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! +const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), +]; +const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; +const worker_blob = new Blob([worker_str], { type: "text/javascript" }); +const worker = new Worker(URL.createObjectURL(worker_blob)); + +/////// SEARCH MAIN /////// + +// Whether the worker is currently handling a search. This is a boolean +// as the worker only ever handles 1 or 0 searches at a time. +var worker_is_running = false; + +// The last search text that was sent to the worker. This is used to determine +// if the worker should be launched again when it reports back results. +var last_search_text = ""; + +// The results of the last search. This, in combination with the state of the filters +// in the DOM, is used compute the results to display on calls to update_search. +var unfiltered_results = []; + +// Which filter is currently selected +var selected_filter = ""; + +$(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } +}); + +function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); +} + +worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } + + unfiltered_results = e.data; + update_search(); +}; + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + selected_filter = ""; + } else { + selected_filter = $(this).text().toLowerCase(); + } + + // This updates search results and toggles classes for UI: + update_search(); +}); + +/** + * Make/Update the search component + */ +function update_search() { + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + if (selected_filter == "") { + results = unfiltered_results; + } else { + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); + } + + let search_result_container = ``; + let modal_filters = make_modal_body_filters(); + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; + links.push(result.location); + } + } + + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
${count_str}
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(` +
Type something to get started!
+ `); + } +} + +/** + * Make the modal filter html + * + * @returns string + */ +function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); + + return ` +
+ Filters: + ${str} +
`; +} }) //////////////////////////////////////////////////////////////////////////////// require(['jquery'], function($) { // Modal settings dialog -$(document).ready(function() { - var settings = $('#documenter-settings'); - $('#documenter-settings-button').click(function(){ - settings.toggleClass('is-active'); +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); }); // Close the dialog if X is clicked - $('#documenter-settings button.delete').click(function(){ - settings.removeClass('is-active'); + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); }); // Close dialog if ESC is pressed - $(document).keyup(function(e) { - if (e.keyCode == 27) settings.removeClass('is-active'); + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+ `; + + let initial_search_body = ` +
Type something to get started!
+ `; + + let search_modal_footer = ` + + `; + + $(document.body).append( + ` + + ` + ); + + document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); + }); + + document + .querySelector(".close-search-modal") + .addEventListener("click", () => { + closeModal(); + }); + + $(document).on("click", ".search-result-link", function () { + closeModal(); + }); + + document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; }); + + // Functions to open and close a modal + function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); + } + + function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); + } + + document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); }); }) @@ -169,150 +899,139 @@ $(document).ready(function() { require(['jquery'], function($) { // Manages the showing and hiding of the sidebar. -$(document).ready(function() { +$(document).ready(function () { var sidebar = $("#documenter > .docs-sidebar"); - var sidebar_button = $("#documenter-sidebar-button") - sidebar_button.click(function(ev) { + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { ev.preventDefault(); - sidebar.toggleClass('visible'); - if (sidebar.hasClass('visible')) { + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { // Makes sure that the current menu item is visible in the sidebar. $("#documenter .docs-menu a.is-active").focus(); } }); - $("#documenter > .docs-main").bind('click', function(ev) { + $("#documenter > .docs-main").bind("click", function (ev) { if ($(ev.target).is(sidebar_button)) { return; } - if (sidebar.hasClass('visible')) { - sidebar.removeClass('visible'); + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); } }); -}) +}); // Resizes the package name / sitename in the sidebar if it is too wide. // Inspired by: https://github.com/davatron5000/FitText.js -$(document).ready(function() { +$(document).ready(function () { e = $("#documenter .docs-autofit"); function resize() { - var L = parseInt(e.css('max-width'), 10); + var L = parseInt(e.css("max-width"), 10); var L0 = e.width(); - if(L0 > L) { - var h0 = parseInt(e.css('font-size'), 10); - e.css('font-size', L * h0 / L0); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); // TODO: make sure it survives resizes? } } // call once and then register events resize(); $(window).resize(resize); - $(window).on('orientationchange', resize); + $(window).on("orientationchange", resize); }); // Scroll the navigation bar to the currently selected menu item -$(document).ready(function() { +$(document).ready(function () { var sidebar = $("#documenter .docs-menu").get(0); var active = $("#documenter .docs-menu .is-active").get(0); - if(typeof active !== 'undefined') { + if (typeof active !== "undefined") { sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; } -}) +}); }) //////////////////////////////////////////////////////////////////////////////// require(['jquery'], function($) { -function set_theme(theme) { - var active = null; - var disabled = []; - for (var i = 0; i < document.styleSheets.length; i++) { - var ss = document.styleSheets[i]; - var themename = ss.ownerNode.getAttribute("data-theme-name"); - if(themename === null) continue; // ignore non-theme stylesheets - // Find the active theme - if(themename === theme) active = ss; - else disabled.push(ss); - } - if(active !== null) { - active.disabled = false; - if(active.ownerNode.getAttribute("data-theme-primary") === null) { - document.getElementsByTagName('html')[0].className = "theme--" + theme; - } else { - document.getElementsByTagName('html')[0].className = ""; - } - disabled.forEach(function(ss){ - ss.disabled = true; - }); - } - - // Store the theme in localStorage - if(typeof(window.localStorage) !== "undefined") { - window.localStorage.setItem("documenter-theme", theme); - } else { - console.error("Browser does not support window.localStorage"); - } -} - // Theme picker setup -$(document).ready(function() { +$(document).ready(function () { // onchange callback - $('#documenter-themepicker').change(function themepick_callback(ev){ - var themename = $('#documenter-themepicker option:selected').attr('value'); - set_theme(themename); + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); }); // Make sure that the themepicker displays the correct theme when the theme is retrieved // from localStorage - if(typeof(window.localStorage) !== "undefined") { - var theme = window.localStorage.getItem("documenter-theme"); - if(theme !== null) { - $('#documenter-themepicker option').each(function(i,e) { - e.selected = (e.value === theme); - }) - } else { - $('#documenter-themepicker option').each(function(i,e) { - e.selected = $("html").hasClass(`theme--${e.value}`); - }) + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); } } -}) +}); }) //////////////////////////////////////////////////////////////////////////////// require(['jquery'], function($) { // update the version selector with info from the siteinfo.js and ../versions.js files -$(document).ready(function() { +$(document).ready(function () { // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the // siteinfo.js file, we just return immediately and not display the version selector. - if (typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === 'boolean' && DOCUMENTER_VERSION_SELECTOR_DISABLED) { + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { return; } var version_selector = $("#documenter .docs-version-selector"); var version_selector_select = $("#documenter .docs-version-selector select"); - version_selector_select.change(function(x) { - target_href = version_selector_select.children("option:selected").get(0).value; + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; window.location.href = target_href; }); // add the current version to the selector based on siteinfo.js, but only if the selector is empty - if (typeof DOCUMENTER_CURRENT_VERSION !== 'undefined' && $('#version-selector > option').length == 0) { - var option = $(""); + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); version_selector_select.append(option); } - if (typeof DOC_VERSIONS !== 'undefined') { + if (typeof DOC_VERSIONS !== "undefined") { var existing_versions = version_selector_select.children("option"); - var existing_versions_texts = existing_versions.map(function(i,x){return x.text}); - DOC_VERSIONS.forEach(function(each) { - var version_url = documenterBaseURL + "/../" + each; + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; var existing_id = $.inArray(each, existing_versions_texts); // if not already in the version selector, add it as a new option, // otherwise update the old option with the URL and enable it if (existing_id == -1) { - var option = $(""); + var option = $( + "" + ); version_selector_select.append(option); } else { var option = existing_versions[existing_id]; @@ -326,6 +1045,6 @@ $(document).ready(function() { if (version_selector_select.children("option").length > 0) { version_selector.toggleClass("visible"); } -}) +}); }) diff --git a/dev/assets/search.js b/dev/assets/search.js deleted file mode 100644 index c133f741..00000000 --- a/dev/assets/search.js +++ /dev/null @@ -1,267 +0,0 @@ -// Generated by Documenter.jl -requirejs.config({ - paths: { - 'lunr': 'https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min', - 'lodash': 'https://cdnjs.cloudflare.com/ajax/libs/lodash.js/4.17.21/lodash.min', - 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.6.0/jquery.min', - } -}); -//////////////////////////////////////////////////////////////////////////////// -require(['jquery', 'lunr', 'lodash'], function($, lunr, _) { - -$(document).ready(function() { - // parseUri 1.2.2 - // (c) Steven Levithan - // MIT License - function parseUri (str) { - var o = parseUri.options, - m = o.parser[o.strictMode ? "strict" : "loose"].exec(str), - uri = {}, - i = 14; - - while (i--) uri[o.key[i]] = m[i] || ""; - - uri[o.q.name] = {}; - uri[o.key[12]].replace(o.q.parser, function ($0, $1, $2) { - if ($1) uri[o.q.name][$1] = $2; - }); - - return uri; - }; - parseUri.options = { - strictMode: false, - key: ["source","protocol","authority","userInfo","user","password","host","port","relative","path","directory","file","query","anchor"], - q: { - name: "queryKey", - parser: /(?:^|&)([^&=]*)=?([^&]*)/g - }, - parser: { - strict: /^(?:([^:\/?#]+):)?(?:\/\/((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?))?((((?:[^?#\/]*\/)*)([^?#]*))(?:\?([^#]*))?(?:#(.*))?)/, - loose: /^(?:(?![^:@]+:[^:@\/]*@)([^:\/?#.]+):)?(?:\/\/)?((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?)(((\/(?:[^?#](?![^?#\/]*\.[^?#\/.]+(?:[?#]|$)))*\/?)?([^?#\/]*))(?:\?([^#]*))?(?:#(.*))?)/ - } - }; - - $("#search-form").submit(function(e) { - e.preventDefault() - }) - - // list below is the lunr 2.1.3 list minus the intersect with names(Base) - // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) - // ideally we'd just filter the original list but it's not available as a variable - lunr.stopWordFilter = lunr.generateStopWordFilter([ - 'a', - 'able', - 'about', - 'across', - 'after', - 'almost', - 'also', - 'am', - 'among', - 'an', - 'and', - 'are', - 'as', - 'at', - 'be', - 'because', - 'been', - 'but', - 'by', - 'can', - 'cannot', - 'could', - 'dear', - 'did', - 'does', - 'either', - 'ever', - 'every', - 'from', - 'got', - 'had', - 'has', - 'have', - 'he', - 'her', - 'hers', - 'him', - 'his', - 'how', - 'however', - 'i', - 'if', - 'into', - 'it', - 'its', - 'just', - 'least', - 'like', - 'likely', - 'may', - 'me', - 'might', - 'most', - 'must', - 'my', - 'neither', - 'no', - 'nor', - 'not', - 'of', - 'off', - 'often', - 'on', - 'or', - 'other', - 'our', - 'own', - 'rather', - 'said', - 'say', - 'says', - 'she', - 'should', - 'since', - 'so', - 'some', - 'than', - 'that', - 'the', - 'their', - 'them', - 'then', - 'there', - 'these', - 'they', - 'this', - 'tis', - 'to', - 'too', - 'twas', - 'us', - 'wants', - 'was', - 'we', - 'were', - 'what', - 'when', - 'who', - 'whom', - 'why', - 'will', - 'would', - 'yet', - 'you', - 'your' - ]) - - // add . as a separator, because otherwise "title": "Documenter.Anchors.add!" - // would not find anything if searching for "add!", only for the entire qualification - lunr.tokenizer.separator = /[\s\-\.]+/ - - // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names - lunr.trimmer = function (token) { - return token.update(function (s) { - return s.replace(/^[^a-zA-Z0-9@!]+/, '').replace(/[^a-zA-Z0-9@!]+$/, '') - }) - } - - lunr.Pipeline.registerFunction(lunr.stopWordFilter, 'juliaStopWordFilter') - lunr.Pipeline.registerFunction(lunr.trimmer, 'juliaTrimmer') - - var index = lunr(function () { - this.ref('location') - this.field('title',{boost: 100}) - this.field('text') - documenterSearchIndex['docs'].forEach(function(e) { - this.add(e) - }, this) - }) - var store = {} - - documenterSearchIndex['docs'].forEach(function(e) { - store[e.location] = {title: e.title, category: e.category, page: e.page} - }) - - $(function(){ - searchresults = $('#documenter-search-results'); - searchinfo = $('#documenter-search-info'); - searchbox = $('#documenter-search-query'); - searchform = $('.docs-search'); - sidebar = $('.docs-sidebar'); - function update_search(querystring) { - tokens = lunr.tokenizer(querystring) - results = index.query(function (q) { - tokens.forEach(function (t) { - q.term(t.toString(), { - fields: ["title"], - boost: 100, - usePipeline: true, - editDistance: 0, - wildcard: lunr.Query.wildcard.NONE - }) - q.term(t.toString(), { - fields: ["title"], - boost: 10, - usePipeline: true, - editDistance: 2, - wildcard: lunr.Query.wildcard.NONE - }) - q.term(t.toString(), { - fields: ["text"], - boost: 1, - usePipeline: true, - editDistance: 0, - wildcard: lunr.Query.wildcard.NONE - }) - }) - }) - searchinfo.text("Number of results: " + results.length) - searchresults.empty() - results.forEach(function(result) { - data = store[result.ref] - link = $(''+data.title+'') - link.attr('href', documenterBaseURL+'/'+result.ref) - if (data.category != "page"){ - cat = $('('+data.category+', '+data.page+')') - } else { - cat = $('('+data.category+')') - } - li = $('
  • ').append(link).append(" ").append(cat) - searchresults.append(li) - }) - } - - function update_search_box() { - querystring = searchbox.val() - update_search(querystring) - } - - searchbox.keyup(_.debounce(update_search_box, 250)) - searchbox.change(update_search_box) - - // Disable enter-key form submission for the searchbox on the search page - // and just re-run search rather than refresh the whole page. - searchform.keypress( - function(event){ - if (event.which == '13') { - if (sidebar.hasClass('visible')) { - sidebar.removeClass('visible'); - } - update_search_box(); - event.preventDefault(); - } - } - ); - - search_query_uri = parseUri(window.location).queryKey["q"] - if(search_query_uri !== undefined) { - search_query = decodeURIComponent(search_query_uri.replace(/\+/g, '%20')) - searchbox.val(search_query) - } - update_search_box(); - }) -}) - -}) diff --git a/dev/assets/themes/documenter-dark.css b/dev/assets/themes/documenter-dark.css index c94a294d..1d717015 100644 --- a/dev/assets/themes/documenter-dark.css +++ b/dev/assets/themes/documenter-dark.css @@ -1,7 +1,7 @@ -@keyframes spinAround{from{transform:rotate(0deg)}to{transform:rotate(359deg)}}html.theme--documenter-dark .tabs,html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark .breadcrumb,html.theme--documenter-dark .file,html.theme--documenter-dark .button,.is-unselectable,html.theme--documenter-dark .modal-close,html.theme--documenter-dark .delete{-webkit-touch-callout:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none}html.theme--documenter-dark .navbar-link:not(.is-arrowless)::after,html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading)::after{border:3px solid rgba(0,0,0,0);border-radius:2px;border-right:0;border-top:0;content:" 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kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content 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figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:left}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 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.button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-dark,html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:hover,html.theme--documenter-dark .content kbd.button:hover,html.theme--documenter-dark .button.is-dark.is-hovered,html.theme--documenter-dark .content kbd.button.is-hovered{background-color:#232829;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus,html.theme--documenter-dark 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kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined,html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-outlined.is-focused{background-color:#282f2f;border-color:#282f2f;color:#fff}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after,html.theme--documenter-dark .content 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html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark 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.docstring>section>a.button.is-outlined.is-hovered.docs-sourcelink,html.theme--documenter-dark .button.is-primary.is-outlined:focus,html.theme--documenter-dark .docstring>section>a.button.is-outlined.docs-sourcelink:focus,html.theme--documenter-dark .button.is-primary.is-outlined.is-focused,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-focused.docs-sourcelink{background-color:#375a7f;border-color:#375a7f;color:#fff}html.theme--documenter-dark .button.is-primary.is-outlined.is-loading::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink::after{border-color:transparent transparent #375a7f #375a7f !important}html.theme--documenter-dark .button.is-primary.is-outlined.is-loading:hover::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink:hover::after,html.theme--documenter-dark .button.is-primary.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark 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.button.is-warning.is-active{background-color:#946e00;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-warning[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning{background-color:#ad8100;border-color:#ad8100;box-shadow:none}html.theme--documenter-dark .button.is-warning.is-inverted{background-color:#fff;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark 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.button.is-danger.is-light{background-color:#fdeeec;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#fce3e0;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#fcd8d5;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons 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Theme: Default Description: Original highlight.js style Author: (c) Ivan Sagalaev @@ -6,4 +6,4 @@ Website: https://highlightjs.org/ License: see project LICENSE Touched: 2021 -*/pre code.hljs{display:block;overflow-x:auto}code.hljs{padding:3px 5px}.hljs{background:#F0F0F0;color:#444}.hljs-comment{color:#888888}.hljs-tag,.hljs-punctuation{color:#444a}.hljs-tag .hljs-name,.hljs-tag .hljs-attr{color:#444}.hljs-keyword,.hljs-attribute,.hljs-selector-tag,.hljs-meta .hljs-keyword,.hljs-doctag,.hljs-name{font-weight:bold}.hljs-type,.hljs-string,.hljs-number,.hljs-selector-id,.hljs-selector-class,.hljs-quote,.hljs-template-tag,.hljs-deletion{color:#880000}.hljs-title,.hljs-section{color:#880000;font-weight:bold}.hljs-regexp,.hljs-symbol,.hljs-variable,.hljs-template-variable,.hljs-link,.hljs-selector-attr,.hljs-operator,.hljs-selector-pseudo{color:#BC6060}.hljs-literal{color:#78A960}.hljs-built_in,.hljs-bullet,.hljs-code,.hljs-addition{color:#397300}.hljs-meta{color:#1f7199}.hljs-meta .hljs-string{color:#4d99bf}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold} +*/pre code.hljs{display:block;overflow-x:auto;padding:1em}code.hljs{padding:3px 5px}.hljs{background:#F3F3F3;color:#444}.hljs-comment{color:#697070}.hljs-tag,.hljs-punctuation{color:#444a}.hljs-tag .hljs-name,.hljs-tag .hljs-attr{color:#444}.hljs-keyword,.hljs-attribute,.hljs-selector-tag,.hljs-meta .hljs-keyword,.hljs-doctag,.hljs-name{font-weight:bold}.hljs-type,.hljs-string,.hljs-number,.hljs-selector-id,.hljs-selector-class,.hljs-quote,.hljs-template-tag,.hljs-deletion{color:#880000}.hljs-title,.hljs-section{color:#880000;font-weight:bold}.hljs-regexp,.hljs-symbol,.hljs-variable,.hljs-template-variable,.hljs-link,.hljs-selector-attr,.hljs-operator,.hljs-selector-pseudo{color:#ab5656}.hljs-literal{color:#695}.hljs-built_in,.hljs-bullet,.hljs-code,.hljs-addition{color:#397300}.hljs-meta{color:#1f7199}.hljs-meta .hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold}.gap-4{gap:1rem} diff --git a/dev/assets/themeswap.js b/dev/assets/themeswap.js index c58e993e..9f5eebe6 100644 --- a/dev/assets/themeswap.js +++ b/dev/assets/themeswap.js @@ -1,20 +1,17 @@ // Small function to quickly swap out themes. Gets put into the tag.. function set_theme_from_local_storage() { - // Intialize the theme to null, which means default + // Initialize the theme to null, which means default var theme = null; // If the browser supports the localstorage and is not disabled then try to get the // documenter theme - if(window.localStorage != null) { + if (window.localStorage != null) { // Get the user-picked theme from localStorage. May be `null`, which means the default // theme. - theme = window.localStorage.getItem("documenter-theme"); + theme = window.localStorage.getItem("documenter-theme"); } - // Check if the browser supports user color preference - var darkPreference = false; // Check if the users preference is for dark color scheme - if(window.matchMedia('(prefers-color-scheme: dark)').matches === true) { - darkPreference = true; - } + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; // Initialize a few variables for the loop: // // - active: will contain the index of the theme that should be active. Note that there @@ -24,43 +21,64 @@ function set_theme_from_local_storage() { // // - disabled: style sheets that should be disabled (i.e. all the theme style sheets // that are not the currently active theme) - var active = null; var disabled = []; var darkTheme = null; + var active = null; + var disabled = []; + var primaryLightTheme = null; + var primaryDarkTheme = null; for (var i = 0; i < document.styleSheets.length; i++) { var ss = document.styleSheets[i]; // The tag of each style sheet is expected to have a data-theme-name attribute // which must contain the name of the theme. The names in localStorage much match this. var themename = ss.ownerNode.getAttribute("data-theme-name"); // attribute not set => non-theme stylesheet => ignore - if(themename === null) continue; + if (themename === null) continue; // To distinguish the default (primary) theme, it needs to have the data-theme-primary // attribute set. - var isprimary = (ss.ownerNode.getAttribute("data-theme-primary") !== null); - // Check if the theme is primary dark theme - var isDarkTheme = (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null); - // If ss is for dark theme then set the value of darkTheme to the name of the theme - if(isDarkTheme) darkTheme = themename; + if (ss.ownerNode.getAttribute("data-theme-primary") !== null) { + primaryLightTheme = themename; + } + // Check if the theme is primary dark theme so that we could store its name in darkTheme + if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) { + primaryDarkTheme = themename; + } // If we find a matching theme (and it's not the default), we'll set active to non-null - if(themename === theme) active = i; + if (themename === theme) active = i; // Store the style sheets of inactive themes so that we could disable them - if(themename !== theme) disabled.push(ss); + if (themename !== theme) disabled.push(ss); } - if(active !== null) { + var activeTheme = null; + if (active !== null) { // If we did find an active theme, we'll (1) add the theme--$(theme) class to - document.getElementsByTagName('html')[0].className = "theme--" + theme; - // and (2) disable all the other theme stylesheets - disabled.forEach(function(ss){ - ss.disabled = true; - }); + document.getElementsByTagName("html")[0].className = "theme--" + theme; + activeTheme = theme; + } else { + // If we did _not_ find an active theme, then we need to fall back to the primary theme + // which can either be dark or light, depending on the user's OS preference. + var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme; + // In case it somehow happens that the relevant primary theme was not found in the + // preceding loop, we abort without doing anything. + if (activeTheme === null) { + console.error("Unable to determine primary theme."); + return; + } + // When switching to the primary light theme, then we must not have a class name + // for the tag. That's only for non-primary or the primary dark theme. + if (darkPreference) { + document.getElementsByTagName("html")[0].className = + "theme--" + activeTheme; + } else { + document.getElementsByTagName("html")[0].className = ""; + } } - else if(darkTheme !== null && darkPreference === true) { - // If we did find an active theme, we'll (1) add the theme--$(theme) class to - document.getElementsByTagName('html')[0].className = "theme--" + darkTheme; - // and (2) disable all the other theme stylesheets - disabled.forEach(function(ss){ - if (ss.ownerNode.getAttribute("data-theme-name") !== darkTheme) { - ss.disabled = true; - } - }); + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // we'll disable all the stylesheets, except for the active one + ss.disabled = !(themename == activeTheme); } } set_theme_from_local_storage(); diff --git a/dev/assets/warner.js b/dev/assets/warner.js index 5531c885..3f6f5d00 100644 --- a/dev/assets/warner.js +++ b/dev/assets/warner.js @@ -1,49 +1,52 @@ -function maybeAddWarning () { - // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE - // in siteinfo.js. - // If either of these are undefined something went horribly wrong, so we abort. - if ( - window.DOCUMENTER_NEWEST === undefined || - window.DOCUMENTER_CURRENT_VERSION === undefined || - window.DOCUMENTER_STABLE === undefined - ) { - return - }; +function maybeAddWarning() { + // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE + // in siteinfo.js. + // If either of these are undefined something went horribly wrong, so we abort. + if ( + window.DOCUMENTER_NEWEST === undefined || + window.DOCUMENTER_CURRENT_VERSION === undefined || + window.DOCUMENTER_STABLE === undefined + ) { + return; + } - // Current version is not a version number, so we can't tell if it's the newest version. Abort. - if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) { - return - }; + // Current version is not a version number, so we can't tell if it's the newest version. Abort. + if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) { + return; + } - // Current version is newest version, so no need to add a warning. - if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) { - return - }; + // Current version is newest version, so no need to add a warning. + if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) { + return; + } - // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs. - if (document.body.querySelector('meta[name="robots"]') === null) { - const meta = document.createElement('meta'); - meta.name = 'robots'; - meta.content = 'noindex'; + // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs. + if (document.body.querySelector('meta[name="robots"]') === null) { + const meta = document.createElement("meta"); + meta.name = "robots"; + meta.content = "noindex"; - document.getElementsByTagName('head')[0].appendChild(meta); - }; + document.getElementsByTagName("head")[0].appendChild(meta); + } - const div = document.createElement('div'); - div.classList.add('outdated-warning-overlay'); - const closer = document.createElement('button'); - closer.classList.add('outdated-warning-closer', 'delete'); - closer.addEventListener('click', function () { - document.body.removeChild(div); - }); - const href = window.documenterBaseURL + '/../' + window.DOCUMENTER_STABLE; - div.innerHTML = 'This documentation is not for the latest stable release, but for either the development version or an older release.
    Click here to go to the documentation for the latest stable release.'; - div.appendChild(closer); - document.body.appendChild(div); -}; + const div = document.createElement("div"); + div.classList.add("outdated-warning-overlay"); + const closer = document.createElement("button"); + closer.classList.add("outdated-warning-closer", "delete"); + closer.addEventListener("click", function () { + document.body.removeChild(div); + }); + const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE; + div.innerHTML = + 'This documentation is not for the latest stable release, but for either the development version or an older release.
    Click here to go to the documentation for the latest stable release.'; + div.appendChild(closer); + document.body.appendChild(div); +} -if (document.readyState === 'loading') { - document.addEventListener('DOMContentLoaded', maybeAddWarning); +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", maybeAddWarning); } else { - maybeAddWarning(); -}; + maybeAddWarning(); +} diff --git a/dev/citing/index.html b/dev/citing/index.html index eacbcbb8..10c7ebf5 100644 --- a/dev/citing/index.html +++ b/dev/citing/index.html @@ -1,9 +1,9 @@ -Citing COSMO · COSMO.jl
    Edit on GitHub

    Citing COSMO

    If you find COSMO useful in your project, we kindly request that you cite the following paper:

    @Article{Garstka_2021,
+
    GitHub
    Citing COSMO
    If you find COSMO useful in your project, we kindly request that you cite the following paper:
    @Article{Garstka_2021,
   author  = {Michael Garstka and Mark Cannon and Paul Goulart},
   journal = {Journal of Optimization Theory and Applications},
   title   = {{COSMO}: A Conic Operator Splitting Method for Convex Conic Problems},
@@ -32,4 +32,4 @@
   pages={435--440},
   year={2022},
   organization={IEEE}
-}
    A preprint can be downloaded here.
    
+}

    A preprint can be downloaded here.

    diff --git a/dev/contributing/index.html b/dev/contributing/index.html index 0d1ea42d..63cbf8b5 100644 --- a/dev/contributing/index.html +++ b/dev/contributing/index.html @@ -1,9 +1,12 @@ -Contributing · COSMO.jl
    Edit on GitHub

    Contributing

    Contributions are always welcome:

    • If you want to contribute features, bug fixes, etc, please take a look at our Code Style Guide below
    • Please report any issues and bugs that you encounter in Issues
    • As an open source project we are also interested in any projects and applications that use COSMO. Please let us know via email to: michael.garstka[at]eng.ox.ac.uk

    Code Style Guide

    The code in this repository follows the naming and style conventions of Julia Base with a few modifications. This style guide is heavily "inspired" by the guides of John Myles White and JuMP.

    Formatting

    • Use one tab when indenting a new block (except module)

    • Use spaces between operators, except for ^, ', and :

    • Use single space after commas and semicolons

    • Don't use spaces around parentheses, or braces

    Bad: f(x,y) = [5*sin(x+y);y'] Good: f(x, y) = [5 * sin(x + y); y']

    • Use spacing with keyword arguments

    Bad: foo(x::Integer=1) Good: foo(x::Integer = 1)

    • Don't parenthesize conditions

    Bad: if (a == b) Good: if a == b

    Naming

    • Modules and Type names use capitilization and camel case, e.g. module LinearAlgebra, struct ConvexSets.
    • Functions are lowercase and use underscores to seperate words, e.g. has_key(x), is_valid(y).
    • Normal variables are lowercase and use underscores like functions, e.g. convex_set
    • Constants are uppercase, e.g. const MY_CONSTANT
    • Always append ! to names of functions that modify their arguments.
    • Function arguments that are mutated come first. Otherwise follow the rules layed out in Julia Base Argument ordering
    • Files are named like functions, e.g. my_new_file.jl

    Syntax

    • Use 1.0 instead of 1.

    Git(hub)-specific conventions

    • Branch names should be prepended with the initials of the creator and a forward slash, e.g. mg/newIdea instead of newIdea
    • Commit messages should have the following format:
    <#IssueId> Short (72 chars or less) summary
+
    GitHub
    Contributing
    This package is currently in maintenance mode. 
    We are trying to keep it compatible with new releases of JuMP/MOI. Contributions for bug fixes or new features are always welcome. Enhancement ideas are tagged as such in the Issues section. If you are unsure how to develop, continue with the section below.
    How-to develop
    Let's assume you found a bug that you want to fix or develop a new feature. You have already downloaded the latest julia version as well as git. Follow these steps to make changes, test the changes, and open your first PR. This assumes that you don't yet have push access to the repository and need to fork.
    • Create a new fork in Github by clicking on Fork at the top right of the repository page and name it e.g. migarstka/COSMO.jl. (If you have repository push rights, you don't need to fork and can clone the repository directly.)
    • Clone the forked repository to your machine with:
    $ git clone git@github.com:migarstka/COSMO.jl.git
    • Navigate into the new folder and create a new branch:
    $ git checkout -b mg/bug_fix
    • Make your code changes by modifying the files in the repository.
    • As a first sanity check that nothing obvious is broken, you can run one of the examples in the examples/ folder. Open julia and navigate into the git repository. Next, we want to activate and instantiate the repositories virtual environment. In the julia REPL type ], then to activate the current environment (i.e. current Project.toml file):
    (@v1.8) pkg>  activate .
    Next, instantiate the environment, i.e. install the solver's dependencies defined in Project.toml:
    (COSMO) pkg> instantiate
    • Since we instantiated the local envrionment, COSMO  now refers to the local version that includes your changes. You can solve an example problem using the modified code:
    julia> include("examples/qp.jl")
    Let's assume you have made some changes and want to check whether the bug is fixed or the new feature broke anything. In both cases you should add some new tests for the change to /test/UnitTest and /test/run_cosmo_tests.jl. If you want to run existing tests follow the steps below.
    How-to run tests
    You can run all package-native tests with the main test file:
    julia> include("test/run_cosmo_tests.jl")
    The full test suite also includes several MathOptInterface test problems that you can run with:
    julia> include("test/runtests.jl")
    These take longer to complete, so I recommend using test/run_cosmo_tests.jl during iterative development and then switch to  test/runtests.jl once the previous tests pass.
    Open a PR
    Once all tests pass, you have added additional tests for your changes, and used docstrings to describe any new functionality in the code, you can open a PR. Commit your changes with a detailed commit message, and push your branch to the (fork) remote.
    $ git add .
+$ git commit 
+[Write commit message]
+$ git push
    Then navigate to https://github.com/oxfordcontrol/COSMO.jl/compare and create a new PR with base repository: oxfordcontrol/COSMO.jl base: master and head repository: [your-fork-repo] base:mg/bug_fix. Ask me or other collaborators to review it, before it can get merged.
    Making a new release
    Changes to master are periodically bundled in a new release version of the package. If you have permissions, you can make a new release by following these steps:
    1. Ensure all tests pass (locally and in Github Actions)
    2. Locally run bumpversion patch (or bumpversion minor or bumpversion major depending on semver convention) to increment the version number across the repository (most importantly in Project.toml).
    3. Check and commit the changes and use the commit message Bump version to X. 
    4. Push changes to remote with git push.
    5. Let the julia package registry know about the new release by commenting @JuliaRegistrator register on the bump commit (like here).
    6. This will open a PR in the JuliaRegistries repository and a new release will typically be approved within hours.
    7. Update the CHANGELOG and describe the changes bundled in the new release.
    Updating COSMO's documentation
    The code for COSMO's documentation resides in the /docs folder of the repository and is created with the Documenter.jl package. There are two main versions of this documentation. Stable is based on the committed changes of the latest tagged release. Dev is based on the latest commit of master.
    The documentation has its own environment defined inside /docs (see docs/Project.toml file). To edit the documentation, edit the files in /docs. It's advised to run the documentation generation process locally do ensure the layout is as expected and that Literate code examples are built without errors. To generate the documentation locally, follow these steps:
    • Clone the repository and make your changes (as described in the section above).
    • Navigate into the /docs folder and start julia.
    • Activate and instantiate the (docs) environment, after ], type:
    (@v1.8) pkg> activate .
    (docs) pkg> instantiate
    • Build the documentation locally into the /build folder with
    include("make.jl")
    (The first build will take several minutes. Subsequent changes will take ~30s.)
    • After the new documentation has been built, you can serve the files in /build in your browser, e.g. using a python webserver. Navigate into /build and run:
    python3 -m http.server --bind localhost
    • View the local documentation in your browser of choice at http://127.0.0.1:8000/
    • If you are satisfied with the results, open a PR for the changes as described in the section above.
    Code Style Guide
    The code in this repository follows the naming and style conventions of Julia Base with a few modifications. This style guide is heavily "inspired" by the guides of John Myles White and JuMP.
    Formatting
    • Use one tab when indenting a new block (except module)
    • Use spaces between operators, except for ^, ', and :
    • Use single space after commas and semicolons
    • Don't use spaces around parentheses, or braces
    Bad: f(x,y) = [5*sin(x+y);y'] Good: f(x, y) = [5 * sin(x + y); y']
    • Use spacing with keyword arguments
    Bad: foo(x::Integer=1) Good: foo(x::Integer = 1)
    • Don't parenthesize conditions
    Bad: if (a == b) Good: if a == b
    Naming
    • Modules and Type names use capitilization and camel case, e.g. module LinearAlgebra, struct ConvexSets.
    • Functions are lowercase and use underscores to seperate words, e.g. has_key(x), is_valid(y).
    • Normal variables are lowercase and use underscores like functions, e.g. convex_set
    • Constants are uppercase, e.g. const MY_CONSTANT
    • Always append ! to names of functions that modify their arguments.
    • Function arguments that are mutated come first. Otherwise follow the rules layed out in Julia Base Argument ordering
    • Files are named like functions, e.g. my_new_file.jl
    Syntax
    • Use 1.0 instead of 1.
    Git(hub)-specific conventions
    • Branch names should be prepended with the initials of the creator and a forward slash, e.g. mg/newIdea instead of newIdea
    • Commit messages should have the following format:
    <#IssueId> Short (72 chars or less) summary
 
 More detailed explanatory text. Wrap it to 72 characters. The blank
 line separating the summary from the body is critical.
@@ -11,4 +14,4 @@
 Imperative style for the commit message: "Fix bug" and not "Fixed
 bug" or "Fixes bug."
 
-The issue id can be ommitted if the commit does not related to a specific open issue
    
+The issue id can be ommitted if the commit does not related to a specific open issue
    diff --git a/dev/decomposition/index.html b/dev/decomposition/index.html index 3611a62e..ec92d7b3 100644 --- a/dev/decomposition/index.html +++ b/dev/decomposition/index.html @@ -1,9 +1,9 @@ -Chordal Decomposition · COSMO.jl
    Edit on GitHub

    Chordal Decomposition

    For very large sparse semidefinite programs (SDPs) it is often helpful to analyse the sparsity structure of the PSD constraint(s). If the equality constraints impose a sparsity structure on the matrix variable, one PSD constraint on a large matrix variable can be decomposed into several smaller constraints. This results usually in a significant speedup and reduction in solve time.

    The following example gives a short overview on chordal decomposition and clique merging. For more details, take a look at our paper on clique merging or watch the corresponding presentation that I gave on the topic.

    Example problem

    Let's consider the following SDP in standard dual form:

    \[\begin{array}{ll} +

    GitHub

    Chordal Decomposition

    For very large sparse semidefinite programs (SDPs) it is often helpful to analyse the sparsity structure of the PSD constraint(s). If the equality constraints impose a sparsity structure on the matrix variable, one PSD constraint on a large matrix variable can be decomposed into several smaller constraints. This results usually in a significant speedup and reduction in solve time.

    The following example gives a short overview on chordal decomposition and clique merging. For more details, take a look at our paper on clique merging or watch the corresponding presentation that I gave on the topic.

    Example problem

    Let's consider the following SDP in standard dual form:

    \[\begin{array}{ll} \text{minimize} & c^\top x \\ \text{subject to} &\displaystyle \sum_{i=1}^m A_i x_i + S = B\\ & S \in \mathbb{S}^n_+, @@ -69,7 +69,7 @@ Status: Solved Iterations: 40 Optimal objective: -1.4134 -Runtime: 0.002s (1.93ms)

    Under sets we can indeed see that COSMO solved a problem with five PSD constraints corresponding to the cliques $\mathcal{C}_1,\ldots, \mathcal{C}_5$ discovered in the sparsity pattern. (Note that the dimension printed is the number of entries in the upper triangle of the matrix block.)

    Clique merging

    After we have found the cliques of the sparsity pattern, we are allowed to merge some of them back together. For the graph of the sparsity pattern this just means adding more edges or treating some structural zeros as numerical zeros. The main reason to merge two cliques is that they might overlap a lot and therefore it is not advantageous to treat them as two different blocks. Consider the two extreme cases below:

    In the left figure we have the ideal case that all the blocks overlap in just one entry. A full decomposition would leave us with a large number of small blocks. The sparsity pattern in the right figure has two large blocks overlapping almost entirely. In this case it would be disadvantageous to decompose the blocks. Instead, we would do the initial decomposition, realize the large overlap, and then merge the two blocks back together. For sparsity patterns that arise from real applications the case is not always as clear and we have to use more sophisticated strategies to decide which blocks to merge.

    COSMO currently provides three different strategies that can be selected by the user:

    COSMO.NoMerge — Type
    NoMerge <: AbstractMergeStrategy

    A strategy that does not merge cliques.

    source
    COSMO.ParentChildMerge — Type
    ParentChildMerge(t_fill = 8, t_size = 8) <: AbstractTreeBasedMerge

    The merge strategy suggested in Sun and Andersen - Decomposition in conic optimization with partially separable structure (2014). The initial clique tree is traversed in topological order and a clique $\mathcal{C}_\ell$ is greedily merged to its parent clique $\mathcal{C}_{par(\ell)}$ if at least one of the two conditions are met

    • $(| \mathcal{C}_{par(\ell)}| -| \eta_\ell|) (|\mathcal{C}_\ell| - |\eta_\ell|) \leq t_{\text{fill}}$ (fill-in condition)
    • $\max \left\{ |\nu_{\ell}|, |\nu_{par(\ell)}| \right\} \leq t_{\text{size}}$ (supernode size condition)
    source
    COSMO.CliqueGraphMerge — Type
    CliqueGraphMerge(edge_weight::AbstractEdgeWeight = ComplexityWeight()) <: AbstractGraphBasedMerge

    The (default) merge strategy based on the reduced clique graph $\mathcal{G}(\mathcal{B}, \xi)$, for a set of cliques $\mathcal{B} = \{ \mathcal{C}_1, \dots, \mathcal{C}_p\}$ where the edge set $\xi$ is obtained by taking the edges of the union of clique trees.

    Moreover, given an edge weighting function $e(\mathcal{C}_i,\mathcal{C}_j) = w_{ij}$, we compute a weight for each edge that quantifies the computational savings of merging the two cliques. After the initial weights are computed, we merge cliques in a loop:

    while clique graph contains positive weights:

    • select two permissible cliques with the highest weight $w_{ij}$
    • merge cliques $\rightarrow$ update clique graph
    • recompute weights for updated clique graph

    Custom edge weighting functions can be used by defining your own CustomEdgeWeight <: AbstractEdgeWeight and a corresponding edge_metric method. By default, the ComplexityWeight <: AbstractEdgeWeight is used which computes the weight based on the cardinalities of the cliques: $e(\mathcal{C}_i,\mathcal{C}_j) = |\mathcal{C}_i|^3 + |\mathcal{C}_j|^3 - |\mathcal{C}_i \cup \mathcal{C}_j|^3$.

    See also: Garstka, Cannon, Goulart - A clique graph based merging strategy for decomposable SDPs (2019)

    source

    In our example problem we have two cliques $\mathcal{C}_3 = \{ 3,6,7,8\}$ and $\mathcal{C}_5 = \{6,7,8,9 \}$ that overlap in three entries. Let's solve the problem again and choose the default clique merging strategy merge_strategy = COSMO.CliqueGraphMerge:

    model = JuMP.Model(with_optimizer(COSMO.Optimizer, decompose = true, merge_strategy = COSMO.CliqueGraphMerge));
+Runtime: 0.002s (1.93ms)

    Under sets we can indeed see that COSMO solved a problem with five PSD constraints corresponding to the cliques $\mathcal{C}_1,\ldots, \mathcal{C}_5$ discovered in the sparsity pattern. (Note that the dimension printed is the number of entries in the upper triangle of the matrix block.)

    Clique merging

    After we have found the cliques of the sparsity pattern, we are allowed to merge some of them back together. For the graph of the sparsity pattern this just means adding more edges or treating some structural zeros as numerical zeros. The main reason to merge two cliques is that they might overlap a lot and therefore it is not advantageous to treat them as two different blocks. Consider the two extreme cases below:

    In the left figure we have the ideal case that all the blocks overlap in just one entry. A full decomposition would leave us with a large number of small blocks. The sparsity pattern in the right figure has two large blocks overlapping almost entirely. In this case it would be disadvantageous to decompose the blocks. Instead, we would do the initial decomposition, realize the large overlap, and then merge the two blocks back together. For sparsity patterns that arise from real applications the case is not always as clear and we have to use more sophisticated strategies to decide which blocks to merge.

    COSMO currently provides three different strategies that can be selected by the user:

    COSMO.NoMerge — Type
    NoMerge <: AbstractMergeStrategy

    A strategy that does not merge cliques.

    source
    COSMO.ParentChildMerge — Type
    ParentChildMerge(t_fill = 8, t_size = 8) <: AbstractTreeBasedMerge

    The merge strategy suggested in Sun and Andersen - Decomposition in conic optimization with partially separable structure (2014). The initial clique tree is traversed in topological order and a clique $\mathcal{C}_\ell$ is greedily merged to its parent clique $\mathcal{C}_{par(\ell)}$ if at least one of the two conditions are met

    • $(| \mathcal{C}_{par(\ell)}| -| \eta_\ell|) (|\mathcal{C}_\ell| - |\eta_\ell|) \leq t_{\text{fill}}$ (fill-in condition)
    • $\max \left\{ |\nu_{\ell}|, |\nu_{par(\ell)}| \right\} \leq t_{\text{size}}$ (supernode size condition)
    source
    COSMO.CliqueGraphMerge — Type
    CliqueGraphMerge(edge_weight::AbstractEdgeWeight = ComplexityWeight()) <: AbstractGraphBasedMerge

    The (default) merge strategy based on the reduced clique graph $\mathcal{G}(\mathcal{B}, \xi)$, for a set of cliques $\mathcal{B} = \{ \mathcal{C}_1, \dots, \mathcal{C}_p\}$ where the edge set $\xi$ is obtained by taking the edges of the union of clique trees.

    Moreover, given an edge weighting function $e(\mathcal{C}_i,\mathcal{C}_j) = w_{ij}$, we compute a weight for each edge that quantifies the computational savings of merging the two cliques. After the initial weights are computed, we merge cliques in a loop:

    while clique graph contains positive weights:

    • select two permissible cliques with the highest weight $w_{ij}$
    • merge cliques $\rightarrow$ update clique graph
    • recompute weights for updated clique graph

    Custom edge weighting functions can be used by defining your own CustomEdgeWeight <: AbstractEdgeWeight and a corresponding edge_metric method. By default, the ComplexityWeight <: AbstractEdgeWeight is used which computes the weight based on the cardinalities of the cliques: $e(\mathcal{C}_i,\mathcal{C}_j) = |\mathcal{C}_i|^3 + |\mathcal{C}_j|^3 - |\mathcal{C}_i \cup \mathcal{C}_j|^3$.

    See also: Garstka, Cannon, Goulart - A clique graph based merging strategy for decomposable SDPs (2019)

    source

    In our example problem we have two cliques $\mathcal{C}_3 = \{ 3,6,7,8\}$ and $\mathcal{C}_5 = \{6,7,8,9 \}$ that overlap in three entries. Let's solve the problem again and choose the default clique merging strategy merge_strategy = COSMO.CliqueGraphMerge:

    model = JuMP.Model(with_optimizer(COSMO.Optimizer, decompose = true, merge_strategy = COSMO.CliqueGraphMerge));
 @variable(model, x[1:2]);
 @objective(model, Min, c' * x )
 @constraint(model, Symmetric(B - A1  .* x[1] - A2 .* x[2] )  in JuMP.PSDCone());
@@ -109,4 +109,4 @@
 Status: Solved
 Iterations: 40
 Optimal objective: -1.4134
-Runtime: 0.003s (2.68ms)

    Unsurprisingly, we can see in the output that COSMO solved a problem with four PSD constraints. One of them is of dimension 15, i.e. a $5\times 5$ block, which correspond to the merged clique $\mathcal{C_3} \cup \mathcal{C}_5 = \{3,6,7,8,9 \}$.

    Completing the dual variable

    After a decomposed problem is solved, we can recover the solution to the original problem by assembling the matrix variable $S$ from its subblocks $S_\ell$:

    \[S = \displaystyle \sum_{\ell = 1}^p T_\ell^\top S_\ell T_\ell,\]

    Following Agler's Theorem, $S$ will be a positive semidefinite matrix. However, this is not true for the corresponding dual variable matrix $Y$. The dual variable returned after solving the decomposed problem will be in the space of PSD completable matrices $Y \in \mathbb{S}_+^n(E,?)$. This means that the entries in $Y$ corresponding to the blocks $S_\ell$ (black dots) have been chosen correctly. The numerical values for all the other entries (corresponding to the zeros in $S$ and denoted with a red dot) have to be chosen in the right way to make $Y$ positive semidefinite.

    For more information about PSD matrix completion and the completion algorithm used in COSMO take a look at Vandenberghe and Andersen - Chordal Graphs and Semidefinite Optimization (Ch.10). To configure COSMO to complete the dual variable after solving the problem you have to set the complete_dual option:

    model = JuMP.Model(with_optimizer(COSMO.Optimizer, complete_dual = true));

    Example Code

    The code used for this example can be found in /examples/chordal_decomposition.jl.

    +Runtime: 0.003s (2.68ms)

    Unsurprisingly, we can see in the output that COSMO solved a problem with four PSD constraints. One of them is of dimension 15, i.e. a $5\times 5$ block, which correspond to the merged clique $\mathcal{C_3} \cup \mathcal{C}_5 = \{3,6,7,8,9 \}$.

    Completing the dual variable

    After a decomposed problem is solved, we can recover the solution to the original problem by assembling the matrix variable $S$ from its subblocks $S_\ell$:

    \[S = \displaystyle \sum_{\ell = 1}^p T_\ell^\top S_\ell T_\ell,\]

    Following Agler's Theorem, $S$ will be a positive semidefinite matrix. However, this is not true for the corresponding dual variable matrix $Y$. The dual variable returned after solving the decomposed problem will be in the space of PSD completable matrices $Y \in \mathbb{S}_+^n(E,?)$. This means that the entries in $Y$ corresponding to the blocks $S_\ell$ (black dots) have been chosen correctly. The numerical values for all the other entries (corresponding to the zeros in $S$ and denoted with a red dot) have to be chosen in the right way to make $Y$ positive semidefinite.

    For more information about PSD matrix completion and the completion algorithm used in COSMO take a look at Vandenberghe and Andersen - Chordal Graphs and Semidefinite Optimization (Ch.10). To configure COSMO to complete the dual variable after solving the problem you have to set the complete_dual option:

    model = JuMP.Model(with_optimizer(COSMO.Optimizer, complete_dual = true));

    Example Code

    The code used for this example can be found in /examples/chordal_decomposition.jl.

    diff --git a/dev/examples/closest_correlation_matrix/index.html b/dev/examples/closest_correlation_matrix/index.html index 472fd328..23896ade 100644 --- a/dev/examples/closest_correlation_matrix/index.html +++ b/dev/examples/closest_correlation_matrix/index.html @@ -1,9 +1,9 @@ -Closest Correlation Matrix · COSMO.jl
    Edit on GitHub

    The source files for all examples can be found in /examples.

    Closest Correlation Matrix

    We consider the problem of finding the closest correlation matrix $X$ to a given random matrix $C$. With closest correlation matrix we mean a positive semidefinite matrix with ones on the diagonal. The problem is given by:

    \[\begin{array}{ll} \text{minimize} & \frac{1}{2}||X - C||_F^2\\ +

    GitHub

    The source files for all examples can be found in /examples.

    Closest Correlation Matrix

    We consider the problem of finding the closest correlation matrix $X$ to a given random matrix $C$. With closest correlation matrix we mean a positive semidefinite matrix with ones on the diagonal. The problem is given by:

    \[\begin{array}{ll} \text{minimize} & \frac{1}{2}||X - C||_F^2\\ \text{subject to} & X_{ii} = 1, \quad i=1,\dots,n \\ & X \succeq 0. \end{array}\]

    Notice that we use JuMP to model the problem. COSMO is chosen as the backend solver. And COSMO-specific settings are passed using the optimizer_with_attributes() function.

    using COSMO, JuMP, LinearAlgebra, SparseArrays, Test, Random
    rng = Random.MersenneTwister(12345);
@@ -43,7 +43,7 @@
 Acc:      Anderson Type2{QRDecomp},
           Memory size = 15, RestartedMemory,
           Safeguarded: true, tol: 2.0
-Setup Time: 1006.87ms
+Setup Time: 994.87ms
 
 Iter:	Objective:	Primal Res:	Dual Res:	Rho:
 1	 9.8585e+00	5.9778e-01	7.6572e-01	1.0000e-01
@@ -54,7 +54,7 @@
 Status: Solved
 Iterations: 25
 Optimal objective: 2.21
-Runtime: 5.389s (5388.81ms)

    Double check result against known solution:

    known_opt_val = 12.5406
+Runtime: 5.357s (5356.77ms)

    Double check result against known solution:

    known_opt_val = 12.5406
 known_solution =  [
   1.0         0.732562   -0.319491   -0.359985   -0.287543   -0.15578     0.0264044  -0.271438;
   0.732562    1.0         0.0913246  -0.0386357   0.299199   -0.122733    0.126612   -0.187489;
@@ -64,4 +64,4 @@
  -0.15578    -0.122733    0.461783    0.250601   -0.0875199   1.0        -0.731556    0.0841783;
   0.0264044   0.126612   -0.248641    0.141151    0.137518   -0.731556    1.0        -0.436274;
  -0.271438   -0.187489   -0.395299    0.286088    0.0262425   0.0841783  -0.436274    1.0  ];
-@test isapprox(obj_val, known_opt_val , atol=1e-3)
    Test Passed
    @test norm(X_sol - known_solution, Inf) < 1e-3
    Test Passed

    This page was generated using Literate.jl.

    +@test isapprox(obj_val, known_opt_val , atol=1e-3)
    Test Passed
    @test norm(X_sol - known_solution, Inf) < 1e-3
    Test Passed

    This page was generated using Literate.jl.

    diff --git a/dev/examples/index.html b/dev/examples/index.html index aac54b46..2c0a0674 100644 --- a/dev/examples/index.html +++ b/dev/examples/index.html @@ -1,9 +1,9 @@ -Examples · COSMO.jl
    Edit on GitHub

    Examples

    Some example problems that are solved with COSMO can be found in the /examples folder.

    In the first example the native solver interface is used to define and solve a Linear Program. In the second example JuMP is used to describe the problem and COSMO is set as the solver backend.

    Linear Program

    We want to solve the following linear program with decision variable x:

    \[\begin{array}{ll} \text{minimize} & c^\top x\\ +

    GitHub

    Examples

    Some example problems that are solved with COSMO can be found in the /examples folder.

    In the first example the native solver interface is used to define and solve a Linear Program. In the second example JuMP is used to describe the problem and COSMO is set as the solver backend.

    Linear Program

    We want to solve the following linear program with decision variable x:

    \[\begin{array}{ll} \text{minimize} & c^\top x\\ \text{subject to} & A x \leq b \\ & x \geq 1 \\ & x_2 \geq 5 \\ @@ -68,4 +68,4 @@ # solve and get results status = JuMP.optimize!(m) obj_val = JuMP.objective_value(m) -X_sol = JuMP.value.(X)

    Logistic Regression

    Logistic regression problems can be solved using exponential cone constraints. An example on how to use COSMO to solve a logistic regression problem is presented in /examples/logistic_regression_regularization.ipynb.

    +X_sol = JuMP.value.(X)

    Logistic Regression

    Logistic regression problems can be solved using exponential cone constraints. An example on how to use COSMO to solve a logistic regression problem is presented in /examples/logistic_regression_regularization.ipynb.

    diff --git a/dev/examples/logistic_regression/10279931.svg b/dev/examples/logistic_regression/10279931.svg new file mode 100644 index 00000000..3387e8ae --- /dev/null +++ b/dev/examples/logistic_regression/10279931.svg @@ -0,0 +1,349 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/examples/logistic_regression/f5fc5625.svg b/dev/examples/logistic_regression/f5fc5625.svg new file mode 100644 index 00000000..76064110 --- /dev/null +++ b/dev/examples/logistic_regression/f5fc5625.svg @@ -0,0 +1,341 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/examples/logistic_regression/index.html b/dev/examples/logistic_regression/index.html index 464a34de..951e610e 100644 --- a/dev/examples/logistic_regression/index.html +++ b/dev/examples/logistic_regression/index.html @@ -1,9 +1,9 @@ -Logistic Regression · COSMO.jl
    Edit on GitHub

    The source files for all examples can be found in /examples.

    Logistic Regression

    The presented example is adapted from the Machine Learning - course by Andrew Ng.

    In this example we use logistic regression to estimate the parameters $\theta_i$ of a logistic model in a classification problem. We will first transform the logistic regression problem into an exponential cone optimisation problem. We will then solve the optimisation problem with COSMO and determine the model parameters and the decision boundary.

    Visualizing the data

    Before we start, let's load and take a look at the example data from examples/chip_data.txt:

    using LinearAlgebra, SparseArrays, COSMO, JuMP
+
    GitHub
    The source files for all examples can be found in /examples.
    Logistic Regression
    The presented example is adapted from the Machine Learning - course by Andrew Ng.
    In this example we use logistic regression to estimate the parameters $\theta_i$ of a logistic model in a classification problem. We will first  transform the logistic regression problem into an exponential cone optimisation problem. We will then solve the optimisation problem with COSMO  and determine the model parameters and the decision boundary.
    Visualizing the data
    Before we start, let's load and take a look at the example data from examples/chip_data.txt:
    using LinearAlgebra, SparseArrays, COSMO, JuMP
 using Plots
 
 # load example data
@@ -25,348 +25,7 @@
 
 # visualize data
 plot(x1[1:n_half-1], x2[1:n_half-1], color = :blue, st=:scatter, markershape = :cross, aspect_ratio=:equal, label = "Accepted", xlabel = "x1 - Microchip Test Score 1", ylabel = "x2 - Microchip Test Score 2")
-plot!(x1[n_half:end], x2[n_half:end], color = :red, st=:scatter, markershape = :circle, label = "Rejected")
    
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    The plot shows two test scores of $n$ microchip samples from a fabrication plant and whether the chip passed the quality check. Based on this data we would like to build a logistic model that takes into account the test scores and helps us predict the likelihood of a chip being accepted.
    Defining the logistic model
    The logistic regression hypothesis is given by
    \[h_\theta(x) = g(\theta^\top x)\]
    where $g$ is the sigmoid function:
    \[g(\theta^\top x) = \frac{1}{1+\exp(-\theta^\top x)}.\]
    The vector $x$ represents the independent variables and $\theta$ represents the model parameters. For our samples we set the dependent variable $y =1$ if the chip was accepted and $y = 0$ otherwise.
    The function $h_\theta(x)$ can be interpreted as the probability of the outcome being true rather than false. We want to find the parameters $\theta$ that maximize the log-likelihood over all (independently Bernoulli distributed) observations
    \[J(\theta) = \sum_{i, y_i = 1} \log h_\theta(x_i) + \sum_{i, y_i = 0} \log (1-h_\theta(x_i)).\]
    Consequently, we want to solve the following optimization problem:
    \[\text{minimize} \quad -J(\theta) + \mu \|\theta \|_2,\]
    where we added a regularization term with parameter $\mu$ to prevent overfitting.
    Feature mapping
    As our dataset only has two independent variables (the test scores) our model $y = \theta_0 + \theta_1 x_1 + \theta_2 x_2$ will have the form of a straight line. Looking at the plot one can see that a line will not perform well in separating the samples. Therefore, we will create more features based on each data point by mapping the original features ($x_1$, $x_2$) into all polynomial terms of $x_1$ and $x_2$ up to the 6th power:
    \[\text{map\_feature}(x_1,x_2) = [1, x_1, x_2, x_1^2, x_1x_2, x_2^2, x_1^3, \dots, x_1x_2^5, x_2^6 ]\]
    This will create 28 features for each sample.
    function map_feature(x1, x2)
+plot!(x1[n_half:end], x2[n_half:end], color = :red, st=:scatter, markershape = :circle, label = "Rejected")
    Example block output
    The plot shows two test scores of $n$ microchip samples from a fabrication plant and whether the chip passed the quality check. Based on this data we would like to build a logistic model that takes into account the test scores and helps us predict the likelihood of a chip being accepted.
    Defining the logistic model
    The logistic regression hypothesis is given by
    \[h_\theta(x) = g(\theta^\top x)\]
    where $g$ is the sigmoid function:
    \[g(\theta^\top x) = \frac{1}{1+\exp(-\theta^\top x)}.\]
    The vector $x$ represents the independent variables and $\theta$ represents the model parameters. For our samples we set the dependent variable $y =1$ if the chip was accepted and $y = 0$ otherwise.
    The function $h_\theta(x)$ can be interpreted as the probability of the outcome being true rather than false. We want to find the parameters $\theta$ that maximize the log-likelihood over all (independently Bernoulli distributed) observations
    \[J(\theta) = \sum_{i, y_i = 1} \log h_\theta(x_i) + \sum_{i, y_i = 0} \log (1-h_\theta(x_i)).\]
    Consequently, we want to solve the following optimization problem:
    \[\text{minimize} \quad -J(\theta) + \mu \|\theta \|_2,\]
    where we added a regularization term with parameter $\mu$ to prevent overfitting.
    Feature mapping
    As our dataset only has two independent variables (the test scores) our model $y = \theta_0 + \theta_1 x_1 + \theta_2 x_2$ will have the form of a straight line. Looking at the plot one can see that a line will not perform well in separating the samples. Therefore, we will create more features based on each data point by mapping the original features ($x_1$, $x_2$) into all polynomial terms of $x_1$ and $x_2$ up to the 6th power:
    \[\text{map\_feature}(x_1,x_2) = [1, x_1, x_2, x_1^2, x_1x_2, x_2^2, x_1^3, \dots, x_1x_2^5, x_2^6 ]\]
    This will create 28 features for each sample.
    function map_feature(x1, x2)
   deg = 6
   x_new = ones(length(x1))
   for i = 1:deg, j = 0:i
@@ -440,7 +99,7 @@
 Acc:      Anderson Type2{QRDecomp},
           Memory size = 15, RestartedMemory,
           Safeguarded: true, tol: 2.0
-Setup Time: 28.37ms
+Setup Time: 23.45ms
 
 Iter:	Objective:	Primal Res:	Dual Res:	Rho:
 1	-1.2883e+03	2.5314e+01	1.7951e+00	1.0000e-01
@@ -453,7 +112,7 @@
 Status: Solved
 Iterations: 75
 Optimal objective: 51.93
-Runtime: 0.962s (961.71ms)
    theta = value.(θ)
    28-element Vector{Float64}:
+Runtime: 0.889s (889.49ms)
    theta = value.(θ)
    28-element Vector{Float64}:
   2.6744881022317313
   1.7749123100576332
   2.9342821763027476
@@ -481,356 +140,7 @@
     z[i, j] = dot(map_feature(u[i], v[j]), theta);
 end
    To add the decision boundary we have to plot the line indicating $50\%$ probability of acceptance, i.e. $g(\theta^\top x) = g(z)  = 0.5$ which we get at $z=0$.
    plot(x1[1:n_half-1], x2[1:n_half-1], color = :blue, st = :scatter, markershape = :cross, aspect_ratio=:equal, label = "Accepted", xlabel = "x1 - Microchip Test Score 1", ylabel = "x2 - Microchip Test Score 2")
 plot!(x1[n_half:end], x2[n_half:end], color = :red, st = :scatter, markershape = :circle, label = "Rejected")
-contour!(u, v, z', levels = [0.], c = :black, linewidth = 2)
    
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    Solving the optimisation problem directly with COSMO
    We can solve the problem directly in COSMO by using its modeling interface. The problem will have $nn = 5 n + n_\theta + 1$ variables. Let us define the cost function  $\frac{1}{2}x^\top P x + q^\top x$:
    nn = 5 * n + n_theta +  1
+contour!(u, v, z', levels = [0.], c = :black, linewidth = 2)
    Example block output
    Solving the optimisation problem directly with COSMO
    We can solve the problem directly in COSMO by using its modeling interface. The problem will have $nn = 5 n + n_\theta + 1$ variables. Let us define the cost function  $\frac{1}{2}x^\top P x + q^\top x$:
    nn = 5 * n + n_theta +  1
 P = spzeros(nn, nn)
 q = zeros(nn)
 q[1] = μ
@@ -909,7 +219,7 @@ 
    Literate.jl.
    
+@test norm(theta_cosmo - theta) < 1e-4
    Test Passed

    This page was generated using Literate.jl.

    diff --git a/dev/examples/lovasz_petersen/abe5edc1.svg b/dev/examples/lovasz_petersen/abe5edc1.svg new file mode 100644 index 00000000..f89235ee --- /dev/null +++ b/dev/examples/lovasz_petersen/abe5edc1.svg @@ -0,0 +1,95 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/examples/lovasz_petersen/index.html b/dev/examples/lovasz_petersen/index.html index 9cdd528a..32a932c5 100644 --- a/dev/examples/lovasz_petersen/index.html +++ b/dev/examples/lovasz_petersen/index.html @@ -1,9 +1,9 @@ -Lovász Theta Function · COSMO.jl
    Edit on GitHub

    The source files for all examples can be found in /examples.

    Lovász Theta Function

    The Lovász theta function is an important concept in graph theory. Consider an undirected graph $G = (V,E)$ with vertex set $V = \{1,\dots,n\}$ and edge set $E$ with $E_{ij} = 1$ if there exists an edge between the vertices $i$ and $j$. A stable set (or independent set) is a subset of $V$ such that the induced subgraph does not contain edges. The stability number $\alpha(G)$ of the graph is equal to the cardinality of the largest stable set. It is closely related to the Shannon capacity $\Theta(G)$ that models the amount of information that a noisy communication channel can carry if certain signal values can be confused with each other. Unfortunately, the determination of $\alpha(G)$ is an NP-hard problem and the computational complexity to determine $\Theta(G)$ remains unknown. The Lovász theta function $\vartheta(G)$ was introduced in [1], can be computed in polynomial time, and represents an upper bound on both the stability number and the Shannon capacity:

    \[\alpha(G) \leq \Theta(G) \leq \vartheta(G).\]

    The value of $\vartheta(G)$ can be determined by computing the optimal value $p^*$ of the following SDP:

    \[\begin{array}{ll} \text{maximize} & \text{Tr}(JX)\\ +

    GitHub

    The source files for all examples can be found in /examples.

    Lovász Theta Function

    The Lovász theta function is an important concept in graph theory. Consider an undirected graph $G = (V,E)$ with vertex set $V = \{1,\dots,n\}$ and edge set $E$ with $E_{ij} = 1$ if there exists an edge between the vertices $i$ and $j$. A stable set (or independent set) is a subset of $V$ such that the induced subgraph does not contain edges. The stability number $\alpha(G)$ of the graph is equal to the cardinality of the largest stable set. It is closely related to the Shannon capacity $\Theta(G)$ that models the amount of information that a noisy communication channel can carry if certain signal values can be confused with each other. Unfortunately, the determination of $\alpha(G)$ is an NP-hard problem and the computational complexity to determine $\Theta(G)$ remains unknown. The Lovász theta function $\vartheta(G)$ was introduced in [1], can be computed in polynomial time, and represents an upper bound on both the stability number and the Shannon capacity:

    \[\alpha(G) \leq \Theta(G) \leq \vartheta(G).\]

    The value of $\vartheta(G)$ can be determined by computing the optimal value $p^*$ of the following SDP:

    \[\begin{array}{ll} \text{maximize} & \text{Tr}(JX)\\ \text{subject to} & \text{Tr}(X) = 1, \\ & X_{ij} = 0, \quad (i, j) \in E, \\ & X \succeq 0, @@ -31,102 +31,7 @@ for θ = 90:-72:-198 push!(coordinates, [ri * cosd(θ), ri * sind(θ)]) end -graphplot(E, names = 1:n, x = getindex.(coordinates, 1), y = getindex.(coordinates, 2), fontsize = 10, nodesize = 1, nodeshape =:circle, curvature = 0.) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

    Let's solve the SDP with COSMO and JuMP:

    model = JuMP.Model(COSMO.Optimizer);
+graphplot(E, names = 1:n,  x = getindex.(coordinates, 1), y = getindex.(coordinates, 2), fontsize = 10, nodesize = 1, nodeshape =:circle, curvature = 0.)
    Example block output

    Let's solve the SDP with COSMO and JuMP:

    model = JuMP.Model(COSMO.Optimizer);
 
 @variable(model, X[1:n, 1:n], PSD)
 x = vec(X)
@@ -162,7 +67,7 @@
 Acc:      Anderson Type2{QRDecomp},
           Memory size = 15, RestartedMemory,
           Safeguarded: true, tol: 2.0
-Setup Time: 0.15ms
+Setup Time: 0.14ms
 
 Iter:	Objective:	Primal Res:	Dual Res:	Rho:
 1	-1.2711e+03	2.1055e+01	1.0052e+00	1.0000e-01
@@ -173,4 +78,4 @@
 Status: Solved
 Iterations: 25
 Optimal objective: -4
-Runtime: 0.47s (469.82ms)

    The optimal objective is given by:

    JuMP.objective_value(model)
    4.000000007423991

    Which is the correct known value for the Petersen Graph.

    References

    [1] Lovász - On the Shannon Capacity of a Graph, IEEE Transactions on Information Theory (1979)

    This page was generated using Literate.jl.

    +Runtime: 0.461s (460.58ms)

    The optimal objective is given by:

    JuMP.objective_value(model)
    4.000000007423991

    Which is the correct known value for the Petersen Graph.

    References

    [1] Lovász - On the Shannon Capacity of a Graph, IEEE Transactions on Information Theory (1979)

    This page was generated using Literate.jl.

    diff --git a/dev/examples/lp/index.html b/dev/examples/lp/index.html index a846b38f..dac7b996 100644 --- a/dev/examples/lp/index.html +++ b/dev/examples/lp/index.html @@ -1,9 +1,9 @@ -Linear Program · COSMO.jl
    Edit on GitHub

    The source files for all examples can be found in /examples.

    Linear Program

    We want to solve the following linear program with decision variable x:

    \[\begin{array}{ll} \text{minimize} & c^\top x\\ +

    GitHub

    The source files for all examples can be found in /examples.

    Linear Program

    We want to solve the following linear program with decision variable x:

    \[\begin{array}{ll} \text{minimize} & c^\top x\\ \text{subject to} & A x \leq b \\ & x \geq 1 \\ & x_2 \geq 5 \\ @@ -24,4 +24,4 @@ model = COSMO.Model(); assemble!(model, P, q, [c1; c2; c3; c4], settings = settings); res = COSMO.optimize!(model)

    >>> COSMO - Results
-Status:

    Compare the result to the known solution:

    @test isapprox(res.x[1:4], [3; 5; 1; 1], atol=1e-2, norm = (x -> norm(x, Inf)))
    Test Passed
    @test isapprox(res.obj_val, 20.0, atol=1e-2)
    Test Passed

    This page was generated using Literate.jl.

    +Status:

    Compare the result to the known solution:

    @test isapprox(res.x[1:4], [3; 5; 1; 1], atol=1e-2, norm = (x -> norm(x, Inf)))
    Test Passed
    @test isapprox(res.obj_val, 20.0, atol=1e-2)
    Test Passed

    This page was generated using Literate.jl.

    diff --git a/dev/examples/maxcut/b98ec54e.svg b/dev/examples/maxcut/b98ec54e.svg new file mode 100644 index 00000000..91f464b9 --- /dev/null +++ b/dev/examples/maxcut/b98ec54e.svg @@ -0,0 +1,47 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/examples/maxcut/index.html b/dev/examples/maxcut/index.html index 41857065..adffc0d1 100644 --- a/dev/examples/maxcut/index.html +++ b/dev/examples/maxcut/index.html @@ -1,9 +1,9 @@ -Maximum Cut Problem · COSMO.jl
    Edit on GitHub

    The source files for all examples can be found in /examples.

    Maximum Cut Problem

    We are interested in solving an approximation to the maximum cut problem using semidefinite programming. Consider the graph $G(V,E)$ with weights $w_{ij}$ shown below:

    # create the weighted adjacency matrix of the graph and plot the graph
+
    GitHub
    The source files for all examples can be found in /examples.
    Maximum Cut Problem
    We are interested in solving an approximation to the maximum cut problem using semidefinite programming. Consider the graph $G(V,E)$ with weights $w_{ij}$ shown below:
    # create the weighted adjacency matrix of the graph and plot the graph
 using LinearAlgebra, Plots, GraphRecipes
 n = 4;
 W = zeros(n, n);
@@ -11,54 +11,7 @@
 W[2, 3] = 2; W[2, 4] = 10;
 W[3, 4] = 6;
 W = Symmetric(W)
-graphplot(W, names = 1:n, edgelabel = W,  x = [0; 1; 1; 0], y = [1; 1; 0; 0], fontsize = 12, nodeshape =:circle)
    
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    The maximum cut problem tries to find a cut or a partition of the graph's vertices into two complementary sets $S$ and $\bar{S}$ such that the total weight of the edges between the two sets is maximized. For this small graph the problem is trivial. The optimal solution is $S = \{1,2,3 \}$ and $\bar{S}=\{4\}$. Formally, this problem can be written as a mixed-integer optimisation problem:
    \[\begin{array}{ll} \text{maximize} &  \frac{1}{2} \sum_{i < j} w_{ij}(1 - y_i y_j)\\
+graphplot(W, names = 1:n, edgelabel = W,  x = [0; 1; 1; 0], y = [1; 1; 0; 0], fontsize = 12, nodeshape =:circle)
    Example block output

    The maximum cut problem tries to find a cut or a partition of the graph's vertices into two complementary sets $S$ and $\bar{S}$ such that the total weight of the edges between the two sets is maximized. For this small graph the problem is trivial. The optimal solution is $S = \{1,2,3 \}$ and $\bar{S}=\{4\}$. Formally, this problem can be written as a mixed-integer optimisation problem:

    \[\begin{array}{ll} \text{maximize} & \frac{1}{2} \sum_{i < j} w_{ij}(1 - y_i y_j)\\ \text{subject to} & y_i \in \, \{-1, 1 \}, \quad \forall i \in V, \end{array}\]

    where $y_i = 1$ indicates that $v_i \in S$ and $y_i = -1$ indicates that $v_i \in \bar{S}$. This problem is of interest in the field of integrated circuit design, where one tries to minimize the number of cross-layer connections in a circuit under layout constraints.

    For more complicated graphs this problem quickly becomes hard to solve to optimality and in fact is known to be NP-hard. For this example we are interested in the randomized approximation algorithm that relaxes the problem to an SDP and was devised in Goemans and Williamson (1995) [1] (see for more details).

    The approach can be divided into three steps:

    1. Solve a relaxed SDP to obtain $Y^*$
    2. Recover an approximate solution $V$ via a Cholesky factorisation $Y^* = V^\top V$
    3. Round the approximate solution using a random vector $r$ from the unit sphere.

    The authors showed that this approach guarantees a solution of at least $0.87856$ times the optimal solution.

    Solving the primal SDP

    Before we formulate the SDP, let's compute the Laplacian matrix $L$:

    using LinearAlgebra, Random, SparseArrays, COSMO, JuMP
 
@@ -111,7 +64,7 @@
 Status: Solved
 Iterations: 176 (incl. 26 safeguarding iter)
 Optimal objective: -24
-Runtime: 0.278s (278.29ms)
    Yopt = JuMP.value.(Y);
+Runtime: 0.276s (276.19ms)
    Yopt = JuMP.value.(Y);
 obj_val = JuMP.objective_value(model)
    23.9999999938528

    Solving the dual SDP

    Notice that the decision matrix $Y$ is generally dense (as correctly classified in the solver output above). Therefore, we won't be able to utilize COSMO's chordal decomposition features. However, assuming strong duality, it turns out that we can also solve the dual problem, which is given by:

    \[\begin{array}{ll} \text{minimize} & \sum_i \gamma_i \\ \text{subject to} & \text{diag}(\gamma) - \frac{1}{4} L = S \\ & S \in \mathbf{S}_{+}^n. @@ -139,4 +92,4 @@ 1.0 1.0 1.0 - -1.0

    For larger graphs this rounding step could be repeated several times to improve the rounding. As expected $S = \{1, 2, 3\}$ and $\bar{S}=\{ 4\}$.

    References

    [1] Goemans and Williamson - Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programs (1995)

    [2] Post-processing code steps 2 and 3 from this JuMP example

    This page was generated using Literate.jl.

    + -1.0

    For larger graphs this rounding step could be repeated several times to improve the rounding. As expected $S = \{1, 2, 3\}$ and $\bar{S}=\{ 4\}$.

    References

    [1] Goemans and Williamson - Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programs (1995)

    [2] Post-processing code steps 2 and 3 from this JuMP example

    This page was generated using Literate.jl.

    diff --git a/dev/examples/portfolio_optimisation/11509310.svg b/dev/examples/portfolio_optimisation/11509310.svg new file mode 100644 index 00000000..1df95949 --- /dev/null +++ b/dev/examples/portfolio_optimisation/11509310.svg @@ -0,0 +1,47 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/examples/portfolio_optimisation/index.html b/dev/examples/portfolio_optimisation/index.html index 5a269f80..60d8fddc 100644 --- a/dev/examples/portfolio_optimisation/index.html +++ b/dev/examples/portfolio_optimisation/index.html @@ -1,9 +1,9 @@ -Portfolio Optimisation · COSMO.jl
    Edit on GitHub

    The source files for all examples can be found in /examples.

    Portfolio Optimisation

    We consider a single-period Markowitz portfolio optimisation example. Assume that we have a portfolio with $n$ assets at the beginning of time period $t$. Given some forecasts on risks and expected returns we try to find the optimal trade vector that rebalances the portfolio to achieve a good balance between expected risk (variance) $x^\top \Sigma x$ and returns $\mu^\top x$. In it's most simple form we want to solve:

    \[\begin{array}{ll} \text{maximize} & \mu^\top x - \gamma (x^\top \Sigma x)\\ +

    GitHub

    The source files for all examples can be found in /examples.

    Portfolio Optimisation

    We consider a single-period Markowitz portfolio optimisation example. Assume that we have a portfolio with $n$ assets at the beginning of time period $t$. Given some forecasts on risks and expected returns we try to find the optimal trade vector that rebalances the portfolio to achieve a good balance between expected risk (variance) $x^\top \Sigma x$ and returns $\mu^\top x$. In it's most simple form we want to solve:

    \[\begin{array}{ll} \text{maximize} & \mu^\top x - \gamma (x^\top \Sigma x)\\ \text{subject to} & 1^\top x = d + 1^\top x^0 \\ & x \geq 0, \end{array}\]

    with variable $x \in \mathbf{R}^n$, $\mu$ forecasted (expected) returns, $\gamma > 0 $ risk aversion parameter. $x^0_i$ represents the initial investment in asset $i$ and $d$ represents the cash reserve. Consequently, the equality constraint tells us that the sum of the new allocation vector $x$ has to equal the initial allocation plus the cash reserve. Furthermore, the covariance matrix of our risk model is given by $\Sigma \in \mathbf{S}_+^n$. Here we assume a factor risk model, i.e. we can write $\Sigma$ as $\Sigma = D + F F^\top$ where $D$ is diagonal and the factor matrix $F$ has a lower rank $k < n$. This approach allows us to reduce the number of nonzeros in the problem. Furthermore, note that we don't consider shortselling in this example. Let's generate some problem data:

    using LinearAlgebra, SparseArrays, Random, COSMO, JuMP, Test
@@ -51,7 +51,7 @@
 Acc:      Anderson Type2{QRDecomp},
           Memory size = 15, RestartedMemory,
           Safeguarded: true, tol: 2.0
-Setup Time: 0.14ms
+Setup Time: 0.16ms
 
 Iter:	Objective:	Primal Res:	Dual Res:	Rho:
 1	-1.0930e-01	6.0222e-01	6.5188e-02	1.0000e-01
@@ -62,7 +62,7 @@
 Status: Solved
 Iterations: 25
 Optimal objective: -0.07952
-Runtime: 0.001s (0.76ms)

    After solving the problem, we can calculate the expected return and risk $\sigma= \sqrt{x^{* \top} \Sigma x^*}$:

    x_opt = JuMP.value.(x);
+Runtime: 0.001s (0.81ms)

    After solving the problem, we can calculate the expected return and risk $\sigma= \sqrt{x^{* \top} \Sigma x^*}$:

    x_opt = JuMP.value.(x);
 y_opt = JuMP.value.(y);
 expected_return_basic = dot(μ, x_opt)
    0.09960959937666913
    expected_risk_basic = sqrt(dot(y_opt, y_opt))
    0.013048755030479095

    Using standard deviation in the model

    It is pointed out in [1] that above problem formulation can lead to numerical problems, e.g. if $\Sigma$ is not strictly positive semidefinite. Another option is to formulate the risk constraint in terms of the standard deviation $\|M^\top x \|$ where $M M^\top = D + F F^\top$ and bound it using a second-order cone constraint:

    \[\begin{array}{ll} \text{minimize} & - \mu^\top x \\ \text{subject to} & \|M^\top x\| \leq \gamma \\ @@ -99,7 +99,7 @@ Acc: Anderson Type2{QRDecomp}, Memory size = 15, RestartedMemory, Safeguarded: true, tol: 2.0 -Setup Time: 0.15ms +Setup Time: 0.18ms Iter: Objective: Primal Res: Dual Res: Rho: 1 -2.9234e-01 6.0111e-01 6.3843e-02 1.0000e-01 @@ -122,8 +122,8 @@ Status: Solved Iterations: 326 (incl. 1 safeguarding iter) Optimal objective: -0.1189 -Runtime: 0.006s (5.78ms)

    Note that the result is different from the example above because $\gamma$ scales the problem in a different way. Here it can be seen as an upper bound on the standard deviation of the portfolio.

    x_opt = JuMP.value.(x);
-expected_return = dot(μ, x_opt)
    0.11886638687579054

    Let us verify that the bound holds:

    @test norm(Mt * x_opt) <= γ
    Test Passed

    Pareto-optimal front

    The above portfolio optimisation approach yields the optimal expected return for a given level of risk. The result is obviously impacted by the risk aversion $\gamma$ parameter. To visualise the trade-off and present the investor with an efficient Pareto optimal portfolio for their risk appetite we can compute the optimal portfolio for many choices of $\gamma$ and plot the corresponding risk-return trade-off curve.

    gammas = [ 0.001, 0.01, 0.1,  0.5,  1., 3., 10, 100, 1000]
+Runtime: 0.007s (7.0ms)

    Note that the result is different from the example above because $\gamma$ scales the problem in a different way. Here it can be seen as an upper bound on the standard deviation of the portfolio.

    x_opt = JuMP.value.(x);
+expected_return = dot(μ, x_opt)
    0.11886638687579054

    Let us verify that the bound holds:

    @test norm(Mt * x_opt) <= γ
    Test Passed

    Pareto-optimal front

    The above portfolio optimisation approach yields the optimal expected return for a given level of risk. The result is obviously impacted by the risk aversion $\gamma$ parameter. To visualise the trade-off and present the investor with an efficient Pareto optimal portfolio for their risk appetite we can compute the optimal portfolio for many choices of $\gamma$ and plot the corresponding risk-return trade-off curve.

    gammas = [ 0.001, 0.01, 0.1,  0.5,  1., 3., 10, 100, 1000]
 risks = zeros(length(gammas))
 returns = zeros(length(gammas))
 model = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, "verbose" => false));
@@ -144,54 +144,7 @@
     returns[k] = dot(μ, x_opt)
     risks[k] = sqrt(dot(y_opt, y_opt))
 end

    We can now plot the risk-return trade-off curve:

    using Plots
-Plots.plot(risks, returns, xlabel = "Standard deviation (risk)", ylabel = "Expected return", title = "Risk-return trade-off for efficient portfolios", legend = false)
    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    Note

    When the model is updated in JuMP as above the JuMP.model is copied in full to COSMO. We are trying to improve the interface with respect to model updates in the future. Until then you can use Model Updates in COSMOs native interface.

    Transaction costs

    In the model above we assume that trading the assets is free and does not impact the market. However, this is clearly not the case in reality. To make the example more realistic consider the following cost $c_j$ associated with the trade $δ_j = x_j - x_j^0$:

    \[c_j(\delta_j) = a_j |\delta_j| + b_j |\delta_j|^{3/2},\]

    where the first term models the bid-ask spread and broker fees for asset $j$. The second term models the impact on the market that our trade has. This is obviously only a factor if the volume of our trade is significant. The constant $b_j$ is a function of the total volume traded in the considered time periode and the price volatility of the asset and has to be estimated by the trader. To make this example simple we consider the same coefficients $a$ and $b$ for every asset. The $|\delta_j|^{3/2}$ term can be easily modeled using a power cone constraint $\mathcal{K}_{pow} = \{(x, y, z) \mid x^\alpha y^{(1-\alpha)} \geq |z|, x \geq 0, y \geq 0, 0 \leq \alpha \leq 1 \}$. In fact this can be used to model any market impact function with exponent greater than 1. We can write the total transaction cost $a^\top s + b^\top t$ where $s_j$ bounds the absolute value of $\delta_j$ and $t_{j}$ is used to bound the term $|x_j - x_j^0|^{3/2} \leq t_{j}$ using a power cone formulation: $(t_{j}, 1, x_j - x_j^0) \in \mathcal{K}_{pow}(2/3)$.

    a = 1e-3
+Plots.plot(risks, returns, xlabel = "Standard deviation (risk)", ylabel = "Expected return", title = "Risk-return trade-off for efficient portfolios", legend = false)
    Example block output
    Note

    When the model is updated in JuMP as above the JuMP.model is copied in full to COSMO. We are trying to improve the interface with respect to model updates in the future. Until then you can use Model Updates in COSMOs native interface.

    Transaction costs

    In the model above we assume that trading the assets is free and does not impact the market. However, this is clearly not the case in reality. To make the example more realistic consider the following cost $c_j$ associated with the trade $δ_j = x_j - x_j^0$:

    \[c_j(\delta_j) = a_j |\delta_j| + b_j |\delta_j|^{3/2},\]

    where the first term models the bid-ask spread and broker fees for asset $j$. The second term models the impact on the market that our trade has. This is obviously only a factor if the volume of our trade is significant. The constant $b_j$ is a function of the total volume traded in the considered time periode and the price volatility of the asset and has to be estimated by the trader. To make this example simple we consider the same coefficients $a$ and $b$ for every asset. The $|\delta_j|^{3/2}$ term can be easily modeled using a power cone constraint $\mathcal{K}_{pow} = \{(x, y, z) \mid x^\alpha y^{(1-\alpha)} \geq |z|, x \geq 0, y \geq 0, 0 \leq \alpha \leq 1 \}$. In fact this can be used to model any market impact function with exponent greater than 1. We can write the total transaction cost $a^\top s + b^\top t$ where $s_j$ bounds the absolute value of $\delta_j$ and $t_{j}$ is used to bound the term $|x_j - x_j^0|^{3/2} \leq t_{j}$ using a power cone formulation: $(t_{j}, 1, x_j - x_j^0) \in \mathcal{K}_{pow}(2/3)$.

    a = 1e-3
 b = 1e-1
 γ = 1.0;
 model = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, "eps_abs" => 1e-5, "eps_rel" => 1e-5));
@@ -235,7 +188,7 @@
 Acc:      Anderson Type2{QRDecomp},
           Memory size = 15, RestartedMemory,
           Safeguarded: true, tol: 2.0
-Setup Time: 3.73ms
+Setup Time: 3.81ms
 
 Iter:	Objective:	Primal Res:	Dual Res:	Rho:
 1	-1.5637e-01	1.1398e+00	6.3250e-02	1.0000e-01
@@ -248,8 +201,8 @@
 Status: Solved
 Iterations: 80 (incl. 5 safeguarding iter)
 Optimal objective: -0.07786
-Runtime: 0.088s (87.95ms)

    Let's look at the expected return and the total transaction cost:

    x_opt = JuMP.value.(x);
+Runtime: 0.094s (93.66ms)

    Let's look at the expected return and the total transaction cost:

    x_opt = JuMP.value.(x);
 y_opt = JuMP.value.(y);
 s_opt = JuMP.value.(s);
 t_opt = JuMP.value.(t);
-expected_return = dot(μ, x_opt)
    0.09665905111704713
    expected_risk = dot(y_opt, y_opt)
    0.00016794910539706204
    transaction_cost = a * sum(s_opt) + b * sum( t_opt)
    0.02732439918367407

    References

    [1] Mosek Case Studies

    This page was generated using Literate.jl.

    +expected_return = dot(μ, x_opt)
    0.09665905111704713
    expected_risk = dot(y_opt, y_opt)
    0.00016794910539706204
    transaction_cost = a * sum(s_opt) + b * sum( t_opt)
    0.02732439918367407

    References

    [1] Mosek Case Studies

    This page was generated using Literate.jl.

    diff --git a/dev/examples/qp/index.html b/dev/examples/qp/index.html index 566efa71..29330beb 100644 --- a/dev/examples/qp/index.html +++ b/dev/examples/qp/index.html @@ -1,9 +1,9 @@ -Quadratic Program · COSMO.jl
    Edit on GitHub

    The source files for all examples can be found in /examples.

    Quadratic Program

    We want to solve the following quadratic program with decision variable x:

    \[\begin{array}{ll} \text{minimize} & 1/2 x^\top P x + q^\top x \\ +

    GitHub

    The source files for all examples can be found in /examples.

    Quadratic Program

    We want to solve the following quadratic program with decision variable x:

    \[\begin{array}{ll} \text{minimize} & 1/2 x^\top P x + q^\top x \\ \text{subject to} & l \leq A x \leq u \end{array}\]

    The problem can be solved with COSMO in the following way. Start by defining the problem data

    using COSMO, SparseArrays, LinearAlgebra, Test
 
@@ -38,7 +38,7 @@
 Acc:      Anderson Type2{QRDecomp},
           Memory size = 8, RestartedMemory,
           Safeguarded: true, tol: 2.0
-Setup Time: 0.07ms
+Setup Time: 0.08ms
 
 Iter:	Objective:	Primal Res:	Dual Res:	Rho:
 1	-2.0835e-01	1.2796e+00	2.3510e-01	1.0000e-01
@@ -49,7 +49,7 @@
 Status: Solved
 Iterations: 30 (incl. 5 safeguarding iter)
 Optimal objective: 1.88
-Runtime: 0.001s (0.52ms)

    Alternatively, we can also use two-sided constraints with COSMO.Box:

    constraint1 = COSMO.Constraint(A, zeros(3), COSMO.Box(l, u));
+Runtime: 0.001s (0.58ms)

    Alternatively, we can also use two-sided constraints with COSMO.Box:

    constraint1 = COSMO.Constraint(A, zeros(3), COSMO.Box(l, u));
 
 model = COSMO.Model();
 assemble!(model, P, q, constraint1, settings = settings);
@@ -86,11 +86,11 @@
 Status: Solved
 Iterations: 26 (incl. 1 safeguarding iter)
 Optimal objective: 1.88
-Runtime: 0.004s (4.42ms)

    Let's check that the solution is correct:

    @testset "QP Problem" begin
+Runtime: 0.004s (4.43ms)

    Let's check that the solution is correct:

    @testset "QP Problem" begin
   @test norm(res.x[1:2] - [0.3; 0.7], Inf) < 1e-3
   @test norm(res_box.x[1:2] - [0.3; 0.7], Inf) < 1e-3
   @test abs(res.obj_val - 1.88) < 1e-3
   @test abs(res_box.obj_val - 1.88) < 1e-3
 end
 nothing
    Test Summary: | Pass  Total  Time
-QP Problem    |    4      4  0.1s

    This page was generated using Literate.jl.

    +QP Problem | 4 4 0.1s

    This page was generated using Literate.jl.

    diff --git a/dev/examples/sum_abs_k_eigenvalues/index.html b/dev/examples/sum_abs_k_eigenvalues/index.html index dc5c93a2..c60d8bb2 100644 --- a/dev/examples/sum_abs_k_eigenvalues/index.html +++ b/dev/examples/sum_abs_k_eigenvalues/index.html @@ -1,9 +1,9 @@ -Minimizing the sum of the k-largest λ · COSMO.jl
    Edit on GitHub

    The source files for all examples can be found in /examples.

    Minimizing the sum of the k-largest λ

    We show how to find the sum of absolute value of the k largest eigenvalues of a symmetric matrix $A \in \mathbb{S}^n$. This problem can be solved as a semidefinite program. The primal and dual forms are stated in Alizadeh [1]:

    \[\begin{array}{llll} \text{maximize} & \text{Tr}(AY) - \text{Tr}(AW) & \text{minimize} & kz + Tr(U) + Tr(V) \\ +

    GitHub

    The source files for all examples can be found in /examples.

    Minimizing the sum of the k-largest λ

    We show how to find the sum of absolute value of the k largest eigenvalues of a symmetric matrix $A \in \mathbb{S}^n$. This problem can be solved as a semidefinite program. The primal and dual forms are stated in Alizadeh [1]:

    \[\begin{array}{llll} \text{maximize} & \text{Tr}(AY) - \text{Tr}(AW) & \text{minimize} & kz + Tr(U) + Tr(V) \\ \text{subject to} & \text{Tr}(Y + W) = k & \text{subject to} & zI + V - A \succeq 0 \\ & 0 \preceq Y \preceq I & & zI + U + A \succeq 0 \\ & 0 \preceq W \preceq I & & U, V \succeq 0, @@ -58,7 +58,7 @@ Acc: Anderson Type2{QRDecomp}, Memory size = 15, RestartedMemory, Safeguarded: true, tol: 2.0 -Setup Time: 0.14ms +Setup Time: 0.15ms Iter: Objective: Primal Res: Dual Res: Rho: 1 -6.3802e+04 1.7856e+02 6.3659e+00 1.0000e-01 @@ -71,7 +71,7 @@ Status: Solved Iterations: 89 (incl. 14 safeguarding iter) Optimal objective: -79.28 -Runtime: 0.011s (10.81ms)

    opt_objective = JuMP.objective_value(model)
    79.28110570563726

    Now, we can check the solution by computing the sum of the absolute value of the 3-largest eigenvalues:

    k_λ_abs = sum(sort(abs.(eigen(A).values), rev = true)[1:k])
    79.28110570563712

    Solve the dual

    Alternatively, we can solve the dual problem:

    model = JuMP.Model(with_optimizer(COSMO.Optimizer, verbose=true));
+Runtime: 0.011s (10.69ms)
    opt_objective = JuMP.objective_value(model)
    79.28110570563726

    Now, we can check the solution by computing the sum of the absolute value of the 3-largest eigenvalues:

    k_λ_abs = sum(sort(abs.(eigen(A).values), rev = true)[1:k])
    79.28110570563712

    Solve the dual

    Alternatively, we can solve the dual problem:

    model = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, "verbose" => true));
 @variable(model, V[1:n, 1:n], PSD);
 @variable(model, U[1:n, 1:n], PSD);
 @variable(model, z);
@@ -79,9 +79,45 @@
 @objective(model, Min, k * z + tr(V) + tr(U));
 @constraint(model, Symmetric(z .* diagm(0 => ones(n)) + V - A) in PSDCone());
 @constraint(model, Symmetric(z .* diagm(0 => ones(n)) + U + A) in PSDCone());
-status = JuMP.optimize!(model)
    opt_objective = JuMP.objective_value(model)
    79.28110570563726

    This gives the same result.

    Problem with A as variable

    Above problems are mostly helpful for illustrative purpose. It is obviously easier to find the sum of the k-largest eigenvalues by simply computing the eigenvalues of $A$. However, above results become useful if finding $A$ itself is part of the problem. For example, assume we want to find a valid matrix $A$ under the constraints: $C\, \text{vec}(A) = b$ with the minimum sum of absolute values of the k-largest eigenvalues. We can then solve the equivalent problem:

    \[\begin{array}{ll} \text{minimize} & kz + Tr(U) + Tr(V) \\ +status = JuMP.optimize!(model)

    ------------------------------------------------------------------
+          COSMO v0.8.9 - A Quadratic Objective Conic Solver
+                         Michael Garstka
+                University of Oxford, 2017 - 2022
+------------------------------------------------------------------
+
+Problem:  x ∈ R^{111},
+          constraints: A ∈ R^{220x111} (240 nnz),
+          matrix size to factor: 331x331,
+          Floating-point precision: Float64
+Sets:     DensePsdConeTriangle of dim: 55 (10x10)
+          DensePsdConeTriangle of dim: 55 (10x10)
+          DensePsdConeTriangle of dim: 55 (10x10)
+          DensePsdConeTriangle of dim: 55 (10x10)
+Settings: ϵ_abs = 1.0e-05, ϵ_rel = 1.0e-05,
+          ϵ_prim_inf = 1.0e-04, ϵ_dual_inf = 1.0e-04,
+          ρ = 0.1, σ = 1e-06, α = 1.6,
+          max_iter = 5000,
+          scaling iter = 10 (on),
+          check termination every 25 iter,
+          check infeasibility every 40 iter,
+          KKT system solver: QDLDL
+Acc:      Anderson Type2{QRDecomp},
+          Memory size = 15, RestartedMemory,
+          Safeguarded: true, tol: 2.0
+Setup Time: 0.16ms
+
+Iter:	Objective:	Primal Res:	Dual Res:	Rho:
+1	-3.1509e+02	2.7426e+01	8.3187e+00	1.0000e-01
+25	 7.9281e+01	1.9374e-08	9.9322e-10	1.0000e-01
+
+------------------------------------------------------------------
+>>> Results
+Status: Solved
+Iterations: 25
+Optimal objective: 79.28
+Runtime: 0.003s (2.81ms)
    opt_objective = JuMP.objective_value(model)
    79.28110569379851

    This gives the same result.

    Problem with A as variable

    Above problems are mostly helpful for illustrative purpose. It is obviously easier to find the sum of the k-largest eigenvalues by simply computing the eigenvalues of $A$. However, above results become useful if finding $A$ itself is part of the problem. For example, assume we want to find a valid matrix $A$ under the constraints: $C\, \text{vec}(A) = b$ with the minimum sum of absolute values of the k-largest eigenvalues. We can then solve the equivalent problem:

    \[\begin{array}{ll} \text{minimize} & kz + Tr(U) + Tr(V) \\ \text{subject to} & C \text{vec}(A) = b \\ & zI + V - A \succeq 0 \\ & zI + U + A \succeq 0 \\ & U, V \succeq 0. -\end{array}\]

    References

    [1] Alizadeh - Interior point methods in semidefinite programming with applications to combinatorial optimization (1995)

    This page was generated using Literate.jl.

    +\end{array}\]

    References

    [1] Alizadeh - Interior point methods in semidefinite programming with applications to combinatorial optimization (1995)

    This page was generated using Literate.jl.

    diff --git a/dev/examples/svm_primal/2f3f2fa2.svg b/dev/examples/svm_primal/2f3f2fa2.svg new file mode 100644 index 00000000..34e928fc --- /dev/null +++ b/dev/examples/svm_primal/2f3f2fa2.svg @@ -0,0 +1,144 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/examples/svm_primal/ba6f4b3f.svg b/dev/examples/svm_primal/ba6f4b3f.svg new file mode 100644 index 00000000..ea51b001 --- /dev/null +++ b/dev/examples/svm_primal/ba6f4b3f.svg @@ -0,0 +1,148 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/examples/svm_primal/index.html b/dev/examples/svm_primal/index.html index 6267cdf5..e523d870 100644 --- a/dev/examples/svm_primal/index.html +++ b/dev/examples/svm_primal/index.html @@ -1,9 +1,9 @@ -Support Vector Machine · COSMO.jl
    Edit on GitHub

    The source files for all examples can be found in /examples.

    Support Vector Machine

    We are showing how to solve a support vector machine problem with COSMO (and JuMP).

    Generating the Dataset

    We want to classify the points in this example dataset with $m = 100$ samples and $n = 2$ features:

    using Distributions: MvNormal
+
    GitHub
    The source files for all examples can be found in /examples.
    Support Vector Machine
    We are showing how to solve a support vector machine problem with COSMO (and JuMP).
    Generating the Dataset
    We want to classify the points in this example dataset with $m = 100$ samples and $n = 2$ features:
    using Distributions: MvNormal
 using Plots, LinearAlgebra, SparseArrays, Random, Test
    # Generate dataset
 rng = Random.MersenneTwister(123);
 num_samples = 100;
@@ -12,151 +12,7 @@
 ypos = ones(div(num_samples, 2));
 yneg = -ones(div(num_samples, 2));
    # Plot dataset
 plot(Xpos[:, 1], Xpos[:, 2], color = :red, st=:scatter, markershape = :rect, label = "positive", xlabel = "x1", ylabel = "x2")
-plot!(Xneg[:, 1], Xneg[:, 2], color = :blue, st=:scatter, markershape = :circle, label = "negative")
    
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    with samples $(x_1, x_2, \ldots, x_m) \in \mathbb{R}^2$ and labels $y_i \in \{-1,1\}$.
    Solving SVM as a QP
    We want to compute the weights $w$ and bias term $b$ of the (soft-margin) SVM classifier:
    \[\begin{array}{ll}
+plot!(Xneg[:, 1], Xneg[:, 2], color = :blue, st=:scatter, markershape = :circle, label = "negative")
    Example block output

    with samples $(x_1, x_2, \ldots, x_m) \in \mathbb{R}^2$ and labels $y_i \in \{-1,1\}$.

    Solving SVM as a QP

    We want to compute the weights $w$ and bias term $b$ of the (soft-margin) SVM classifier:

    \[\begin{array}{ll} \text{minimize} & \|w\|^2 + \lambda \sum_{i=1}^m \text{max}(0, 1 - y_i(w^\top x_i - b)), \end{array}\]

    where $\lambda$ is a hyperparameter. This problem can be solved as a quadratic program. We can rewrite above problem into an optimisation problem in primal form by introducing the auxiliary slack variables $t_i$:

    \[t_i = \text{max}(0, 1 - y_i(w^T x_i - b)), \quad t_i \geq 0.\]

    This allows us to write the problems in standard QP format:

    \[\begin{array}{ll} \text{minimize} & \|w\|^2 + \lambda \sum_{i=1}^m t_i\\ @@ -194,7 +50,7 @@ Acc: Anderson Type2{QRDecomp}, Memory size = 15, RestartedMemory, Safeguarded: true, tol: 2.0 -Setup Time: 0.18ms +Setup Time: 0.17ms Iter: Objective: Primal Res: Dual Res: Rho: 1 -1.2711e+03 1.8883e+01 9.5020e+00 1.0000e-01 @@ -217,160 +73,12 @@ Status: Solved Iterations: 335 (incl. 10 safeguarding iter) Optimal objective: 15.55 -Runtime: 0.003s (3.42ms)

    The optimal weights $w = [w_0, w_1, w_2]^\top$ (where $w_0 = b$) are:

    w_opt = JuMP.value.(w)
    3-element Vector{Float64}:
+Runtime: 0.003s (3.44ms)

    The optimal weights $w = [w_0, w_1, w_2]^\top$ (where $w_0 = b$) are:

    w_opt = JuMP.value.(w)
    3-element Vector{Float64}:
  0.1531242891284758
  1.0341323254762473
  0.520268546984156

    Plotting the hyperplane

    The separating hyperplane is defined by $w^\top x - b = 0$. To plot the hyperplane, we calculate $x_2$ over a range of $x_1$ values:

    \[x_2 = (-w_1 x_1 - w_0) / w_2, \text{ where } w_0 = b.\]

    x1 = -4:0.1:4;
 x2 = (-w_opt[2] * x1  .- w_opt[1]) / w_opt[3]
-plot!(x1, x2, label = "SVM separator", legend = :topleft)
    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

    Modelling with COSMO

    The problem can also be solved by transforming it directly into COSMO's problem format. Define COSMO`s $x$-variable to be $x=[w, t]^\top$ and choose $P$, $q$, accordingly:

    P = blockdiag(spdiagm(0 => ones(n)), spzeros(m, m));
+plot!(x1, x2, label = "SVM separator", legend = :topleft)
    Example block output

    Modelling with COSMO

    The problem can also be solved by transforming it directly into COSMO's problem format. Define COSMO`s $x$-variable to be $x=[w, t]^\top$ and choose $P$, $q$, accordingly:

    P = blockdiag(spdiagm(0 => ones(n)), spzeros(m, m));
 q = [zeros(n); 0.5 * λ * ones(m)];

    Next we transform the first constraint $y_i (w^\top x_i - b) \geq 1 - t_i, \quad \text{for } i = 1,\ldots, m$ into COSMO's constraint format: $Ax + b \in \mathcal{K}$.

    A1 = [(spdiagm(0 => y) * X) spdiagm(0 => ones(m))];
 b1 = -ones(m);
 cs1 = COSMO.Constraint(A1, b1, COSMO.Nonnegatives);

    It remains to specify the constraint $t_i \geq 0, \quad \text{for } i = 1,\ldots, m$:

    A2 = spdiagm(0 => ones(m));
@@ -379,4 +87,4 @@ 
    Test Passed

    This page was generated using Literate.jl.

    +@test norm(w_opt2 - w_opt, Inf) < 1e-3
    Test Passed

    This page was generated using Literate.jl.

    diff --git a/dev/examples/two_way_partitioning/index.html b/dev/examples/two_way_partitioning/index.html index 45214ea3..6f8997e9 100644 --- a/dev/examples/two_way_partitioning/index.html +++ b/dev/examples/two_way_partitioning/index.html @@ -1,9 +1,9 @@ -Relaxed Two-Way Partitioning Problem · COSMO.jl
    Edit on GitHub

    The source files for all examples can be found in /examples.

    Relaxed Two-Way Partitioning Problem

    We consider the (nonconvex) two-way partitioning problem from Boyd and Vandenberghe (2004), p.219 [1]:

    \[\begin{array}{ll} \text{minimize} & x^\top W x \\ +

    GitHub

    The source files for all examples can be found in /examples.

    Relaxed Two-Way Partitioning Problem

    We consider the (nonconvex) two-way partitioning problem from Boyd and Vandenberghe (2004), p.219 [1]:

    \[\begin{array}{ll} \text{minimize} & x^\top W x \\ \text{subject to} & x_i^2 = 1, \quad i=1,\dots, n, \end{array}\]

    with $x \in \mathbf{R}^n$ and $W \in \mathbf{S}^n$. The problem can be interpreted as finding a partition of the points $i = 1,\dots, n$ into two sets, where the cost of two points in the same set is $W_{ij}$ and $-W_{ij}$ otherwise. Brute forcing the solution $x^*$ takes $2^n$ attempts and becomes quickly intractable, e.g. for $n \geq 30$. However, a lower-bound for this problem can be computed by solving the convex dual of the problem:

    \[\begin{array}{ll} \text{maximize} & -\sum_i \nu \\ \text{subject to} & W + \text{diag}(\nu) \in \mathbf{S}_+^n. @@ -69,7 +69,7 @@ Acc: Anderson Type2{QRDecomp}, Memory size = 15, RestartedMemory, Safeguarded: true, tol: 2.0 -Setup Time: 0.09ms +Setup Time: 0.1ms Iter: Objective: Primal Res: Dual Res: Rho: 1 -6.1146e+02 1.7072e+01 5.9998e-01 1.0000e-01 @@ -103,7 +103,7 @@ Status: Solved Iterations: 677 (incl. 77 safeguarding iter) Optimal objective: 25.62 -Runtime: 0.137s (137.37ms)

    Looking at the solver output you can see how the PSD constraint was decomposed into 19 PSD constraints. Let's look at the lower bound:

    JuMP.objective_value(model)
    -25.623709241260592

    As $n$ is small, we can verify our result by finding the optimal solution by trying out all possible combinations:

    function brute_force_optimisation(W, n)
+Runtime: 0.117s (117.5ms)

    Looking at the solver output you can see how the PSD constraint was decomposed into 19 PSD constraints. Let's look at the lower bound:

    JuMP.objective_value(model)
    -25.623709241260592

    As $n$ is small, we can verify our result by finding the optimal solution by trying out all possible combinations:

    function brute_force_optimisation(W, n)
    opt_obj = Inf
    opt_x = Inf * ones(n)
 
@@ -117,4 +117,4 @@
    end
    return opt_obj, opt_x
 end
-opt_obj, opt_x = brute_force_optimisation(W, n)
    (-25.623709242765823, [-1.0, 1.0, -1.0, -1.0, -1.0, 1.0, -1.0, 1.0, 1.0, 1.0, -1.0, -1.0, -1.0, -1.0, -1.0, -1.0, 1.0, -1.0, 1.0, 1.0])

    References

    [1] Boyd and Vandenberghe - Convex Optimization, Cambridge University Press (2004)

    This page was generated using Literate.jl.

    +opt_obj, opt_x = brute_force_optimisation(W, n)
    (-25.623709242765823, [-1.0, 1.0, -1.0, -1.0, -1.0, 1.0, -1.0, 1.0, 1.0, 1.0, -1.0, -1.0, -1.0, -1.0, -1.0, -1.0, 1.0, -1.0, 1.0, 1.0])

    References

    [1] Boyd and Vandenberghe - Convex Optimization, Cambridge University Press (2004)

    This page was generated using Literate.jl.

    diff --git a/dev/getting_started/index.html b/dev/getting_started/index.html index 2953b36a..9a5a684e 100644 --- a/dev/getting_started/index.html +++ b/dev/getting_started/index.html @@ -1,13 +1,13 @@ -Getting Started · COSMO.jl
    Edit on GitHub

    Getting Started

    This user guide describes the basic structures and functions to define an optimisation problem, to solve the problem and to analyse the result. If you want to use JuMP to describe the problem, see the JuMP Interface section.

    COSMO solves optimisation problems in the following format:

    \[\begin{array}{ll} \text{minimize} & \textstyle{\frac{1}{2}}x^\top Px + q^\top x\\ \text{subject to} & Ax + s = b \\ & s \in \mathcal{K}, \end{array}\]

    with decision variables $x \in \mathbb{R}^n$, $s \in \mathbb{R}^m$ and data matrices $P=P^\top \succeq 0$, $q \in \mathbb{R}^n$, $A \in \mathbb{R}^{m \times n}$, and $b \in \mathbb{R}^m$. The convex set $\mathcal{K}$ is a composition of convex sets and cones.

    Model

    The problem data, user settings and workspace variables are all stored in a Model. To get started define an empty model:

    model = COSMO.Model()

    To initialize the model with an optimisation problem we need to define three more things:

    • the objective function, i.e. the matrix P and the vector q in $\frac{1}{2}x^\top P x + q^\top x$
    • an array of constraints
    • a Settings object that specifies how COSMO solves the problem (optional)

    Objective Function

    To set the objective function of your optimisation problem simply define the square positive semidefinite matrix $P \in \mathrm{R}^{n\times n}$ and the vector $q \in \mathrm{R}^{n}$. You might have to transform your optimisation problem into a solver compatible format for this step.

    Constraints

    The COSMO interface expects constraints to have the form $A_i x + b_i \in \mathcal{K}_i$, where $\mathcal{K}_i$ is one of the convex sets defined below:

    Convex SetDescription
    ZeroSetThe set $\{ 0 \}^{dim}$ that contains the origin
    NonnegativesThe nonnegative orthant $\{ x \in \mathbb{R}^{dim} : x_i \ge 0, \forall i=1,\dots,\mathrm{dim} \}$
    Box(l, u)The hyperbox $\{ x \in \mathbb{R}^{dim} : l \leq x \leq u\}$ with vectors $l \in \mathbb{R}^{dim} \cup \{-\infty\}$ and $u \in \mathbb{R}^{dim} \cup \{+\infty\}$
    SecondOrderConeThe second-order (Lorenz) cone $\{ (t,x) \in \mathbb{R}^{dim} : |x|_2 \leq t \}$
    PsdConeThe vectorized positive semidefinite cone $\mathcal{S}_+^{dim}$. $x$ is the vector obtained by stacking the columns of the positive semidefinite matrix $X$, i.e. $X \in \mathbb{S}^{\sqrt{dim}}_+ \rarr \text{vec}(X) = x \in \mathcal{S}_+^{dim}$
    PsdConeTriangleThe vectorized positive semidefinite cone $\mathcal{S}_+^{dim}$. $x$ is the vector obtained by stacking the columns of the upper triangular part of the positive semidefinite matrix $X$, i.e. $X \in \mathbb{S}^{d}_+ \rarr \text{svec}(X) = x \in \mathcal{S}_+^{dim}$ where $d=\sqrt{1/4 + 2 \text{dim}} - 1/2$
    ExponentialConeThe exponential cone $\mathcal{K}_{exp} = \{(x, y, z) \mid y \geq 0, ye^{x/y} ≤ z\} \cup \{ (x,y,z) \mid x \leq 0, y = 0, z \geq 0 \}$
    DualExponentialConeThe dual exponential cone $\mathcal{K}^*_{exp} = \{(x, y, z) \mid x < 0, -xe^{y/x} \leq e^1 z \} \cup \{ (0,y,z) \mid y \geq 0, z \geq 0 \}$
    PowerCone(alpha)The 3d power cone $\mathcal{K}_{pow} = \{(x, y, z) \mid x^\alpha y^{(1-\alpha)} \geq |z|, x \geq 0, y \geq 0 \}$ with $0 < \alpha < 1$
    DualPowerCone(alpha)The 3d dual power cone $\mathcal{K}^*_{pow} = \{(u, v, w) \mid \left( \frac{u}{\alpha}\right)^\alpha \left( \frac{v}{1-\alpha}\right)^{(1-\alpha)} \geq |w|, u \geq 0, v \geq 0 \}$ with $0 < \alpha < 1$

    The constructor for a constraint expects a matrix A, a vector b and a convex_set.

    Lets consider a problem with a decision variable $x \in \mathbb{R}^5$. Suppose we want to create the two constraint $x_2 + 5 \geq 0$ and $x_3 - 3 \geq 0$. We can do this either by creating two constraints and adding them to an array:

      constraint1 = COSMO.Constraint([0.0 1.0 0.0 0.0 0.0], 5.0, COSMO.Nonnegatives)
+
    GitHub
    Getting Started
    This user guide describes the basic structures and functions to define an optimisation problem, to solve the problem and to analyse the result. If you want to use JuMP to describe the problem, see the JuMP Interface section.
    COSMO solves optimisation problems in the following format:
    \[\begin{array}{ll} \text{minimize} & \textstyle{\frac{1}{2}}x^\top Px + q^\top x\\ \text{subject to} & Ax + s = b \\ & s \in \mathcal{K}, \end{array}\]
    with decision variables $x \in \mathbb{R}^n$, $s \in \mathbb{R}^m$ and data matrices $P=P^\top \succeq 0$, $q \in \mathbb{R}^n$, $A \in \mathbb{R}^{m \times n}$, and $b \in \mathbb{R}^m$. The convex set $\mathcal{K}$  is a composition of convex sets and cones.
    Model
    The problem data, user settings and workspace variables are all stored in a Model. To get started define an empty model:
    model = COSMO.Model()
    To initialize the model with an optimisation problem we need to define three more things:
    • the objective function, i.e. the matrix P and the vector q in $\frac{1}{2}x^\top P x + q^\top x$
    • an array of constraints
    • a Settings object that specifies how COSMO solves the problem (optional)
    Objective Function
    To set the objective function of your optimisation problem simply define the square positive semidefinite matrix $P \in \mathrm{R}^{n\times n}$ and the vector $q \in \mathrm{R}^{n}$. You might have to transform your optimisation problem into a solver compatible format for this step.
    Constraints
    The COSMO interface expects constraints to have the form $A_i x + b_i \in \mathcal{K}_i$, where $\mathcal{K}_i$ is one of the convex sets defined below:
    Convex SetDescription
    ZeroSetThe set $\{ 0 \}^{dim}$ that contains the origin
    NonnegativesThe nonnegative orthant $\{ x \in \mathbb{R}^{dim} : x_i \ge 0, \forall i=1,\dots,\mathrm{dim} \}$
    Box(l, u)The hyperbox $\{ x \in \mathbb{R}^{dim} : l \leq x \leq u\}$ with vectors $l \in \mathbb{R}^{dim} \cup \{-\infty\}$ and $u \in \mathbb{R}^{dim} \cup \{+\infty\}$
    SecondOrderConeThe second-order (Lorenz) cone $\{ (t,x) \in \mathbb{R}^{dim}  :  |x|_2   \leq t \}$
    PsdConeThe vectorized positive semidefinite cone $\mathcal{S}_+^{dim}$. $x$ is the vector obtained by stacking the columns of the positive semidefinite matrix $X$, i.e. $X \in \mathbb{S}^{\sqrt{dim}}_+ \rarr \text{vec}(X) = x \in \mathcal{S}_+^{dim}$
    PsdConeTriangleThe vectorized positive semidefinite cone $\mathcal{S}_+^{dim}$. $x$ is the vector obtained by stacking the columns of the upper triangular part of the positive semidefinite matrix $X$, i.e. $X \in \mathbb{S}^{d}_+ \rarr \text{svec}(X) = x \in \mathcal{S}_+^{dim}$ where $d=\sqrt{1/4 + 2 \text{dim}} - 1/2$
    ExponentialConeThe exponential cone $\mathcal{K}_{exp} = \{(x, y, z) \mid y \geq 0,  ye^{x/y} ≤ z\} \cup \{ (x,y,z) \mid   x \leq 0, y = 0, z \geq 0 \}$
    DualExponentialConeThe dual exponential cone $\mathcal{K}^*_{exp} = \{(x, y, z) \mid x < 0,  -xe^{y/x} \leq e^1 z \} \cup \{ (0,y,z) \mid   y \geq 0, z \geq 0 \}$
    PowerCone(alpha)The 3d power cone $\mathcal{K}_{pow} = \{(x, y, z) \mid x^\alpha y^{(1-\alpha)} \geq  |z|, x \geq 0, y \geq 0 \}$ with $0 < \alpha < 1$
    DualPowerCone(alpha)The 3d dual power cone $\mathcal{K}^*_{pow} = \{(u, v, w) \mid \left( \frac{u}{\alpha}\right)^\alpha \left( \frac{v}{1-\alpha}\right)^{(1-\alpha)} \geq  |w|, u \geq 0, v \geq 0 \}$ with $0 < \alpha < 1$
    The constructor for a constraint expects a matrix A, a vector b and a convex_set.
    Lets consider a problem with a decision variable $x \in \mathbb{R}^5$. Suppose we want to create the two constraint $x_2 + 5 \geq 0$ and $x_3 - 3 \geq 0$. We can do this either by creating two constraints and adding them to an array:
      constraint1 = COSMO.Constraint([0.0 1.0 0.0 0.0 0.0], 5.0, COSMO.Nonnegatives)
   constraint2 = COSMO.Constraint([0.0 0.0 1.0 0.0 0.0], -3.0, COSMO.Nonnegatives)
   constraints = [constraint1; constraint2]
    The second option is to include both in one constraint:
    constraint1 = COSMO.Constraint([0.0 1.0 0.0 0.0 0.0; 0.0 0.0 1.0 0.0 0.0], [5.0; -3.0], COSMO.Nonnegatives)
    Another way to construct the constraint is to used the optional arguments dim, the dimension of x, and indices, the elements of x that appear in the constraint. When specifying these arguments, A and b only refer to the elements of x in indices:
    constraint1 = COSMO.Constraint([1.0 0.0; 0.0 1.0], [5.0; -3.0], COSMO.Nonnegatives, 5, 2:3)
    Consider as a second example the positive semidefinite constraint on a matrix $X \in  \mathbb{S}_+^{3}$. Our decision variable is the vector $x$ obtained by stacking the columns of $X$. We can specify the constraint on $x$ in the following way:
    \[I_9 x + \{0\}_9 \in \mathcal{S}_+^9,\]
    or in Julia:
    constraint1 = COSMO.Constraint(Matrix(1.0I, 9, 9), zeros(9), COSMO.PsdCone)
    Several constraints can be combined in an array:
    constraints = [constraint_1, constraint_2, ..., constraint_N]
    It is usually enough to pass the convex_set as a type. However, some convex sets like Box, PowerCone and DualPowerCone require more information to be created. In that case you have to pass an object to the constructor, e.g.
    l = -ones(2)
 u = ones(2)
-constraint = COSMO.Constraint(Matrix(1.0I, 2, 2), zeros(2), COSMO.Box(l, u))
    or in the case of a power Cone you specify the alpha:
    constraint = COSMO.Constraint(Matrix(1.0I, 3, 3), zeros(3), COSMO.PowerCone(0.6))
    Settings
    The solver settings are stored in a Settings object and can be adjusted by the user. To create a Settings object just call the constructor:
    COSMO.Settings — Type
    COSMO.Settings{T}(; kwargs) where {T <: AbstractFloat}
    Creates a COSMO settings object that is used to pass user settings to the solver.
    ArgumentDescriptionValues (default)
    rhoADMM rho step0.1
    sigmaADMM sigma step1e-6
    alphaRelaxation parameter1.6
    eps_absAbsolute residual tolerance1e-5
    eps_relRelative residual tolerance1e-5
    nearly_ratioResidual tolerance ratio between MOI.NEARLY_FEASIBLE_POINT and $MOI.FEASIBLE_POINT$100
    eps_prim_infPrimal infeasibility tolerance1e-5
    eps_dual_infDual infeasibility tolerance1e-5
    max_iterMaximum number of iterations5000
    verboseVerbose printingfalse
    verbose_timingVerbose timingfalse
    kkt_solverLinear System solverQdldlKKTSolver
    check_terminationCheck termination interval25
    check_infeasibilityCheck infeasibility interval40
    scalingNumber of scaling iterations10
    adaptive_rhoAutomatic adaptation of step size parametertrue
    adaptiverhomax_adaptionsMax number of rho adaptionstypemax(Int64) (deactivated)
    decomposeActivate to decompose chordal psd constraintstrue
    complete_dualActivate to complete the dual variable after decompositionfalse
    merge_strategyChoose a strategy for clique mergingCliqueGraphMerge
    compact_transformationChoose how a decomposed problem is transformedtrue
    time_limitSet solver time limit in s0 (deactivated)
    acceleratorAcceleration schemeAndersonAccelerator{T, Type2{QRDecomp}, RestartedMemory, NoRegularizer}
    accelerator_activationAccelerator activationImmediateActivation
    safeguardAccelerator safeguardingtrue
    safeguard_tolSafeguarding tolerance2.0
    source
    To adjust those values, either pass your preferred option and parameter as a key-value pair to the constructor or edit the corresponding field afterwards. For example if you want to enable verbose printing and increase the solver accuracy, you can type
    settings = COSMO.Settings(verbose = true, eps_abs = 1e-5, eps_rel = 1e-5)
+constraint = COSMO.Constraint(Matrix(1.0I, 2, 2), zeros(2), COSMO.Box(l, u))
    or in the case of a power Cone you specify the alpha:
    constraint = COSMO.Constraint(Matrix(1.0I, 3, 3), zeros(3), COSMO.PowerCone(0.6))
    Settings
    The solver settings are stored in a Settings object and can be adjusted by the user. To create a Settings object just call the constructor:
    COSMO.Settings — Type
    COSMO.Settings{T}(; kwargs) where {T <: AbstractFloat}
    Creates a COSMO settings object that is used to pass user settings to the solver.
    ArgumentDescriptionValues (default)
    rhoADMM rho step0.1
    sigmaADMM sigma step1e-6
    alphaRelaxation parameter1.6
    eps_absAbsolute residual tolerance1e-5
    eps_relRelative residual tolerance1e-5
    nearly_ratioResidual tolerance ratio between MOI.NEARLY_FEASIBLE_POINT and $MOI.FEASIBLE_POINT$100
    eps_prim_infPrimal infeasibility tolerance1e-5
    eps_dual_infDual infeasibility tolerance1e-5
    max_iterMaximum number of iterations5000
    verboseVerbose printingfalse
    verbose_timingVerbose timingfalse
    kkt_solverLinear System solverQdldlKKTSolver
    check_terminationCheck termination interval25
    check_infeasibilityCheck infeasibility interval40
    scalingNumber of scaling iterations10
    adaptive_rhoAutomatic adaptation of step size parametertrue
    adaptiverhomax_adaptionsMax number of rho adaptionstypemax(Int64) (deactivated)
    decomposeActivate to decompose chordal psd constraintstrue
    complete_dualActivate to complete the dual variable after decompositionfalse
    merge_strategyChoose a strategy for clique mergingCliqueGraphMerge
    compact_transformationChoose how a decomposed problem is transformedtrue
    time_limitSet solver time limit in s0 (deactivated)
    acceleratorAcceleration schemeAndersonAccelerator{T, Type2{QRDecomp}, RestartedMemory, NoRegularizer}
    accelerator_activationAccelerator activationImmediateActivation
    safeguardAccelerator safeguardingtrue
    safeguard_tolSafeguarding tolerance2.0
    source
    To adjust those values, either pass your preferred option and parameter as a key-value pair to the constructor or edit the corresponding field afterwards. For example if you want to enable verbose printing and increase the solver accuracy, you can type
    settings = COSMO.Settings(verbose = true, eps_abs = 1e-5, eps_rel = 1e-5)
 # the following is equivalent
 settings = COSMO.Settings()
 settings.verbose = true
@@ -16,4 +16,4 @@
 y_0 = [1.0; 2.0]
 COSMO.assemble!(model, P, q, constraints, x0 = x_0, y0 = y_0)
    Another option is to use
    COSMO.assemble!(model, P, q, constraints)
 warm_start_primal!(model, x_0)
-warm_start_dual!(model, y_0)
    Solving
    After the model has been assembled, we can solve the problem by typing
    results = COSMO.optimize!(model)
    Once the solver algorithm terminates, it will return a Results object that gives information about the status of the solver. If successful, it contains the optimal objective value and optimal primal and dual variables. For more information see the following section.
    Results
    After attempting to solve the problem, COSMO will return a result object with the following fields:
    COSMO.Result — Type
    Result{T <: AbstractFloat}
    Object returned by the COSMO solver after calling optimize!(model). It has the following fields:
    FieldnameTypeDescription
    xVector{T}Primal variable
    yVector{T}Dual variable
    sVector{T}(Primal) set variable
    obj_valTObjective value
    iterIntTotal number of ADMM iterations (incl. safeguarding_iter)
    safeguarding_iterIntNumber of iterations due to safeguarding of accelerator
    statusSymbolSolution status
    infoCOSMO.ResultInfoStruct with more information
    timesCOSMO.ResultTimesStruct with several measured times
    source
    COSMO.ResultInfo — Type
    ResultInfo{T <: AbstractFloat}
    Object that contains further information about the primal residual, the dual residuals and the rho updates.
    source
    Status Codes
    COSMO will return one of the following statuses:
    Status CodeDescription
    :SolvedAn optimal solution was found
    :UnsolvedDefault value
    :Max_iter_reachedSolver reached iteration limit (set with Settings.max_iter)
    :Time_limit_reachedSolver reached time limit (set with Settings.time_limit)
    :Primal_infeasibleProblem is primal infeasible
    :Dual_infeasibleProblem is dual infeasible
    Timings
    If settings.verbose_timing is set to true, COSMO will report the following times in result.times:
    COSMO.ResultTimes — Type
    ResultTimes
    Part of the Result object returned by the solver. ResultTimes contains timing results for certain parts of the algorithm:
    Time NameDescription
    solver_timeTotal time used to solve the problem
    setup_timeSetup time = graph_time + init_factor_time + scaling_time
    scaling_timeTime to scale the problem data
    graph_timeTime used to perform chordal decomposition
    init_factor_timeTime used for initial factorisation of the system of linear equations
    factor_update_timeSum of times used to refactor the system of linear equations due to rho
    iter_timeTime spent in iteration loop
    proj_timeTime spent in projection functions
    post_timeTime used for post processing
    update_timeTime spent in the update! function of the accelerator
    accelerate_timeTime spent in the accelerate! function of the accelerator
    By default COSMO only measures solver_time, setup_time and proj_time. To measure the other times set verbose_timing = true.
    source
    It holds: solver_time = setup_time+ iter_time + factor_update_time + post_time,
    setup_time = graph_time+ init_factor_time + scaling_time,
    proj_time is a subset of iter_time.
    Updating the model
    In some cases we want to solve a large number of similar models. COSMO allows you to update the model problem data vectors q and b after the first call of optimize!(). After changing the problem data, COSMO can reuse the factorisation step of the KKT matrix from the previous problem which can save a lot of time in the case of LPs and QPs.
    COSMO.update! — Function
    update!(model, q = nothing, b = nothing)
    Updates the model data for b or q. This can be done without refactoring the KKT matrix. The vectors will be appropriatly scaled.
    source
    
+warm_start_dual!(model, y_0)

    Solving

    After the model has been assembled, we can solve the problem by typing

    results = COSMO.optimize!(model)

    Once the solver algorithm terminates, it will return a Results object that gives information about the status of the solver. If successful, it contains the optimal objective value and optimal primal and dual variables. For more information see the following section.

    Results

    After attempting to solve the problem, COSMO will return a result object with the following fields:

    COSMO.Result — Type
    Result{T <: AbstractFloat}

    Object returned by the COSMO solver after calling optimize!(model). It has the following fields:

    FieldnameTypeDescription
    xVector{T}Primal variable
    yVector{T}Dual variable
    sVector{T}(Primal) set variable
    obj_valTObjective value
    iterIntTotal number of ADMM iterations (incl. safeguarding_iter)
    safeguarding_iterIntNumber of iterations due to safeguarding of accelerator
    statusSymbolSolution status
    infoCOSMO.ResultInfoStruct with more information
    timesCOSMO.ResultTimesStruct with several measured times
    source
    COSMO.ResultInfo — Type
    ResultInfo{T <: AbstractFloat}

    Object that contains further information about the primal residual, the dual residuals and the rho updates.

    source

    Status Codes

    COSMO will return one of the following statuses:

    Status CodeDescription
    :SolvedAn optimal solution was found
    :UnsolvedDefault value
    :Max_iter_reachedSolver reached iteration limit (set with Settings.max_iter)
    :Time_limit_reachedSolver reached time limit (set with Settings.time_limit)
    :Primal_infeasibleProblem is primal infeasible
    :Dual_infeasibleProblem is dual infeasible

    Timings

    If settings.verbose_timing is set to true, COSMO will report the following times in result.times:

    COSMO.ResultTimes — Type
    ResultTimes

    Part of the Result object returned by the solver. ResultTimes contains timing results for certain parts of the algorithm:

    Time NameDescription
    solver_timeTotal time used to solve the problem
    setup_timeSetup time = graph_time + init_factor_time + scaling_time
    scaling_timeTime to scale the problem data
    graph_timeTime used to perform chordal decomposition
    init_factor_timeTime used for initial factorisation of the system of linear equations
    factor_update_timeSum of times used to refactor the system of linear equations due to rho
    iter_timeTime spent in iteration loop
    proj_timeTime spent in projection functions
    post_timeTime used for post processing
    update_timeTime spent in the update! function of the accelerator
    accelerate_timeTime spent in the accelerate! function of the accelerator

    By default COSMO only measures solver_time, setup_time and proj_time. To measure the other times set verbose_timing = true.

    source

    It holds: solver_time = setup_time+ iter_time + factor_update_time + post_time,

    setup_time = graph_time+ init_factor_time + scaling_time,

    proj_time is a subset of iter_time.

    Updating the model

    In some cases we want to solve a large number of similar models. COSMO allows you to update the model problem data vectors q and b after the first call of optimize!(). After changing the problem data, COSMO can reuse the factorisation step of the KKT matrix from the previous problem which can save a lot of time in the case of LPs and QPs.

    COSMO.update! — Function
    update!(model, q = nothing, b = nothing)

    Updates the model data for b or q. This can be done without refactoring the KKT matrix. The vectors will be appropriatly scaled.

    source
    diff --git a/dev/index.html b/dev/index.html index ef4983df..55086bd0 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,10 +1,10 @@ -Home · COSMO.jl
    Edit on GitHub

    COSMO.jl is a Julia implementation of the Conic Operator Splitting Method. The underlying ADMM-algorithm is well-suited for large convex conic problems. COSMO solves the following problem:

    \[\begin{array}{ll} \text{minimize} & \textstyle{\frac{1}{2}}x^\top Px + q^\top x\\ \text{subject to} & Ax + s = b \\ & s \in \mathcal{K}, -\end{array}\]

    with decision variables $x \in \mathbb{R}^n$, $s \in \mathbb{R}^m$ and data matrices $P=P^\top \succeq 0$, $q \in \mathbb{R}^n$, $A \in \mathbb{R}^{m \times n}$, and $b \in \mathbb{R}^m$. The convex set $\mathcal{K}$ is a composition of convex sets and cones.

    Features

    • Versatile: COSMO solves linear programs, quadratic programs, second-order cone programs, semidefinite programs and problems involving exponential and power cones
    • Quad SDPs: Positive semidefinite programs with quadratic objective functions are natively supported
    • Safeguarded acceleration: robust and faster convergence to higher precision using COSMOAccelerators
    • Infeasibility detection: Infeasible problems are detected without a homogeneous self-dual embedding of the problem
    • JuMP / Convex.jl support: We provide an interface to MathOptInterface (MOI), which allows you to describe your problem in JuMP and Convex.jl.
    • Chordal decomposition: COSMO tries to decompose large structured PSD constraints into multiple smaller PSD constraints using chordal decomposition techniques. This often results in a significant speedup compared to solving the original problem.
    • Smart clique merging: After an initial decomposition of a structured SDP, COSMO recombines overlapping cliques/blocks to speed up the algorithm.
    • Warm starting: COSMO supports warm starting of the decision variables
    • Arbitrary precision types: You can solve problems with any floating point precision.
    • Open Source: Our code is available on GitHub and distributed under the Apache 2.0 Licence

    !!! Audience: User

    COSMO has both a native interface and can also be interfaced through JuMP. If you are interested in using COSMO to solve your optimization problem, we recommend using JuMP to define your problem. Install this package as described below, take a look at our [Examples](@ref) and the [JuMP documentation](https://jump.dev/JuMP.jl/stable/) for inspiration.

    !!! Audience: Researcher / Developer

    If you are interested in COSMO's algorithm, chordal decomposition or acceleration methods take a look at [Method](@ref), [Chordal Decomposition](@ref), and [Acceleration](@ref). If you want low-level control over the algorithm, it can make sense to start with COSMO's native interface.

    Installation

    COSMO can be installed using the Julia package manager for Julia v1.6 and higher. Inside the Julia REPL, type ] to enter the Pkg REPL mode then run

    pkg> add COSMO

    If you want to install the latest version from master run

    pkg> add COSMO#master

    Quick Example

    Consider the following 2x2 semidefinite program with decision variable X:

    \[\begin{array}{ll} \text{minimize} & \text{tr}(CX)\\ +

    GitHub

    COSMO.jl is a Julia implementation of the Conic Operator Splitting Method. The underlying ADMM-algorithm is well-suited for large convex conic problems. COSMO solves the following problem:

    \[\begin{array}{ll} \text{minimize} & \textstyle{\frac{1}{2}}x^\top Px + q^\top x\\ \text{subject to} & Ax + s = b \\ & s \in \mathcal{K}, +\end{array}\]

    with decision variables $x \in \mathbb{R}^n$, $s \in \mathbb{R}^m$ and data matrices $P=P^\top \succeq 0$, $q \in \mathbb{R}^n$, $A \in \mathbb{R}^{m \times n}$, and $b \in \mathbb{R}^m$. The convex set $\mathcal{K}$ is a composition of convex sets and cones.

    Features

    • Versatile: COSMO solves linear programs, quadratic programs, second-order cone programs, semidefinite programs and problems involving exponential and power cones
    • Quad SDPs: Positive semidefinite programs with quadratic objective functions are natively supported
    • Safeguarded acceleration: robust and faster convergence to higher precision using COSMOAccelerators
    • Infeasibility detection: Infeasible problems are detected without a homogeneous self-dual embedding of the problem
    • JuMP / Convex.jl support: We provide an interface to MathOptInterface (MOI), which allows you to describe your problem in JuMP and Convex.jl.
    • Chordal decomposition: COSMO tries to decompose large structured PSD constraints into multiple smaller PSD constraints using chordal decomposition techniques. This often results in a significant speedup compared to solving the original problem.
    • Smart clique merging: After an initial decomposition of a structured SDP, COSMO recombines overlapping cliques/blocks to speed up the algorithm.
    • Warm starting: COSMO supports warm starting of the decision variables
    • Arbitrary precision types: You can solve problems with any floating point precision.
    • Open Source: Our code is available on GitHub and distributed under the Apache 2.0 Licence
    Tip

    For users: COSMO has both a native interface and can also be interfaced through JuMP. If you are interested in using COSMO to solve your optimization problem, we recommend using JuMP to define your problem. Install this package as described below, take a look at our Examples and the JuMP documentation for inspiration.

    Tip

    For researchers and developers: If you are interested in COSMO's algorithm, chordal decomposition or acceleration methods take a look at Method, Chordal Decomposition, and Acceleration. If you want low-level control over the algorithm, it can make sense to start with COSMO's native interface. Moreover, take a look at Contributing if you want to propose changes.

    Installation

    COSMO can be installed using the Julia package manager for Julia v1.6 and higher. Inside the Julia REPL, type ] to enter the Pkg REPL mode then run

    pkg> add COSMO

    If you want to install the latest version from master run

    pkg> add COSMO#master

    Quick Example

    Consider the following 2x2 semidefinite program with decision variable X:

    \[\begin{array}{ll} \text{minimize} & \text{tr}(CX)\\ \text{subject to} & \text{tr}(A X) = b \\ & X \succeq 0, \end{array}\]

    with problem data A, b and C:

    \[A = \begin{bmatrix} 1 & 5 \\ 5 & 2\end{bmatrix}, @@ -46,4 +46,4 @@ status = JuMP.termination_status(m) X_sol = JuMP.value.(X) -obj_value = JuMP.objective_value(m)

    Credits

    The following people are involved in the development of COSMO:

    *all contributors are affiliated with the University of Oxford.

    If this project is useful for your work please consider

    Licence

    COSMO.jl is licensed under the Apache License 2.0. For more details click here.

    +obj_value = JuMP.objective_value(m)

    Credits

    The following people are involved in the development of COSMO:

    *all contributors are affiliated with the University of Oxford.

    If this project is useful for your work please consider

    Licence

    COSMO.jl is licensed under the Apache License 2.0. For more details click here.

    diff --git a/dev/jump/index.html b/dev/jump/index.html index 009680dc..f20319bf 100644 --- a/dev/jump/index.html +++ b/dev/jump/index.html @@ -1,9 +1,9 @@ -JuMP Interface · COSMO.jl
    Edit on GitHub

    JuMP Interface

    Our JuMP interface allows you to describe and modify your optimisation problem with JuMP and use COSMO as the backend solver. The interface is defined in /src/MOIWrapper.jl.

    Note

    We assume here that the latest JuMP release (~0.21.0) is used.

    Use COSMO

    To specify COSMO as the solver for your JuMP model, load the solver module with using COSMO and then pass a COSMO.Optimizer when initialising the JuMP model:

    m = JuMP.Model(COSMO.Optimizer);

    Specify Solver Settings

    Solver-specific settings can be passed using the optimizer_with_attributes() function. For example, if you want to adjust the maximum number of iterations and turn on verbose printing use

    m = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, "max_iter" => 5000, "verbose" => true));

    Note that the attributes are passed as key-value pairs and the keys are strings. This is slightly different to using the native COSMO interface. Equivalently, one can also use:

    m = JuMP.Model(COSMO.Optimizer);
+
    GitHub
    JuMP Interface
    Our JuMP interface allows you to describe and modify your optimisation problem with JuMP and use COSMO as the backend solver. The interface is defined in /src/MOIWrapper.jl.
    Note
    We assume here that the latest JuMP release (~0.21.0) is used.
    Use COSMO
    To specify COSMO as the solver for your JuMP model, load the solver module with using COSMO and then pass a COSMO.Optimizer when initialising the JuMP model:
    m = JuMP.Model(COSMO.Optimizer);
    Specify Solver Settings
    Solver-specific settings can be passed using the optimizer_with_attributes() function. For example, if you want to adjust the maximum number of iterations and turn on verbose printing use
    m = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, "max_iter" => 5000, "verbose" => true));
    Note that the attributes are passed as key-value pairs and the keys are strings. This is slightly different to using the native COSMO interface. Equivalently, one can also use:
    m = JuMP.Model(COSMO.Optimizer);
 set_optimizer_attribute(m, "max_iter", 5000)
 set_optimizer_attribute(m, "verbose", true)
    The full list of available settings can be found in the Settings section. All of them should be compatible with JuMP.
    Results
    After solving the problem the result can be obtained using the standard JuMP commands. To see if the optimisation was successful use
    JuMP.termination_status(m)
 JuMP.primal_status(m)
    If a solution is available, the optimal objective value can be queried using
    JuMP.objective_value(m)
    and the value of a decision variable x can be obtained with
    JuMP.value.(x)
    To query the number of iterations of the algorithm use:
    iter = MOI.get(m, COSMO.ADMMIterations())
    Feasibility of solution
    In case JuMP.termination_status(m) is MOI.ITERATION_LIMIT, JuMP.primal_status(m) (resp. JuMP.dual_status(m)) should be checked to determine the feasibility status of the primal (resp. dual) solutions. A primal (resp. dual) solution is considered feasible (resported as MOI.FEASIBLE_POINT) if
    \[r < \epsilon_{\text{abs}} + s\epsilon_{\text{rel}}\]
    where
    • $r$ is the primal (resp. dual) residual r_prim $= \lVert Ax + s - b \rVert_\infty$
    (resp. r_dual $= \lVert P x + q - A^\top * μ \rVert_\infty$),
    • $s$ is the primal (resp. dual) scaling of this residual
    max_norm_prim $= \max(\lVert Ax \rVert_\infty, \lVert s \rVert_\infty, \lVert b \rVert_\infty)$ (resp. max_norm_dual $= \max(\lVert P x \rVert_\infty, \lVert q \rVert_\infty, \lVert A^\top * μ \rVert_\infty)$).
    • $\epsilon_{\text{abs}}$ is the optimizer attribute eps_abs and
    • $\epsilon_{\text{rel}}$ is the optimizer attribute eps_rel.
    A primal (resp. dual solution) is considered nearly feasible (resported as MOI.NEARLY_FEASIBLE_POINT) if it is not feasible but the same inequality is satisfied where the right-hand side is multiplied by the optimizer attribute nearly_ratio.
    Otherwise, the solution is considered infeasible (resported as MOI.INFEASIBLE_POINT). Note that this does not mean that the problem is infeasible, it only means that the current primal (resp. dual) solution is infeasible.
    Note that, in case JuMP.termination_status(m) is MOI.ITERATION_LIMIT, the feasibility of the solution is checked with the current optimizer attributes, eps_abs, eps_rel and nearly_ratio. This means that if they are changed after optimize!, the status may change! This can be used to check whether the solution is feasible up to a relaxed tolerance. For instance, the following could happen:
    optimize!(m)
@@ -14,4 +14,4 @@
 # This means that `1 <= r_prim / (eps_abs * max_norm_prim * eps_rel) < nearly_ratio`
    The values of r_prim, max_norm_prim, r_dual and max_norm_dual can also be accessed as the fields of the res_info struct that can be obtained as follows
    res_info = MOI.get(m, COSMO.RawResult()).info
    Then, the feasibility can either be checked manually as in
    res_info.r_prim < eps_abs + res_info.max_norm_prim * eps_rel
    or using one of the following:
    COSMO.is_primal_feasible(res_info, eps_abs, eps_rel)
 COSMO.is_primal_nearly_feasible(res_info, eps_abs, eps_rel)
 COSMO.is_dual_feasible(res_info, eps_abs, eps_rel)
-COSMO.is_dual_nearly_feasible(res_info, eps_abs, eps_rel)
    For more information on how to use JuMP check the JuMP documentation.
    
+COSMO.is_dual_nearly_feasible(res_info, eps_abs, eps_rel)

    For more information on how to use JuMP check the JuMP documentation.

    diff --git a/dev/lin_solver/index.html b/dev/lin_solver/index.html index 1704a64c..7298b77a 100644 --- a/dev/lin_solver/index.html +++ b/dev/lin_solver/index.html @@ -1,9 +1,9 @@ -Linear System Solver · COSMO.jl
    Edit on GitHub

    Linear System Solver

    One major step of COSMO's ADMM algorithm is solving a linear system of equations at each iteration. Fortunately, the left-hand matrix is only dependent on the problem data and therefore only needs to be factored once. Depending on the problem class this factorisation can be the computationally most expensive step of the algorithm (LPs, QPs). See the Method section for a more detailed description of the linear system.

    COSMO allows you to specify the linear system solver that performs the factorisation and back-substitution. We also support indirect system solver which are useful for very large problems where a factorisation becomes inefficient. The table below shows the currently supported linear system solver:

    TypeSolverDescription
    CholmodKKTSolverCholmodJulia's default linear system solver (from SuiteSparse)
    QdldlKKTSolverQDLDLFor more information QDLDL.jl
    PardisoDirectKKTSolverPardiso (direct)Pardiso 6.0 direct solver
    PardisoIndirectKKTSolverPardiso (indirect)Pardiso 6.0 indirect solver
    MKLPardisoKKTSolverIntel MKL PardisoPardiso optimised for Intel platforms
    CGIndirectKKTSolverIterativeSolvers.jlConjugate Gradients on the reduced KKT linear system.
    MINRESIndirectKKTSolverIterativeSolvers.jlMINRES on the (full) KKT linear system.
    Note

    To use the Pardiso and Intel MKL Pardiso solver, you have to install the respective libraries and the corresponding Julia wrapper. For more information about installing these, visit the Pardiso.jl repository page. Likewise in order to use Indirect(Reduced)KKTSolver you have to install IterativeSolvers.jl (v0.9+) and LinearMaps.jl. We are using the Requires package for lazy loading of code related to Pardiso and IterativeSolvers. This means in order to use Pardiso / IterativeSolvers, you'll have to load these packages alongside COSMO, i.e. using Pardiso and using IterativeSolvers, LinearMaps.

    COSMO uses the Cholmod linear system solver by default. You can specify a different solver in the settings by using the kkt_solver keyword and the respective type:

    settings = COSMO.Settings(kkt_solver = CholmodKKTSolver)
+
    GitHub
    Linear System Solver
    One major step of COSMO's ADMM algorithm is solving a linear system of equations at each iteration. Fortunately, the left-hand matrix is only dependent on the problem data and therefore only needs to be factored once. Depending on the problem class this factorisation can be the computationally most expensive step of the algorithm (LPs, QPs). See the Method section for a more detailed description of the linear system.
    COSMO allows you to specify the linear system solver that performs the factorisation and back-substitution. We also support indirect system solver which are useful for very large problems where a factorisation becomes inefficient. The table below shows the currently supported linear system solver:
    TypeSolverDescription
    CholmodKKTSolverCholmodJulia's default linear system solver (from SuiteSparse)
    QdldlKKTSolverQDLDLFor more information QDLDL.jl
    PardisoDirectKKTSolverPardiso (direct)Pardiso 6.0 direct solver
    PardisoIndirectKKTSolverPardiso (indirect)Pardiso 6.0 indirect solver
    MKLPardisoKKTSolverIntel MKL PardisoPardiso optimised for Intel platforms
    CGIndirectKKTSolverIterativeSolvers.jlConjugate Gradients on the reduced KKT linear system.
    MINRESIndirectKKTSolverIterativeSolvers.jlMINRES on the (full) KKT linear system.
    Note
    To use the Pardiso and Intel MKL Pardiso solver, you have to install the respective libraries and the corresponding Julia wrapper. For more information about installing these, visit the Pardiso.jl repository page. Likewise in order to use Indirect(Reduced)KKTSolver you have to install IterativeSolvers.jl (v0.9+) and LinearMaps.jl. We are using the Requires package for lazy loading of code related to Pardiso and IterativeSolvers. This means in order to use Pardiso / IterativeSolvers, you'll have to load these packages alongside COSMO, i.e. using Pardiso and using IterativeSolvers, LinearMaps.
    COSMO uses the Cholmod linear system solver by default. You can specify a different solver in the settings by using the kkt_solver keyword and the respective type:
    settings = COSMO.Settings(kkt_solver = CholmodKKTSolver)
 
    COSMO also allows you to pass in solver-specific options with the with_options(solver_type, args...; kwargs...) syntax. For example, if you want to use Pardiso with verbose printing use the following syntax:
    settings = COSMO.Settings(kkt_solver = with_options(PardisoDirectKKTSolver, msg_level_on = true))
    Likewise, CGIndirectKKTSolver and MINRESIndirectKKTSolver are also parameterizable with with_options(solver_type, args...; kwargs...) and accept the following arguments:
    ArgumentDescriptionValues (default)
    tol_constant::T and tol_exponent::TParameter that defines the solution tolerance for the iterative solvers across iterations. In particular, the solution tolerance at every iteration is defined as $\text{tol\_constant} /  \text{iteration}^{\text{tol\_exponent}}$1.0, 1.5
    This also works if you want to use this configuration with JuMP:
    model = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, "kkt_solver" => with_options(PardisoDirectKKTSolver, msg_level_on = true));
 
    Or alternatively:
    model = JuMP.Model(COSMO.Optimizer);
-set_optimizer_attribute(model, "kkt_solver", with_options(PardisoDirectKKTSolver, msg_level_on = true));
    
+set_optimizer_attribute(model, "kkt_solver", with_options(PardisoDirectKKTSolver, msg_level_on = true));
    diff --git a/dev/literate/build/arbitrary_precision/index.html b/dev/literate/build/arbitrary_precision/index.html index 03bfd022..e9255614 100644 --- a/dev/literate/build/arbitrary_precision/index.html +++ b/dev/literate/build/arbitrary_precision/index.html @@ -1,9 +1,9 @@ -Arbitrary Precision · COSMO.jl
    Edit on GitHub

    Arbitrary Precision

    COSMO allows you to solve problems with arbitrary floating-point precision, e.g. by using BigFloat problem data. To do this, the desired floating point type has to be consistent across the model COSMO.Model{<: AbstractFloat}, the input data and the (optional) settings object COSMO.Settings{<: AbstractFloat}. As an example, assume we want to solve the following quadratic program:

    \[\begin{array}{ll} \text{minimize} & 1/2 x^\top P x + q^\top x \\ +

    GitHub

    Arbitrary Precision

    COSMO allows you to solve problems with arbitrary floating-point precision, e.g. by using BigFloat problem data. To do this, the desired floating point type has to be consistent across the model COSMO.Model{<: AbstractFloat}, the input data and the (optional) settings object COSMO.Settings{<: AbstractFloat}. As an example, assume we want to solve the following quadratic program:

    \[\begin{array}{ll} \text{minimize} & 1/2 x^\top P x + q^\top x \\ \text{subject to} & l \leq A x \leq u \end{array}\]

    where $P = \begin{bmatrix} 4 & 1 \\ 1 & 2\end{bmatrix}$, $q = [1, 1]^\top$, $A = \begin{bmatrix} 1 & 1 \\ 1 & 0 \\ 0 & 1\end{bmatrix}$ and $l= [1,0, 0]^\top$, $u=[1, 0.7, 0.7]^\top$. We start by creating the model with the desired precision:

    using COSMO, LinearAlgebra, SparseArrays
 model = COSMO.Model{BigFloat}()
    A COSMO Model with Float precision: BigFloat
@@ -39,7 +39,7 @@
 Acc:      Anderson Type2{QRDecomp},
           Memory size = 5, RestartedMemory,
           Safeguarded: true, tol: 2.0
-Setup Time: 1135.32ms
+Setup Time: 1118.09ms
 
 Iter:	Objective:	Primal Res:	Dual Res:	Rho:
 1	-7.8808e-03	1.0079e+00	2.0033e+02	1.0000e-01
@@ -50,5 +50,5 @@
 Status: Solved
 Iterations: 25
 Optimal objective: 1.88
-Runtime: 3.247s (3247.3ms)

    Moreover, notice that when no type parameter is specified, all objects default to Float64:

    model = COSMO.Model()
    A COSMO Model with Float precision: Float64
-

    Two limitations to arbitrary precision:

    • Since we call the LAPACK function syevr for eigenvalue decompositions, we currently only support solving problems with PSD constraints in Float32 and Float64.
    • We suggest to use the pure Julia QDLDL linear system solver (kkt_solver = QdldlKKTSolver) when working with arbitrary precision types as some of the other available solvers don't support all available precisions.
    Note

    If you want to use COSMO directly with MathOptInterface, you can use: COSMO.Optimizer{<: AbstractFloat} as your optimizer. Again, the problem data precision of your MathOptInterface-model has to agree with the optimizer's precision.

    This page was generated using Literate.jl.

    +Runtime: 3.248s (3247.58ms)

    Moreover, notice that when no type parameter is specified, all objects default to Float64:

    model = COSMO.Model()
    A COSMO Model with Float precision: Float64
+

    Two limitations to arbitrary precision:

    • Since we call the LAPACK function syevr for eigenvalue decompositions, we currently only support solving problems with PSD constraints in Float32 and Float64.
    • We suggest to use the pure Julia QDLDL linear system solver (kkt_solver = QdldlKKTSolver) when working with arbitrary precision types as some of the other available solvers don't support all available precisions.
    Note

    If you want to use COSMO directly with MathOptInterface, you can use: COSMO.Optimizer{<: AbstractFloat} as your optimizer. Again, the problem data precision of your MathOptInterface-model has to agree with the optimizer's precision.

    This page was generated using Literate.jl.

    diff --git a/dev/literate/build/custom_cone/index.html b/dev/literate/build/custom_cone/index.html index 133f16a7..c0c90dd2 100644 --- a/dev/literate/build/custom_cone/index.html +++ b/dev/literate/build/custom_cone/index.html @@ -1,9 +1,9 @@ -Custom Cone Constraint · COSMO.jl
    Edit on GitHub

    Custom Cone Constraint

    This example demonstrate how the user can extend COSMO with his own custom convex cones and use them in constraints. To make things easy, we will implement the simple cone of Nonpositives, i.e. $\mathbb{R}_{-}^n = \{x \in \mathbb{R}^n \mid x_i \leq 0\}$.

    Cone definition and projection function

    We start by creating a concrete subtype of our cone interface-type: COSMO.AbstractConvexCone{T}, where T is a parameter for the floating-point precision. The only field that is strictly required for our cone definition is dim, which is used by the solver to store information about the cone dimension. However, you can also use the object to store other information, or allocate the workspace necessary for the projection step.

    using COSMO, LinearAlgebra, SparseArrays, Test
+
    GitHub
    Custom Cone Constraint
    This example demonstrate how the user can extend COSMO with his own custom convex cones and use them in constraints. To make things easy, we will implement the simple cone of Nonpositives, i.e. $\mathbb{R}_{-}^n = \{x \in   \mathbb{R}^n \mid x_i \leq 0\}$.
    Cone definition and projection function
    We start by creating a concrete subtype of our cone interface-type: COSMO.AbstractConvexCone{T}, where T is a parameter for the floating-point precision. The only field that is strictly required for our cone definition is dim, which is used by the solver to store information about the cone dimension. However, you can also use the object to store other information, or allocate the workspace necessary for the projection step.
    using COSMO, LinearAlgebra, SparseArrays, Test
 
 # define new cone type
 struct Nonpositives{T} <: COSMO.AbstractConvexCone{T}
@@ -53,7 +53,7 @@
 
 model = COSMO.Model();
 assemble!(model, P, q, [ci]);
-res = COSMO.optimize!(model);
    @test res.status == :Dual_infeasible
    Test Passed
    which is what we would expect.
    Using new cone with JuMP
    The nice composibility of Julia-code allows us to use our new cone definition even when the problem is modelled with JuMP. For this to work, we define a concrete subtype of MOI.AbstractSet, that JuMP can use to build a constraint.
    using MathOptInterface, JuMP
+res = COSMO.optimize!(model);
    @test res.status == :Dual_infeasible
    Test Passed
    which is what we would expect.
    Using new cone with JuMP
    The nice composibility of Julia-code allows us to use our new cone definition even when the problem is modelled with JuMP. For this to work, we define a concrete subtype of MOI.AbstractSet, that JuMP can use to build a constraint.
    using MathOptInterface, JuMP
 import Base.copy
 const MOI = MathOptInterface
 
@@ -93,7 +93,7 @@
 Acc:      Anderson Type2{QRDecomp},
           Memory size = 6, RestartedMemory,
           Safeguarded: true, tol: 2.0
-Setup Time: 0.09ms
+Setup Time: 0.08ms
 
 Iter:	Objective:	Primal Res:	Dual Res:	Rho:
 1	-2.7216e+01	1.7200e+01	6.0000e-01	1.0000e-01
@@ -104,7 +104,7 @@
 Status: Solved
 Iterations: 26 (incl. 1 safeguarding iter)
 Optimal objective: -7
-Runtime: 0.015s (15.0ms)
    x_opt = JuMP.value.(x)
    3-element Vector{Float64}:
+Runtime: 0.015s (15.22ms)
    x_opt = JuMP.value.(x)
    3-element Vector{Float64}:
  2.999990000199988
  2.000000000116415
- 2.0000099998433143
    You can see in the solver output that indeed a set Nonpositives of dim: 2 was used.
    The discussed cone of Nonpositives is of course trivial, but the ability to define new cones and constraints for COSMO can be very powerful to model complex problems.
    References
    [1] Garstka et al. - COSMO: A conic operator splitting method for convex conic problems (2020)
    This page was generated using Literate.jl.
    
+ 2.0000099998433143

    You can see in the solver output that indeed a set Nonpositives of dim: 2 was used.

    The discussed cone of Nonpositives is of course trivial, but the ability to define new cones and constraints for COSMO can be very powerful to model complex problems.

    References

    [1] Garstka et al. - COSMO: A conic operator splitting method for convex conic problems (2020)

    This page was generated using Literate.jl.

    diff --git a/dev/literate/build/portfolio_model_updates/ab8679a2.svg b/dev/literate/build/portfolio_model_updates/ab8679a2.svg new file mode 100644 index 00000000..1c5e216b --- /dev/null +++ b/dev/literate/build/portfolio_model_updates/ab8679a2.svg @@ -0,0 +1,47 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/literate/build/portfolio_model_updates/index.html b/dev/literate/build/portfolio_model_updates/index.html index 9da04215..70bf1e62 100644 --- a/dev/literate/build/portfolio_model_updates/index.html +++ b/dev/literate/build/portfolio_model_updates/index.html @@ -1,9 +1,9 @@ -Model Updates · COSMO.jl
    Edit on GitHub

    Model Updates

    When a pareto-optimal front in portfolio optimisation is computed for different risk-aversion parameters $\gamma$, we are repeatedly solving the same model with only a slight change in the cost function. This allows us to reuse the KKT factorisation from previous solves and recompute the solution very efficiently, see Algorithm. For more information about solving portfolio optimsiation problems with COSMO see Portfolio Optimisation. We are planning to solve the following QP for different values of $\gamma$:

    \[\begin{array}{ll} \text{minimize} & x^\top D x + y^\top y - \gamma^{-1} \mu^\top x \\ +

    GitHub

    Model Updates

    When a pareto-optimal front in portfolio optimisation is computed for different risk-aversion parameters $\gamma$, we are repeatedly solving the same model with only a slight change in the cost function. This allows us to reuse the KKT factorisation from previous solves and recompute the solution very efficiently, see Algorithm. For more information about solving portfolio optimsiation problems with COSMO see Portfolio Optimisation. We are planning to solve the following QP for different values of $\gamma$:

    \[\begin{array}{ll} \text{minimize} & x^\top D x + y^\top y - \gamma^{-1} \mu^\top x \\ \text{subject to} & y = F^\top x \\ & 1^\top x = d + 1^\top x^0 \\ & x \geq 0, @@ -66,51 +66,4 @@ end end solve_repeatedly!(returns, risks, gammas, model, k, n, μ);

    We can now plot the risk-return trade-off curve:

    using Plots
-Plots.plot(risks, returns, xlabel = "Standard deviation (risk)", ylabel = "Expected return", title = "Risk-return trade-off for efficient portfolios", legend = false)
    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

    This page was generated using Literate.jl.

    +Plots.plot(risks, returns, xlabel = "Standard deviation (risk)", ylabel = "Expected return", title = "Risk-return trade-off for efficient portfolios", legend = false)Example block output

    This page was generated using Literate.jl.

    diff --git a/dev/logistic_regression/index.html b/dev/logistic_regression/index.html index 4da4c4a5..869f7c7d 100644 --- a/dev/logistic_regression/index.html +++ b/dev/logistic_regression/index.html @@ -1,6 +1,6 @@ -Logistic Regression · COSMO.jl + diff --git a/dev/method/index.html b/dev/method/index.html index 232b8d59..dc31a428 100644 --- a/dev/method/index.html +++ b/dev/method/index.html @@ -1,9 +1,9 @@ -Method · COSMO.jl
    Edit on GitHub

    Method

    This section describes COSMO's underlying ADMM algorithm and how the user can use the settings to adjust this algorithm. For a more detailed explanation take a look at the associated publication in Citing COSMO.

    COSMO solves problems with quadratic objective function and a number of conic constraints in the following form:

    \[\begin{array}{ll} \text{minimize} & \textstyle{\frac{1}{2}}x^\top Px + q^\top x\\ \text{subject to} & Ax + s = b \\ & s \in \mathcal{K}, \end{array}\]

    with primal decision variable $x \in \mathbb{R}^n$, primal slack variable $s \in \mathbb{R}^m$. The objective function is defined by positive semidefinite matrix $P=P^\top \succeq 0$ and vector $q \in \mathbb{R}^n$. The constraints are defined by matrix $A \in \mathbb{R}^{m \times n}$, vector $b \in \mathbb{R}^m$ and a non-empty, closed convex set $\mathcal{K}$. The convex set itself can be a Cartesian product of convex sets in the form:

    \[ \mathcal{K} = \mathcal{K}_1^{m_1} \times \mathcal{K}_2^{m_2} \times \cdots \times \mathcal{K}_N^{m_N}.\]

    Accordingly, by an appropriate choice of convex sets one can represent any LP, QP, SOCP or SDP.

    Dual problem

    The dual problem of the optimisation problem above is given by:

    \[\begin{array}{ll} +

    GitHub

    Method

    This section describes COSMO's underlying ADMM algorithm and how the user can use the settings to adjust this algorithm. For a more detailed explanation take a look at the associated publication in Citing COSMO.

    COSMO solves problems with quadratic objective function and a number of conic constraints in the following form:

    \[\begin{array}{ll} \text{minimize} & \textstyle{\frac{1}{2}}x^\top Px + q^\top x\\ \text{subject to} & Ax + s = b \\ & s \in \mathcal{K}, \end{array}\]

    with primal decision variable $x \in \mathbb{R}^n$, primal slack variable $s \in \mathbb{R}^m$. The objective function is defined by positive semidefinite matrix $P=P^\top \succeq 0$ and vector $q \in \mathbb{R}^n$. The constraints are defined by matrix $A \in \mathbb{R}^{m \times n}$, vector $b \in \mathbb{R}^m$ and a non-empty, closed convex set $\mathcal{K}$. The convex set itself can be a Cartesian product of convex sets in the form:

    \[ \mathcal{K} = \mathcal{K}_1^{m_1} \times \mathcal{K}_2^{m_2} \times \cdots \times \mathcal{K}_N^{m_N}.\]

    Accordingly, by an appropriate choice of convex sets one can represent any LP, QP, SOCP or SDP.

    Dual problem

    The dual problem of the optimisation problem above is given by:

    \[\begin{array}{ll} \text{maximize} & -\textstyle{\frac{1}{2}}x^\top Px - b^\top y - \text{sup}_{s \in \mathcal{K}}(-y^\top s ) \\ \text{subject to} & Px + A^\top y = -q\\ & y \in (\mathcal{K}^\infty)^* , @@ -33,4 +33,4 @@ \end{aligned}\]

    The absolute and relative tolerances $\epsilon_{\mathrm{abs}}$and $\epsilon_{\mathrm{rel}}$ can be set by the user by specifying eps_abs and eps_rel. Furthermore, the user can adjust the number of iterations after which the convergence criteria are checked (check_termination).

    Infeasibility detection

    COSMO uses conditions based on separating hyperplanes to detect infeasible problems. The conditions for COSMO's problem format have been developed in [1]. Define the convex set $\mathcal{C} = \mathcal{-K} + \{b\}$ then we can use the following infeasibility conditions:

    \[\begin{aligned} \mathcal{P} &= \left\{x \in \mathbb{R}^n \mid Px = 0, \, Ax \in \mathcal{C}^{\infty}, \, \langle q,x \rangle < 0 \right\} \\ \mathcal{D} &= \left\{y \in \mathbb{R}^m \mid A^\top y = 0, \, \sigma_\mathcal{C}(y) < 0 \right\}. -\end{aligned}\]

    The existence of some $y \in \mathcal{D}$ is a certificate that the problem is primal infeasible, while the existence of some $x \in \mathcal{P}$ is a certificate for dual infeasibility. COSMO regularly checks above conditions to detect infeasible problems. If the detection is successful, the solver terminates and returns the status codes :Primal_infeasible or :Dual_infeasible. COSMO checks the conditions every check_infeasibility iterations, which can be adjusted by the user.

    References

    [1] Banjac, G. et al. Infeasibility detection in the alternating direction method of multipliers for convex optimization. Preprint, 2017.

    +\end{aligned}\]

    The existence of some $y \in \mathcal{D}$ is a certificate that the problem is primal infeasible, while the existence of some $x \in \mathcal{P}$ is a certificate for dual infeasibility. COSMO regularly checks above conditions to detect infeasible problems. If the detection is successful, the solver terminates and returns the status codes :Primal_infeasible or :Dual_infeasible. COSMO checks the conditions every check_infeasibility iterations, which can be adjusted by the user.

    References

    [1] Banjac, G. et al. Infeasibility detection in the alternating direction method of multipliers for convex optimization. Preprint, 2017.

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    Edit on GitHub

    Performance Tips

    There are a number of ways to improve the performance of the solver given a particular problem. If you are not satisfied with the performance there are a number of things you have to determine first. Is the solver slow because

    • it's the first time you ran it in the current Julia session or
    • is it because the solver needs a lot of iterations (convergence) or
    • is each iteration or the initial factorisation slow (computational performance)?

    Let's see how each point can be addressed.

    First run

    Whenever a new Julia session is started, the first run will trigger a compilation of all functions based on their arguments used in your script. This will slow the first execution of COSMO down. After that Julia will call the fast compiled functions. To get around this, you can either keep your current Julia session open and discard the first run. Alternatively, if your problem is very large, you could solve a small version of your problem first to trigger the compilation. Another option is to use PackageCompiler to save compiled functions into a sysimage that can be loaded at startup.

    Solver Timings

    It is often instructive to look at the detailed solver timing results for your problem. This can reveal where most of the time is spent. To achieve this, run COSMO with the setting verbose_timing = true. After solving the problem with result = COSMO.optimize!(model) you can look at result.times for a breakdown of the times spent in different parts of the algorithm, see Timings for more details. Especially take a look at the ratio of factorisation time and iteration time. If you use JuMP to solve the problem, you can take a look at the timings with backend(model).optimizer.model.optimizer.results.times.

    Convergence

    It is possible that COSMO converges slowly, i.e. needs a large number of iterations, for your problem given its default parameters.

    Parameters

    You could try changing any of the following parameters:

    • rho: The initial algorithm step parameter has a large influence on the convergence. Try different values between 1e-5 and 10.
    • adaptive_rho = false: You can try to disable the automatic rho adaption and use different rho values.
    • adaptive_rho_interval: This specifies after how many iterations COSMO tries to adapt the rho parameter. You can also set adaptive_rho_interval = 0 which adapts the rho parameter after the time spent iterating passes 40% of the factorisation time. This is currently the default in OSQP and works well with QPs.
    • alpha = 1.0: This disables the over-relaxation that is used in the algorithm. We recommend values between 1.0 - 1.6.
    • scaling = 0: This disables the problem scaling.
    • eps_abs and eps_rel: Check the impact of modifying the stopping accuracies.

    Use warm starting

    The number of iterations can be dramatically decreased by providing a good initial guess for x, s and y. Examples where warm starting is commonly used are model predictive control and portfolio backtests, see Warm starting.

    Computational performance

    If the convergence of the algorithm is not an issue, there are still a number of steps you can take to make COSMO faster.

    Intel MKL BLAS/LAPACK

    We experienced significant performance improvements on Intel CPUs if Julia is compiled with MKL BLAS. This is because Julia's linear algebra function will use Intel MKL BLAS and LAPACK functions that are optimised for Intel hardware. The effect is especially significant for SDPs because most of the time is spent in the LAPACK function syevr. If you are running Julia on Intel hardware, an easy way to compile Julia with MKL is to add and build the MKL package, see MKL.jl. To verify your current BLAS vendor you can use julia> LinearAlgebra.BLAS.vendor().

    Linear system solver

    COSMO uses QDLDL.jl as the default linear system solver. In our experience this seems to be a competitive choice until about 1e5 - 1e6 nonzeros in the constraint matrix. After that it is worth trying one of the indirect system solvers, such as CG or MINRES. Furthermore, we also recommend trying Pardiso (or MKLPardiso) for problems of that dimension. More details can be found here: Linear System Solver.

    Custom cones

    In some cases the computations can be speed-up if certain constraints in the problem allow the implementation of a fast projection function. We allow the user to define their own custom convex cone with a corresponding projection function. The custom cone has to be defined as struct CustomCone{T} <: COSMO.AbstractConvexSet{T}. Furthermore, the user has to define a function that projects an input vector x onto the custom cone, i.e. function COSMO.project!(x::AbstractVector{T}, C::CustomCone{T}) where {T} ... end.

    Multithreading

    COSMO allows the execution of the projection step for multiple constraints in parallel using Julia's multithreading features. This is currently not enabled in the tagged release because of stability issues in earlier Julia versions. To use multithreading checkout the branch with_multi_threading, which we keep in sync with the latest tagged release. This can be installed via Julia's package manager with pkg> add COSMO#with_multi_threading. Afterwards, before starting Julia, set export JULIA_NUM_THREADS=[NUMBER_PHYSICAL_CORES_HERE]. In Julia you can verify the number of threads with julia> Threads.nthreads().

    Notice that the extra overhead for multithreading can slow the solver down if the problem is small. However, we noticed significant performance improvements if the problem contained multiple positive semidefinite constraints or when one large constraint was decomposed. In that case it also helps to restrict the number of BLAS threads per Julia thread with julia> BLAS.set_num_threads(1) to prevent oversubscription of the available cores.

    Multithreading can also be used in the factorisation step if the Pardiso or MKLPardiso solver are selected. This is only advisable for constraint matrices with more than 1e5 nonzeros.

    Chordal decomposition and Clique merging

    When solving large structured and sparse SDPs significant performance improvements are achievable if the problem is passed to COSMO in the right way. This means the solver has to be able to infer the structure of the positive semidefinite variable from the constraint. See the section on Chordal Decomposition for more details. In some cases the primal SDP doesn't allow decomposition but the dual SDP does, consider the Maximum Cut Problem and the Relaxed Two-Way Partitioning Problem for examples.

    If the problem is decomposable it is also worth experimenting with different clique merging strategies to see how they impact the performance. More details can be found here: Clique merging.

    +
    GitHub

    Performance Tips

    There are a number of ways to improve the performance of the solver given a particular problem. If you are not satisfied with the performance there are a number of things you have to determine first. Is the solver slow because

    • it's the first time you ran it in the current Julia session or
    • is it because the solver needs a lot of iterations (convergence) or
    • is each iteration or the initial factorisation slow (computational performance)?

    Let's see how each point can be addressed.

    First run

    Whenever a new Julia session is started, the first run will trigger a compilation of all functions based on their arguments used in your script. This will slow the first execution of COSMO down. After that Julia will call the fast compiled functions. To get around this, you can either keep your current Julia session open and discard the first run. Alternatively, if your problem is very large, you could solve a small version of your problem first to trigger the compilation. Another option is to use PackageCompiler to save compiled functions into a sysimage that can be loaded at startup.

    Solver Timings

    It is often instructive to look at the detailed solver timing results for your problem. This can reveal where most of the time is spent. To achieve this, run COSMO with the setting verbose_timing = true. After solving the problem with result = COSMO.optimize!(model) you can look at result.times for a breakdown of the times spent in different parts of the algorithm, see Timings for more details. Especially take a look at the ratio of factorisation time and iteration time. If you use JuMP to solve the problem, you can take a look at the timings with backend(model).optimizer.model.optimizer.results.times.

    Convergence

    It is possible that COSMO converges slowly, i.e. needs a large number of iterations, for your problem given its default parameters.

    Parameters

    You could try changing any of the following parameters:

    • rho: The initial algorithm step parameter has a large influence on the convergence. Try different values between 1e-5 and 10.
    • adaptive_rho = false: You can try to disable the automatic rho adaption and use different rho values.
    • adaptive_rho_interval: This specifies after how many iterations COSMO tries to adapt the rho parameter. You can also set adaptive_rho_interval = 0 which adapts the rho parameter after the time spent iterating passes 40% of the factorisation time. This is currently the default in OSQP and works well with QPs.
    • alpha = 1.0: This disables the over-relaxation that is used in the algorithm. We recommend values between 1.0 - 1.6.
    • scaling = 0: This disables the problem scaling.
    • eps_abs and eps_rel: Check the impact of modifying the stopping accuracies.

    Use warm starting

    The number of iterations can be dramatically decreased by providing a good initial guess for x, s and y. Examples where warm starting is commonly used are model predictive control and portfolio backtests, see Warm starting.

    Computational performance

    If the convergence of the algorithm is not an issue, there are still a number of steps you can take to make COSMO faster.

    Intel MKL BLAS/LAPACK

    We experienced significant performance improvements on Intel CPUs if Julia is compiled with MKL BLAS. This is because Julia's linear algebra function will use Intel MKL BLAS and LAPACK functions that are optimised for Intel hardware. The effect is especially significant for SDPs because most of the time is spent in the LAPACK function syevr. If you are running Julia on Intel hardware, an easy way to compile Julia with MKL is to add and build the MKL package, see MKL.jl. To verify your current BLAS vendor you can use julia> LinearAlgebra.BLAS.vendor().

    Linear system solver

    COSMO uses QDLDL.jl as the default linear system solver. In our experience this seems to be a competitive choice until about 1e5 - 1e6 nonzeros in the constraint matrix. After that it is worth trying one of the indirect system solvers, such as CG or MINRES. Furthermore, we also recommend trying Pardiso (or MKLPardiso) for problems of that dimension. More details can be found here: Linear System Solver.

    Custom cones

    In some cases the computations can be speed-up if certain constraints in the problem allow the implementation of a fast projection function. We allow the user to define their own custom convex cone with a corresponding projection function. The custom cone has to be defined as struct CustomCone{T} <: COSMO.AbstractConvexSet{T}. Furthermore, the user has to define a function that projects an input vector x onto the custom cone, i.e. function COSMO.project!(x::AbstractVector{T}, C::CustomCone{T}) where {T} ... end.

    Multithreading

    COSMO allows the execution of the projection step for multiple constraints in parallel using Julia's multithreading features. This is currently not enabled in the tagged release because of stability issues in earlier Julia versions. To use multithreading checkout the branch with_multi_threading, which we keep in sync with the latest tagged release. This can be installed via Julia's package manager with pkg> add COSMO#with_multi_threading. Afterwards, before starting Julia, set export JULIA_NUM_THREADS=[NUMBER_PHYSICAL_CORES_HERE]. In Julia you can verify the number of threads with julia> Threads.nthreads().

    Notice that the extra overhead for multithreading can slow the solver down if the problem is small. However, we noticed significant performance improvements if the problem contained multiple positive semidefinite constraints or when one large constraint was decomposed. In that case it also helps to restrict the number of BLAS threads per Julia thread with julia> BLAS.set_num_threads(1) to prevent oversubscription of the available cores.

    Multithreading can also be used in the factorisation step if the Pardiso or MKLPardiso solver are selected. This is only advisable for constraint matrices with more than 1e5 nonzeros.

    Chordal decomposition and Clique merging

    When solving large structured and sparse SDPs significant performance improvements are achievable if the problem is passed to COSMO in the right way. This means the solver has to be able to infer the structure of the positive semidefinite variable from the constraint. See the section on Chordal Decomposition for more details. In some cases the primal SDP doesn't allow decomposition but the dual SDP does, consider the Maximum Cut Problem and the Relaxed Two-Way Partitioning Problem for examples.

    If the problem is decomposable it is also worth experimenting with different clique merging strategies to see how they impact the performance. More details can be found here: Clique merging.

    diff --git a/dev/search/index.html b/dev/search/index.html deleted file mode 100644 index 24b7052a..00000000 --- a/dev/search/index.html +++ /dev/null @@ -1,6 +0,0 @@ - -Search · COSMO.jl diff --git a/dev/search_index.js b/dev/search_index.js index 01e7ca16..be49ed41 100644 --- a/dev/search_index.js +++ b/dev/search_index.js @@ -1,3 +1,3 @@ var documenterSearchIndex = {"docs": -[{"location":"logistic_regression/#Logistic-Regression","page":"Logistic Regression","title":"Logistic Regression","text":"","category":"section"},{"location":"logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"Logistic regression problems can be solved using exponential cone constraints. An example on how to use COSMO to solve a logistic regression problem is presented in /examples/logistic_regression_regularization.ipynb.","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"EditURL = \"../../../examples/logistic_regression.jl\"","category":"page"},{"location":"examples/logistic_regression/#Logistic-Regression","page":"Logistic Regression","title":"Logistic Regression","text":"","category":"section"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"The presented example is adapted from the Machine Learning - course by Andrew Ng.","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"In this example we use logistic regression to estimate the parameters theta_i of a logistic model in a classification problem. We will first transform the logistic regression problem into an exponential cone optimisation problem. We will then solve the optimisation problem with COSMO and determine the model parameters and the decision boundary.","category":"page"},{"location":"examples/logistic_regression/#Visualizing-the-data","page":"Logistic Regression","title":"Visualizing the data","text":"","category":"section"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"Before we start, let's load and take a look at the example data from examples/chip_data.txt:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"using LinearAlgebra, SparseArrays, COSMO, JuMP\nusing Plots\n\n# load example data\nf = open(joinpath(@__DIR__, \"chip_data.txt\"))\nlines = readlines(f)\nclose(f)\nn_data = length(lines)\nn_half = div(n_data, 2)\nx1 = zeros(n_data)\nx2 = zeros(n_data)\ny = zeros(Float64, n_data)\nfor (i, l) in enumerate(lines)\n s = split(l, \",\")\n x1[i] = parse(Float64, s[1])\n x2[i] = parse(Float64, s[2])\n y[i] = parse(Float64, s[3])\nend\n\n\n# visualize data\nplot(x1[1:n_half-1], x2[1:n_half-1], color = :blue, st=:scatter, markershape = :cross, aspect_ratio=:equal, label = \"Accepted\", xlabel = \"x1 - Microchip Test Score 1\", ylabel = \"x2 - Microchip Test Score 2\")\nplot!(x1[n_half:end], x2[n_half:end], color = :red, st=:scatter, markershape = :circle, label = \"Rejected\")","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"The plot shows two test scores of n microchip samples from a fabrication plant and whether the chip passed the quality check. Based on this data we would like to build a logistic model that takes into account the test scores and helps us predict the likelihood of a chip being accepted.","category":"page"},{"location":"examples/logistic_regression/#Defining-the-logistic-model","page":"Logistic Regression","title":"Defining the logistic model","text":"","category":"section"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"The logistic regression hypothesis is given by","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"h_theta(x) = g(theta^top x)","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"where g is the sigmoid function:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"g(theta^top x) = frac11+exp(-theta^top x)","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"The vector x represents the independent variables and theta represents the model parameters. For our samples we set the dependent variable y =1 if the chip was accepted and y = 0 otherwise.","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"The function h_theta(x) can be interpreted as the probability of the outcome being true rather than false. We want to find the parameters theta that maximize the log-likelihood over all (independently Bernoulli distributed) observations","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"J(theta) = sum_i y_i = 1 log h_theta(x_i) + sum_i y_i = 0 log (1-h_theta(x_i))","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"Consequently, we want to solve the following optimization problem:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"textminimize quad -J(theta) + mu theta _2","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"where we added a regularization term with parameter mu to prevent overfitting.","category":"page"},{"location":"examples/logistic_regression/#Feature-mapping","page":"Logistic Regression","title":"Feature mapping","text":"","category":"section"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"As our dataset only has two independent variables (the test scores) our model y = theta_0 + theta_1 x_1 + theta_2 x_2 will have the form of a straight line. Looking at the plot one can see that a line will not perform well in separating the samples. Therefore, we will create more features based on each data point by mapping the original features (x_1, x_2) into all polynomial terms of x_1 and x_2 up to the 6th power:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"textmap_feature(x_1x_2) = 1 x_1 x_2 x_1^2 x_1x_2 x_2^2 x_1^3 dots x_1x_2^5 x_2^6 ","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"This will create 28 features for each sample.","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"function map_feature(x1, x2)\n deg = 6\n x_new = ones(length(x1))\n for i = 1:deg, j = 0:i\n x_new = hcat(x_new, x1.^(i-j) .* x2.^j)\n end\n return x_new\nend\n\nX = map_feature(x1, x2);\nsize(X)","category":"page"},{"location":"examples/logistic_regression/#Transformation-into-a-conic-optimisation-problem","page":"Logistic Regression","title":"Transformation into a conic optimisation problem","text":"","category":"section"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"We can rewrite above likelihood maximisation problem as a conic optimisation problem with exponential-cone-, second-order-cone-, equality-, and inequality constraints:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"beginarrayll\ntextminimize sum_i^n epsilon_i + mu v\ntextsubject to theta _2 leq v\n log(1 + exp(-theta^top x_i)) leq epsilon_i quad textif y_i = 1 \n log(1 + exp(theta^top x_i)) leq epsilon_i quadtext otherwise\nendarray","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"Implementing the constraint log(1 + exp(z)) leq epsilon for each of the n samples requires two exponential cone constraints, one inequality constraint and two equality constraints. To see this, take the exponential on both sides and then divide by exp(epsilon) to get:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"exp(-epsilon) + exp(z - epsilon) leq 1","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"This constraint is equivalent to:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"beginarrayll\n(z - epsilon s_1 t_1) in K_textexp \n(-epsilon s_2 t_2) in K_textexp \nt_1 + t_2 leq 1\ns_1 = s_2 = 1\nendarray","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"where we defined the exponential cone as:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"K_textexp = (r s t) mid s 0 s exp(rs) leq t cup r leq 0 s = 0 t geq 0 ","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"Based on this transformation our optimisation problem will have 5n + n_theta + 1 variables, 1 SOCP constraint, 2n exponential cone constraints, n inequality constraints and 2n equality constraints. Let's model the problem with JuMP and COSMO:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"n_theta = size(X, 2)\nn = n_data\nμ = 1.\n\nm = JuMP.Model(COSMO.Optimizer)\n@variable(m, v)\n@variable(m, θ[1:n_theta])\n@variable(m, e[1:n])\n@variable(m, t1[1:n])\n@variable(m, t2[1:n])\n@variable(m, s1[1:n])\n@variable(m, s2[1:n])\n\n@objective(m, Min, μ * v + sum(e))\n@constraint(m, [v; θ] in MOI.SecondOrderCone(n_theta + 1))\n\n# create the constraints for each sample points\nfor i = 1:n\n yi = y[i]\n x = X[i, :]\n yi == 1. ? (a = -1) : (a = 1)\n @constraint(m, [a * dot(θ, x) - e[i]; s1[i]; t1[i] ] in MOI.ExponentialCone())\n @constraint(m, [-e[i]; s2[i]; t2[i]] in MOI.ExponentialCone())\n @constraint(m, t1[i] + t2[i] <= 1)\n @constraint(m, s1[i] == 1)\n @constraint(m, s2[i] == 1)\nend\nJuMP.optimize!(m)","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"theta = value.(θ)","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"Once we have solved the optimisation problem and obtained the parameter vector theta, we can plot the decision boundary. This can be done by evaluating our model over a grid of points (uv) and then plotting the contour line where the function g returns a probability of p=05.","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"# First we evaluate our model over a grid of points z = θ' x\nu = collect(range(-1., stop = 1.5, length = 50))\nv = collect(range(-1., stop = 1.5, length = 50))\nz = zeros(length(u), length(v));\nfor i = 1:length(u), j = 1:length(v)\n z[i, j] = dot(map_feature(u[i], v[j]), theta);\nend","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"To add the decision boundary we have to plot the line indicating 50 probability of acceptance, i.e. g(theta^top x) = g(z) = 05 which we get at z=0.","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"plot(x1[1:n_half-1], x2[1:n_half-1], color = :blue, st = :scatter, markershape = :cross, aspect_ratio=:equal, label = \"Accepted\", xlabel = \"x1 - Microchip Test Score 1\", ylabel = \"x2 - Microchip Test Score 2\")\nplot!(x1[n_half:end], x2[n_half:end], color = :red, st = :scatter, markershape = :circle, label = \"Rejected\")\ncontour!(u, v, z', levels = [0.], c = :black, linewidth = 2)","category":"page"},{"location":"examples/logistic_regression/#Solving-the-optimisation-problem-directly-with-COSMO","page":"Logistic Regression","title":"Solving the optimisation problem directly with COSMO","text":"","category":"section"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"We can solve the problem directly in COSMO by using its modeling interface. The problem will have nn = 5 n + n_theta + 1 variables. Let us define the cost function frac12x^top P x + q^top x:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"nn = 5 * n + n_theta + 1\nP = spzeros(nn, nn)\nq = zeros(nn)\nq[1] = μ\nfor i = 1:n\n q[1 + n_theta + (i - 1) * 5 + 1] = 1.\nend","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"Next we define a function that creates the COSMO.Constraints for a given sample:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"# the order of the variables\n# v, thetas, [e1 t11 t12 s11 s12] [e2 t21 t22 s21 s22] ...\n# for each sample create two exponential cone constraints,\n# 1 nonnegatives constraint, 2 zeroset constraints\nfunction add_log_regression_constraints!(constraint_list, x, y, n, sample_num)\n num_thetas = length(x)\n # 1st exponential cone constraint (zi - ei, s1, t1) in Kexp\n c_start = 1 + num_thetas + (sample_num - 1) * 5 + 1\n A = spzeros(3, n)\n A[1, c_start] = -1.\n y == 1. ? (a = -1) : (a = 1)\n for k = 1:num_thetas\n A[1, 2 + k - 1] = a * x[k]\n end\n A[2, c_start + 3] = 1.\n A[3, c_start + 1] = 1.\n b = zeros(3)\n push!(constraint_list, COSMO.Constraint(A, b, COSMO.ExponentialCone))\n\n # 2nd exponential cone constraint (-e, s2, t2)\n A = spzeros(3, n)\n A[1, c_start] = -1.\n A[2, c_start + 4] = 1.\n A[3, c_start + 2] = 1.\n b = zeros(3)\n push!(constraint_list, COSMO.Constraint(A, b, COSMO.ExponentialCone))\n\n # Nonnegatives constraint t1 + t2 <= 1\n A = spzeros(1, n)\n A[1, c_start + 1] = -1.\n A[1, c_start + 2] = -1.\n b = [1.]\n push!(constraint_list, COSMO.Constraint(A, b, COSMO.Nonnegatives))\n\n # ZeroSet constraint s1 == 1, s2 == 1\n A = spzeros(2, n)\n A[1, c_start + 3] = 1.\n A[2, c_start + 4] = 1.\n b = -1 * ones(2)\n push!(constraint_list, COSMO.Constraint(A, b, COSMO.ZeroSet))\nend\nnothing","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"Now we can use this function to loop over the sample points and add the constraints to our constraint list:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"constraint_list = Array{COSMO.Constraint{Float64}}(undef, 0)\nfor i = 1:n\n add_log_regression_constraints!(constraint_list, X[i, :], y[i], nn, i )\nend","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"It remains to add a second order cone constraint for the regularisation term: theta _2 leq v","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"push!(constraint_list, COSMO.Constraint(Matrix(1.0I, n_theta + 1, n_theta + 1), zeros(n_theta + 1), COSMO.SecondOrderCone, nn, 1:n_theta+1));\nnothing #hide","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"We can now create, assemble, and solve our COSMO.Model:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"model = COSMO.Model()\nassemble!(model, P, q, constraint_list, settings = COSMO.Settings(verbose=true))\nres = COSMO.optimize!(model);\nnothing #hide","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"Let us double check that we get the same theta as in the previous section:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"using Test\ntheta_cosmo = res.x[2:2+n_theta-1]\n@test norm(theta_cosmo - theta) < 1e-10","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"This page was generated using Literate.jl.","category":"page"},{"location":"contributing/#Contributing","page":"Contributing","title":"Contributing","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Contributions are always welcome:","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"If you want to contribute features, bug fixes, etc, please take a look at our Code Style Guide below\nPlease report any issues and bugs that you encounter in Issues\nAs an open source project we are also interested in any projects and applications that use COSMO. Please let us know via email to: michael.garstka[at]eng.ox.ac.uk","category":"page"},{"location":"contributing/#Code-Style-Guide","page":"Contributing","title":"Code Style Guide","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"The code in this repository follows the naming and style conventions of Julia Base with a few modifications. This style guide is heavily \"inspired\" by the guides of John Myles White and JuMP.","category":"page"},{"location":"contributing/#Formatting","page":"Contributing","title":"Formatting","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Use one tab when indenting a new block (except module)\nUse spaces between operators, except for ^, ', and :\nUse single space after commas and semicolons\nDon't use spaces around parentheses, or braces","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Bad: f(x,y) = [5*sin(x+y);y'] Good: f(x, y) = [5 * sin(x + y); y']","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Use spacing with keyword arguments","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Bad: foo(x::Integer=1) Good: foo(x::Integer = 1)","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Don't parenthesize conditions","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Bad: if (a == b) Good: if a == b","category":"page"},{"location":"contributing/#Naming","page":"Contributing","title":"Naming","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Modules and Type names use capitilization and camel case, e.g. module LinearAlgebra, struct ConvexSets.\nFunctions are lowercase and use underscores to seperate words, e.g. has_key(x), is_valid(y).\nNormal variables are lowercase and use underscores like functions, e.g. convex_set\nConstants are uppercase, e.g. const MY_CONSTANT\nAlways append ! to names of functions that modify their arguments.\nFunction arguments that are mutated come first. Otherwise follow the rules layed out in Julia Base Argument ordering\nFiles are named like functions, e.g. my_new_file.jl","category":"page"},{"location":"contributing/#Syntax","page":"Contributing","title":"Syntax","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Use 1.0 instead of 1.","category":"page"},{"location":"contributing/#Git(hub)-specific-conventions","page":"Contributing","title":"Git(hub)-specific conventions","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Branch names should be prepended with the initials of the creator and a forward slash, e.g. mg/newIdea instead of newIdea\nCommit messages should have the following format:","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"<#IssueId> Short (72 chars or less) summary\n\nMore detailed explanatory text. Wrap it to 72 characters. The blank\nline separating the summary from the body is critical.\n\nImperative style for the commit message: \"Fix bug\" and not \"Fixed\nbug\" or \"Fixes bug.\"\n\nThe issue id can be ommitted if the commit does not related to a specific open issue","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"EditURL = \"../custom_cone.jl\"","category":"page"},{"location":"literate/build/custom_cone/#Custom-Cone-Constraint","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"","category":"section"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"This example demonstrate how the user can extend COSMO with his own custom convex cones and use them in constraints. To make things easy, we will implement the simple cone of Nonpositives, i.e. mathbbR_-^n = x in mathbbR^n mid x_i leq 0.","category":"page"},{"location":"literate/build/custom_cone/#Cone-definition-and-projection-function","page":"Custom Cone Constraint","title":"Cone definition and projection function","text":"","category":"section"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"We start by creating a concrete subtype of our cone interface-type: COSMO.AbstractConvexCone{T}, where T is a parameter for the floating-point precision. The only field that is strictly required for our cone definition is dim, which is used by the solver to store information about the cone dimension. However, you can also use the object to store other information, or allocate the workspace necessary for the projection step.","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"using COSMO, LinearAlgebra, SparseArrays, Test\n\n# define new cone type\nstruct Nonpositives{T} <: COSMO.AbstractConvexCone{T}\n dim::Int64\nend","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"Next, we define a projection method for Nonpositives{T} that takes a vector x and projects it onto our custom cone. For the cone of nonpositive vectors the projection simply sets all positive elements of x to zero:","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"function COSMO.project!(x::AbstractVector{T}, C::Nonpositives{T}) where {T <: AbstractFloat}\n x .= min.(x, zero(T))\nend","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"This is all that is required to get a basic constraint working. Let's test our new cone/constraint by solving the following LP:","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"beginarrayll textmaximize x_1 + x_2 + x_3\ntextsubject to x_1 leq 3 \n x_2 leq 2 \n x_1 + x_3 = 5\nendarray","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"We define our decision vector x = (x_1 x_2 x_3) and setup the problem:","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"n = 3\nP = spzeros(n, n)\nq = -ones(n)\n\n# A1 * x + b1 <= 0\nA1 = diagm(0 => ones(2))\nb1 = [-3.; -2.]\nc1 = COSMO.Constraint(A1, b1, Nonpositives, n, 1:2); #here we use our new cone\n\n# x_1 + x_3 == 5\nA2 = [1. 0 1.]\nb2 = [-5.]\nc2 = COSMO.Constraint(A2, b2, COSMO.ZeroSet);\n\n# assemple and solve\nmodel = COSMO.Model();\nassemble!(model, P, q, [c1; c2]);\nres = COSMO.optimize!(model);\nnothing #hide","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"res.x","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"obj_val = -dot(q, res.x)","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"The problem was solved using constraints involving our new cone and yields the expected result. Next, we want to add the ability to detect infeasible problems.","category":"page"},{"location":"literate/build/custom_cone/#Support-infeasibility-detection","page":"Custom Cone Constraint","title":"Support infeasibility detection","text":"","category":"section"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"If no further information about the new cone is provided, the infeasibility detection is disabled. However, by defining the two additional methods in_dual and in_pol_recc we can support infeasibility detection. More information on how infeasible problems are detected in COSMO can be found here [1]. We will have to add the ability for the solver to check whether a vector is in the dual cone mathcalK^* of our new cone mathcalK and whether a vector is in the recession cone of the polar cone mathcalK^circ^infty of our cone. The first function is used to check for primal infeasibility, while the second function is used to check for dual infeasibility. Luckily, for the Nonpositives-cone this is straightforward:","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"mathbbR_-^n^* = mathbbR_-^n quad mathbbR_-^n^circ^infty = mathbbR_+^n","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"function COSMO.in_dual(x::AbstractVector{T}, C::Nonpositives{T}, tol::T) where {T <: AbstractFloat}\n\treturn !any( x-> (x > -tol), x)\nend\n\nfunction COSMO.in_pol_recc(x::AbstractVector{T}, C::Nonpositives{T}, tol::T) where {T <: AbstractFloat}\n\treturn !any( x-> (x < tol), x)\nend","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"Notice that for numerical reasons, we give the check-functions a bit of slack using the tolerance parameter tol. To test the infeasibility detection, we will attempt to use our new cone to solve the clearly dual infeasible problem:","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"beginarrayll textminimize x\ntextsubject to xleq 3\nendarray","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"n = 1\nP = spzeros(n, n)\nq = [1.]\n\n# x <= 3 <=> x - 3 in Nonpositives\nAi = [1.]\nbi = [-3.]\nci = COSMO.Constraint(Ai, bi, Nonpositives);\n\nmodel = COSMO.Model();\nassemble!(model, P, q, [ci]);\nres = COSMO.optimize!(model);\nnothing #hide","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"@test res.status == :Dual_infeasible","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"which is what we would expect.","category":"page"},{"location":"literate/build/custom_cone/#Using-new-cone-with-JuMP","page":"Custom Cone Constraint","title":"Using new cone with JuMP","text":"","category":"section"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"The nice composibility of Julia-code allows us to use our new cone definition even when the problem is modelled with JuMP. For this to work, we define a concrete subtype of MOI.AbstractSet, that JuMP can use to build a constraint.","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"using MathOptInterface, JuMP\nimport Base.copy\nconst MOI = MathOptInterface\n\nstruct NonPos <: MOI.AbstractVectorSet\n\tdimension::Int\nend\nBase.copy(set::NonPos) = set #this is needed for MOI","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"Next, we tell COSMO that whenever JuMP wants to solve a problem with a NonPos-cone it should translate it into a Nonpositives constraint and we also tell MOI that we now support constraints of this new mysterious cone NonPos.","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"function COSMO.processSet!(b::Vector{T}, rows::UnitRange{Int}, cs, s::NonPos) where {T <: AbstractFloat}\n push!(cs, Nonpositives{T}(length(rows)))\n nothing\nend\n#tell MOI, that COSMO supports constraints with this new cone\nMOI.supports_constraint(optimizer::COSMO.Optimizer, ::Type{<:Union{MOI.VectorOfVariables, MOI.VectorAffineFunction{Float64}}}, ::Type{NonPos}) = true","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"We should now be able to model the LP from above in JuMP and solve it internally using our Nonpositives cone and projection function.","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"model = JuMP.Model(COSMO.Optimizer);\n@variable(model, x[1:3]);\n@objective(model, Max, x[1] + x[2] + x[3]);\n@constraint(model, A1 * x[1:2] .+ b1 in NonPos(2));\n@constraint(model, x[1] + x[3] == 5);\nJuMP.optimize!(model)","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"x_opt = JuMP.value.(x)","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"You can see in the solver output that indeed a set Nonpositives of dim: 2 was used.","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"The discussed cone of Nonpositives is of course trivial, but the ability to define new cones and constraints for COSMO can be very powerful to model complex problems.","category":"page"},{"location":"literate/build/custom_cone/#References","page":"Custom Cone Constraint","title":"References","text":"","category":"section"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"[1] Garstka et al. - COSMO: A conic operator splitting method for convex conic problems (2020)","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"This page was generated using Literate.jl.","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"EditURL = \"../../../examples/lp.jl\"","category":"page"},{"location":"examples/lp/#Linear-Program","page":"Linear Program","title":"Linear Program","text":"","category":"section"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"We want to solve the following linear program with decision variable x:","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"beginarrayll textminimize c^top x\ntextsubject to A x leq b \n x geq 1 \n x_2 geq 5 \n x_1 + x_3 geq 4\nendarray","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"The problem can be solved with COSMO in the following way:","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"using COSMO, LinearAlgebra, SparseArrays, Test","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"##Define problem data:\nc = [1; 2; 3; 4.];\nA = Matrix(1.0I, 4, 4);\nb = [10.; 10; 10; 10];\nn = 4;\nnothing #hide","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"Create the constraints Ax + b in mathcalK:","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"# Ax <= b\nc1 = COSMO.Constraint(-A, b, COSMO.Nonnegatives);\n# x >= 1\nc2 = COSMO.Constraint(Matrix(1.0I, n, n), -ones(n), COSMO.Nonnegatives);\n# x2 >= 5\nc3 = COSMO.Constraint(1, -5, COSMO.Nonnegatives, n, 2:2);\n# x1 + x3 >= 4\nc4 = COSMO.Constraint([1 0 1 0], -4, COSMO.Nonnegatives);\nnothing #hide","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"Define matrix P and vector q for the objective function:","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"P = spzeros(4, 4);\nq = c;\nnothing #hide","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"Create, assemble and solve the model:","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"settings = COSMO.Settings(verbose=true, eps_abs = 1e-4, eps_rel = 1e-5);\nmodel = COSMO.Model();\nassemble!(model, P, q, [c1; c2; c3; c4], settings = settings);\nres = COSMO.optimize!(model)","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"Compare the result to the known solution:","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"@test isapprox(res.x[1:4], [3; 5; 1; 1], atol=1e-2, norm = (x -> norm(x, Inf)))","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"@test isapprox(res.obj_val, 20.0, atol=1e-2)","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"This page was generated using Literate.jl.","category":"page"},{"location":"performance/#Performance-Tips","page":"Performance Tips","title":"Performance Tips","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"There are a number of ways to improve the performance of the solver given a particular problem. If you are not satisfied with the performance there are a number of things you have to determine first. Is the solver slow because","category":"page"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"it's the first time you ran it in the current Julia session or\nis it because the solver needs a lot of iterations (convergence) or\nis each iteration or the initial factorisation slow (computational performance)?","category":"page"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"Let's see how each point can be addressed.","category":"page"},{"location":"performance/#First-run","page":"Performance Tips","title":"First run","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"Whenever a new Julia session is started, the first run will trigger a compilation of all functions based on their arguments used in your script. This will slow the first execution of COSMO down. After that Julia will call the fast compiled functions. To get around this, you can either keep your current Julia session open and discard the first run. Alternatively, if your problem is very large, you could solve a small version of your problem first to trigger the compilation. Another option is to use PackageCompiler to save compiled functions into a sysimage that can be loaded at startup.","category":"page"},{"location":"performance/#Solver-Timings","page":"Performance Tips","title":"Solver Timings","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"It is often instructive to look at the detailed solver timing results for your problem. This can reveal where most of the time is spent. To achieve this, run COSMO with the setting verbose_timing = true. After solving the problem with result = COSMO.optimize!(model) you can look at result.times for a breakdown of the times spent in different parts of the algorithm, see Timings for more details. Especially take a look at the ratio of factorisation time and iteration time. If you use JuMP to solve the problem, you can take a look at the timings with backend(model).optimizer.model.optimizer.results.times.","category":"page"},{"location":"performance/#Convergence","page":"Performance Tips","title":"Convergence","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"It is possible that COSMO converges slowly, i.e. needs a large number of iterations, for your problem given its default parameters.","category":"page"},{"location":"performance/#Parameters","page":"Performance Tips","title":"Parameters","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"You could try changing any of the following parameters:","category":"page"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"rho: The initial algorithm step parameter has a large influence on the convergence. Try different values between 1e-5 and 10.\nadaptive_rho = false: You can try to disable the automatic rho adaption and use different rho values.\nadaptive_rho_interval: This specifies after how many iterations COSMO tries to adapt the rho parameter. You can also set adaptive_rho_interval = 0 which adapts the rho parameter after the time spent iterating passes 40% of the factorisation time. This is currently the default in OSQP and works well with QPs.\nalpha = 1.0: This disables the over-relaxation that is used in the algorithm. We recommend values between 1.0 - 1.6.\nscaling = 0: This disables the problem scaling.\neps_abs and eps_rel: Check the impact of modifying the stopping accuracies.","category":"page"},{"location":"performance/#Use-warm-starting","page":"Performance Tips","title":"Use warm starting","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"The number of iterations can be dramatically decreased by providing a good initial guess for x, s and y. Examples where warm starting is commonly used are model predictive control and portfolio backtests, see Warm starting.","category":"page"},{"location":"performance/#Computational-performance","page":"Performance Tips","title":"Computational performance","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"If the convergence of the algorithm is not an issue, there are still a number of steps you can take to make COSMO faster.","category":"page"},{"location":"performance/#Intel-MKL-BLAS/LAPACK","page":"Performance Tips","title":"Intel MKL BLAS/LAPACK","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"We experienced significant performance improvements on Intel CPUs if Julia is compiled with MKL BLAS. This is because Julia's linear algebra function will use Intel MKL BLAS and LAPACK functions that are optimised for Intel hardware. The effect is especially significant for SDPs because most of the time is spent in the LAPACK function syevr. If you are running Julia on Intel hardware, an easy way to compile Julia with MKL is to add and build the MKL package, see MKL.jl. To verify your current BLAS vendor you can use julia> LinearAlgebra.BLAS.vendor().","category":"page"},{"location":"performance/#Linear-system-solver","page":"Performance Tips","title":"Linear system solver","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"COSMO uses QDLDL.jl as the default linear system solver. In our experience this seems to be a competitive choice until about 1e5 - 1e6 nonzeros in the constraint matrix. After that it is worth trying one of the indirect system solvers, such as CG or MINRES. Furthermore, we also recommend trying Pardiso (or MKLPardiso) for problems of that dimension. More details can be found here: Linear System Solver.","category":"page"},{"location":"performance/#Custom-cones","page":"Performance Tips","title":"Custom cones","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"In some cases the computations can be speed-up if certain constraints in the problem allow the implementation of a fast projection function. We allow the user to define their own custom convex cone with a corresponding projection function. The custom cone has to be defined as struct CustomCone{T} <: COSMO.AbstractConvexSet{T}. Furthermore, the user has to define a function that projects an input vector x onto the custom cone, i.e. function COSMO.project!(x::AbstractVector{T}, C::CustomCone{T}) where {T} ... end.","category":"page"},{"location":"performance/#Multithreading","page":"Performance Tips","title":"Multithreading","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"COSMO allows the execution of the projection step for multiple constraints in parallel using Julia's multithreading features. This is currently not enabled in the tagged release because of stability issues in earlier Julia versions. To use multithreading checkout the branch with_multi_threading, which we keep in sync with the latest tagged release. This can be installed via Julia's package manager with pkg> add COSMO#with_multi_threading. Afterwards, before starting Julia, set export JULIA_NUM_THREADS=[NUMBER_PHYSICAL_CORES_HERE]. In Julia you can verify the number of threads with julia> Threads.nthreads().","category":"page"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"Notice that the extra overhead for multithreading can slow the solver down if the problem is small. However, we noticed significant performance improvements if the problem contained multiple positive semidefinite constraints or when one large constraint was decomposed. In that case it also helps to restrict the number of BLAS threads per Julia thread with julia> BLAS.set_num_threads(1) to prevent oversubscription of the available cores.","category":"page"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"Multithreading can also be used in the factorisation step if the Pardiso or MKLPardiso solver are selected. This is only advisable for constraint matrices with more than 1e5 nonzeros.","category":"page"},{"location":"performance/#Chordal-decomposition-and-Clique-merging","page":"Performance Tips","title":"Chordal decomposition and Clique merging","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"When solving large structured and sparse SDPs significant performance improvements are achievable if the problem is passed to COSMO in the right way. This means the solver has to be able to infer the structure of the positive semidefinite variable from the constraint. See the section on Chordal Decomposition for more details. In some cases the primal SDP doesn't allow decomposition but the dual SDP does, consider the Maximum Cut Problem and the Relaxed Two-Way Partitioning Problem for examples.","category":"page"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"If the problem is decomposable it is also worth experimenting with different clique merging strategies to see how they impact the performance. More details can be found here: Clique merging.","category":"page"},{"location":"decomposition/#Chordal-Decomposition","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"","category":"section"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"For very large sparse semidefinite programs (SDPs) it is often helpful to analyse the sparsity structure of the PSD constraint(s). If the equality constraints impose a sparsity structure on the matrix variable, one PSD constraint on a large matrix variable can be decomposed into several smaller constraints. This results usually in a significant speedup and reduction in solve time.","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"The following example gives a short overview on chordal decomposition and clique merging. For more details, take a look at our paper on clique merging or watch the corresponding presentation that I gave on the topic.","category":"page"},{"location":"decomposition/#Example-problem","page":"Chordal Decomposition","title":"Example problem","text":"","category":"section"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Let's consider the following SDP in standard dual form:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"beginarrayll\n textminimize c^top x \n textsubject to displaystyle sum_i=1^m A_i x_i + S = B\n S in mathbbS^n_+\nendarray","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"with problem data matrices A_1 ldots A_m B in mathbbS^n, vector variable x in mathbbR^n, and matrix variable S in mathbbS^n_+.","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Let's look at the following example problem with m=2 and n=9:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"A1 = [-4.0 0.0 -2.0 0.0 0.0 -1.0 0.0 0.0 0.0; 0.0 -3.0 -1.0 0.0 0.0 0.0 0.0 0.0 0.0; -2.0 -1.0 -2.0 0.0 0.0 5.0 4.0 -4.0 0.0; 0.0 0.0 0.0 -4.0 -5.0 0.0 0.0 3.0 0.0; 0.0 0.0 0.0 -5.0 4.0 0.0 0.0 2.0 0.0; -1.0 0.0 5.0 0.0 0.0 5.0 -4.0 -4.0 -5.0; 0.0 0.0 4.0 0.0 0.0 -4.0 -1.0 -1.0 -3.0; 0.0 0.0 -4.0 3.0 2.0 -4.0 -1.0 2.0 -2.0; 0.0 0.0 0.0 0.0 0.0 -5.0 -3.0 -2.0 -3.0];\n\nA2 = [-5.0 0.0 3.0 0.0 0.0 -2.0 0.0 0.0 0.0; 0.0 -3.0 -5.0 0.0 0.0 0.0 0.0 0.0 0.0; 3.0 -5.0 3.0 0.0 0.0 5.0 -4.0 -5.0 0.0; 0.0 0.0 0.0 3.0 2.0 0.0 0.0 -2.0 0.0; 0.0 0.0 0.0 2.0 4.0 0.0 0.0 -3.0 0.0; -2.0 0.0 5.0 0.0 0.0 1.0 -5.0 -2.0 -4.0; 0.0 0.0 -4.0 0.0 0.0 -5.0 -2.0 -3.0 3.0; 0.0 0.0 -5.0 -2.0 -3.0 -2.0 -3.0 5.0 3.0; 0.0 0.0 0.0 0.0 0.0 -4.0 3.0 3.0 -4.0];\n\nB = [-0.11477375644968069 0.0 6.739182490600791 0.0 0.0 -1.2185593245043502 0.0 0.0 0.0; 0.0 1.2827680528587497 -5.136452036888789 0.0 0.0 0.0 0.0 0.0 0.0; 6.739182490600791 -5.136452036888789 7.344770673489607 0.0 0.0 -0.2224400187044442 -10.505300166831221 -1.2627361794562273 0.0; 0.0 0.0 0.0 10.327710040060499 8.91534585379813 0.0 0.0 -6.525873789637007 0.0; 0.0 0.0 0.0 8.91534585379813 0.8370459338528677 0.0 0.0 -6.210900615408826 0.0; -1.2185593245043502 0.0 -0.2224400187044442 0.0 0.0 -3.8185953011245024 -0.994033914192722 2.8156077981712997 1.4524716674219218; 0.0 0.0 -10.505300166831221 0.0 0.0 -0.994033914192722 0.029162208619863517 -2.8123790276830745 7.663416446183705; 0.0 0.0 -1.2627361794562273 -6.525873789637007 -6.210900615408826 2.8156077981712997 -2.8123790276830745 4.71893305728242 6.322431630550857; 0.0 0.0 0.0 0.0 0.0 1.4524716674219218 7.663416446183705 6.322431630550857 0.5026094532322212];\n\nc = [-0.21052661285686525, -1.263324575834677];","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Note that the data matrices A_1dots A_m B all have the same sparsity pattern (common zeros in certain entries). Take a look at A_1:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"A1 = [-4.0 0.0 -2.0 0.0 0.0 -1.0 0.0 0.0 0.0; 0.0 -3.0 -1.0 0.0 0.0 0.0 0.0 0.0 0.0; -2.0 -1.0 -2.0 0.0 0.0 5.0 4.0 -4.0 0.0; 0.0 0.0 0.0 -4.0 -5.0 0.0 0.0 3.0 0.0; 0.0 0.0 0.0 -5.0 4.0 0.0 0.0 2.0 0.0; -1.0 0.0 5.0 0.0 0.0 5.0 -4.0 -4.0 -5.0; 0.0 0.0 4.0 0.0 0.0 -4.0 -1.0 -1.0 -3.0; 0.0 0.0 -4.0 3.0 2.0 -4.0 -1.0 2.0 -2.0; 0.0 0.0 0.0 0.0 0.0 -5.0 -3.0 -2.0 -3.0] # hide\nA1","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Since all the data matrices have zeros in the same places, the equality constraint of the problem tells us that our matrix variable S will also have zeros in these places. The aggregated pattern matrix of S is shown in the following figure, where a dot represents a nonzero entry:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"(Image: )","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Since the matrices are symmetric we only show the lower triangle. We can represent the aggregated sparsity pattern of the problem using a graph G(V E), where the vertex set V is given by the column indices of the matrix and we introduce an edge (ij) in E for every nonzero matrix element S_ij. The graph for the example problem is shown in the right figure. In order to decompose the problem, we have to find the cliques, i.e. completely connected subgraphs, of the graph G. These cliques represent the dense subblocks of non-zero entries in the matrix (see the colored entries in the figure). In order for the theory to work, we also have to require G to be a chordal graph. For the purpose of this example, we just assume that G is chordal and keep in mind that we can always make a graph chordal by adding more edges.","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"COSMO finds the cliques of the graph automatically, if a constraint of type COSMO.PsdCone or COSMO.PsdConeTriangle is present in the problem and additional equality constraints impose a structure on them. For the example problem COSMO finds the following cliques: mathcalC_1=136 mathcalC_2=23 mathcalC_3=3678 mathcalC_4=458 and mathcalC_5=6789.","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Let's denote the set of cliques mathcalB=mathcalC_1ldotsmathcalC_p. To represent the relationship between different cliques, e.g. in terms of overlapping entries, it is helpful to represent them either as a clique tree mathcalT(mathcalB mathcalE) (left) or a clique graph mathcalG(mathcalB xi) (right), shown in the following figures:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"(Image: ) (Image: )","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Once the cliques have been found, we can use the following theorem to decompose the problem (and speed up the solver significantly).","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Theorem (Agler's Theorem)","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Let G(VE) be a chordal graph with a set of maximal cliques mathcalC_1ldotsmathcalC_p . Then S in mathbbS^n_+(E0) if and only if there exist matrices S_ell in S^mathcalC_ell for ell = 1ldotsp such that","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"S = displaystyle sum_ell = 1^p T_ell^top S_ell T_ell","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"where T_ell is an entry-selector matrix that maps the subblocks S_ell into the correct entries of the original matrix S.","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"In other words, instead of solving the original problem and choosing S to be positive semidefinite, we only have to choose the subblocks S_ell of the matrix S to be positive semidefinite and make sure that the individual blocks \"add up\" to S. Thus, we can solve the equivalent problem:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"beginarrayll\n textminimize c^top x \n textsubject to displaystyle sum_i=1^m A_i x + sum_ell = 1^p T_l^top S_ell T_ell = B\n S_ell in mathbbS^mathcalC_ell_+ l = 1 ldots p\nendarray","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Compare this again with the problem at the top of the page. We have replaced the positive semidefinite constraint on the matrix variable S by p constraints on the smaller subblocks S_ell. Why is this useful?","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"When solving an SDP, the major linear algebra operation is a projection onto the constraint set, see Method. This projection is done at each iteration of the algorithm and for positive semidefinite constraints involves an eigenvalue decomposition of the matrix iterate. Unfortunately, an eigenvalue decomposition of a matrix of dimension n has a complexity of roughly mathcalO(n^3).","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"For our example this means that we now project one 2times 2 block, two 3 times 3 blocks and two 4 times 4 blocks instead of one 9 times 9 block. For small problems, this doesn't seem to make much of a difference, but consider of a problem with 1000 11 times 11 blocks along the diagonal and overlapping by one entry. Taking into account that the computational cost scales cubicly, we can improve the performance of the solver by projecting 1000 11times 11 blocks instead of projecting one giant 10001times 10001 block at each iteration.","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Let's go back to our example and solve it with COSMO and JuMP:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"model = JuMP.Model(with_optimizer(COSMO.Optimizer, decompose = true, merge_strategy = COSMO.NoMerge));\n@variable(model, x[1:2]);\n@objective(model, Min, c' * x )\n@constraint(model, Symmetric(B - A1 .* x[1] - A2 .* x[2] ) in JuMP.PSDCone());\n\nJuMP.optimize!(model)","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Notice that we set the solver settings decompose = true to allow COSMO to decompose the problem. Furthermore, we set the clique merging strategy to COSMO.NoMerge. This just means that after we have found the cliques we don't attempt to merge some of them (more about clique merging below). We get the following output from the solver:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"------------------------------------------------------------------\n COSMO v0.5.0 - A Quadratic Objective Conic Solver\n Michael Garstka\n University of Oxford, 2017 - 2019\n------------------------------------------------------------------\n\nProblem: x ∈ R^{13},\n constraints: A ∈ R^{35x13} (70 nnz),\n matrix size to factor: 48x48 (188 nnz)\nSets: PsdConeTriangle of dim: 10\n PsdConeTriangle of dim: 10\n PsdConeTriangle of dim: 6\n PsdConeTriangle of dim: 6\n PsdConeTriangle of dim: 3\nDecomp: Num of original PSD cones: 1\n Num decomposable PSD cones: 1\n Num PSD cones after decomposition: 5\n Merge Strategy: NoMerge\nSettings: ϵ_abs = 1.0e-04, ϵ_rel = 1.0e-04,\n ϵ_prim_inf = 1.0e-06, ϵ_dual_inf = 1.0e-04,\n ρ = 0.1, σ = 1.0e-6, α = 1.6,\n max_iter = 2500,\n scaling iter = 10 (on),\n check termination every 40 iter,\n check infeasibility every 40 iter,\n KKT system solver: QDLDL\nSetup Time: 0.14ms\n\nIter: Objective: Primal Res: Dual Res: Rho:\n40 -1.4134e+00 1.2320e-05 3.0409e-05 1.0000e-01\n\n------------------------------------------------------------------\n>>> Results\nStatus: Solved\nIterations: 40\nOptimal objective: -1.4134\nRuntime: 0.002s (1.93ms)","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Under sets we can indeed see that COSMO solved a problem with five PSD constraints corresponding to the cliques mathcalC_1ldots mathcalC_5 discovered in the sparsity pattern. (Note that the dimension printed is the number of entries in the upper triangle of the matrix block.)","category":"page"},{"location":"decomposition/#Clique-merging","page":"Chordal Decomposition","title":"Clique merging","text":"","category":"section"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"After we have found the cliques of the sparsity pattern, we are allowed to merge some of them back together. For the graph of the sparsity pattern this just means adding more edges or treating some structural zeros as numerical zeros. The main reason to merge two cliques is that they might overlap a lot and therefore it is not advantageous to treat them as two different blocks. Consider the two extreme cases below:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"(Image: ) (Image: )","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"In the left figure we have the ideal case that all the blocks overlap in just one entry. A full decomposition would leave us with a large number of small blocks. The sparsity pattern in the right figure has two large blocks overlapping almost entirely. In this case it would be disadvantageous to decompose the blocks. Instead, we would do the initial decomposition, realize the large overlap, and then merge the two blocks back together. For sparsity patterns that arise from real applications the case is not always as clear and we have to use more sophisticated strategies to decide which blocks to merge.","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"COSMO currently provides three different strategies that can be selected by the user:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"COSMO.NoMerge\nCOSMO.ParentChildMerge\nCOSMO.CliqueGraphMerge","category":"page"},{"location":"decomposition/#COSMO.NoMerge","page":"Chordal Decomposition","title":"COSMO.NoMerge","text":"NoMerge <: AbstractMergeStrategy\n\nA strategy that does not merge cliques.\n\n\n\n\n\n","category":"type"},{"location":"decomposition/#COSMO.ParentChildMerge","page":"Chordal Decomposition","title":"COSMO.ParentChildMerge","text":"ParentChildMerge(t_fill = 8, t_size = 8) <: AbstractTreeBasedMerge\n\nThe merge strategy suggested in Sun and Andersen - Decomposition in conic optimization with partially separable structure (2014). The initial clique tree is traversed in topological order and a clique mathcalC_ell is greedily merged to its parent clique mathcalC_par(ell) if at least one of the two conditions are met\n\n( mathcalC_par(ell) - eta_ell) (mathcalC_ell - eta_ell) leq t_textfill (fill-in condition)\nmax left nu_ell nu_par(ell) right leq t_textsize (supernode size condition)\n\n\n\n\n\n","category":"type"},{"location":"decomposition/#COSMO.CliqueGraphMerge","page":"Chordal Decomposition","title":"COSMO.CliqueGraphMerge","text":"CliqueGraphMerge(edge_weight::AbstractEdgeWeight = ComplexityWeight()) <: AbstractGraphBasedMerge\n\nThe (default) merge strategy based on the reduced clique graph mathcalG(mathcalB xi), for a set of cliques mathcalB = mathcalC_1 dots mathcalC_p where the edge set xi is obtained by taking the edges of the union of clique trees.\n\nMoreover, given an edge weighting function e(mathcalC_imathcalC_j) = w_ij, we compute a weight for each edge that quantifies the computational savings of merging the two cliques. After the initial weights are computed, we merge cliques in a loop:\n\nwhile clique graph contains positive weights:\n\nselect two permissible cliques with the highest weight w_ij\nmerge cliques rightarrow update clique graph\nrecompute weights for updated clique graph\n\nCustom edge weighting functions can be used by defining your own CustomEdgeWeight <: AbstractEdgeWeight and a corresponding edge_metric method. By default, the ComplexityWeight <: AbstractEdgeWeight is used which computes the weight based on the cardinalities of the cliques: e(mathcalC_imathcalC_j) = mathcalC_i^3 + mathcalC_j^3 - mathcalC_i cup mathcalC_j^3.\n\nSee also: Garstka, Cannon, Goulart - A clique graph based merging strategy for decomposable SDPs (2019)\n\n\n\n\n\n","category":"type"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"In our example problem we have two cliques mathcalC_3 = 3678 and mathcalC_5 = 6789 that overlap in three entries. Let's solve the problem again and choose the default clique merging strategy merge_strategy = COSMO.CliqueGraphMerge:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"model = JuMP.Model(with_optimizer(COSMO.Optimizer, decompose = true, merge_strategy = COSMO.CliqueGraphMerge));\n@variable(model, x[1:2]);\n@objective(model, Min, c' * x )\n@constraint(model, Symmetric(B - A1 .* x[1] - A2 .* x[2] ) in JuMP.PSDCone());\n\nJuMP.optimize!(model)","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"------------------------------------------------------------------\n COSMO v0.5.0 - A Quadratic Objective Conic Solver\n Michael Garstka\n University of Oxford, 2017 - 2019\n------------------------------------------------------------------\n\nProblem: x ∈ R^{7},\n constraints: A ∈ R^{30x7} (58 nnz),\n matrix size to factor: 37x37 (153 nnz)\nSets: PsdConeTriangle of dim: 15\n PsdConeTriangle of dim: 6\n PsdConeTriangle of dim: 6\n PsdConeTriangle of dim: 3\nDecomp: Num of original PSD cones: 1\n Num decomposable PSD cones: 1\n Num PSD cones after decomposition: 4\n Merge Strategy: CliqueGraphMerge\nSettings: ϵ_abs = 1.0e-04, ϵ_rel = 1.0e-04,\n ϵ_prim_inf = 1.0e-06, ϵ_dual_inf = 1.0e-04,\n ρ = 0.1, σ = 1.0e-6, α = 1.6,\n max_iter = 2500,\n scaling iter = 10 (on),\n check termination every 40 iter,\n check infeasibility every 40 iter,\n KKT system solver: QDLDL\nSetup Time: 0.26ms\n\nIter: Objective: Primal Res: Dual Res: Rho:\n40 -1.4134e+00 1.7924e-05 1.0416e-04 1.0000e-01\n\n------------------------------------------------------------------\n>>> Results\nStatus: Solved\nIterations: 40\nOptimal objective: -1.4134\nRuntime: 0.003s (2.68ms)","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Unsurprisingly, we can see in the output that COSMO solved a problem with four PSD constraints. One of them is of dimension 15, i.e. a 5times 5 block, which correspond to the merged clique mathcalC_3 cup mathcalC_5 = 36789 .","category":"page"},{"location":"decomposition/#Completing-the-dual-variable","page":"Chordal Decomposition","title":"Completing the dual variable","text":"","category":"section"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"After a decomposed problem is solved, we can recover the solution to the original problem by assembling the matrix variable S from its subblocks S_ell:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"S = displaystyle sum_ell = 1^p T_ell^top S_ell T_ell","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Following Agler's Theorem, S will be a positive semidefinite matrix. However, this is not true for the corresponding dual variable matrix Y. The dual variable returned after solving the decomposed problem will be in the space of PSD completable matrices Y in mathbbS_+^n(E). This means that the entries in Y corresponding to the blocks S_ell (black dots) have been chosen correctly. The numerical values for all the other entries (corresponding to the zeros in S and denoted with a red dot) have to be chosen in the right way to make Y positive semidefinite.","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"(Image: )","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"For more information about PSD matrix completion and the completion algorithm used in COSMO take a look at Vandenberghe and Andersen - Chordal Graphs and Semidefinite Optimization (Ch.10). To configure COSMO to complete the dual variable after solving the problem you have to set the complete_dual option:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"model = JuMP.Model(with_optimizer(COSMO.Optimizer, complete_dual = true));","category":"page"},{"location":"decomposition/#Example-Code","page":"Chordal Decomposition","title":"Example Code","text":"","category":"section"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"The code used for this example can be found in /examples/chordal_decomposition.jl.","category":"page"},{"location":"acceleration/#Acceleration","page":"Acceleration","title":"Acceleration","text":"","category":"section"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"By default COSMO's ADMM algorithm is wrapped in a safeguarded acceleration method to achieve faster convergence to higher precision. COSMO uses accelerators from the COSMOAccelerators.jl package.","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"By default, the solver uses the accelerator type AndersonAccelerator{T, Type2{QRDecomp}, RestartedMemory, NoRegularizer}. This is the classic type-II Anderson acceleration method where the least squares subproblem is solved using an updated QR method. Moreover, the method is restarted, i.e. the history of iterates is deleted, after mem steps and no regularisation for the least-squares method is used.","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"In addition, the method is safeguarded (safeguard = true), i.e. the residual-norm of the accelerated point can not deviate too much from the current point. Otherwise, the point is discarded and the ADMM algorithm performs a normal step instead.","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"The acceleration method can be altered as usual via the solver settings and the accelerator keyword. To deactivate acceleration pass an EmptyAccelerator:","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"settings = COSMO.Settings(accelerator = EmptyAccelerator)","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"To use the default accelerator but with a different memory size (number of stored iterates) use:","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"settings = COSMO.Settings(accelerator = with_options(AndersonAccelerator, mem = 15))","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"To turn the safeguarding off use:","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"settings = COSMO.Settings(safeguard = false)","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"To use an Anderson Accelerator of Type-I with a rolling-memory (oldest iterate replaced by newest) approach, use:","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"settings = COSMO.Settings(accelerator = AndersonAccelerator{Float64, Type1, RollingMemory, NoRegularizer})","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"For more fine-grained control look at the implementation of the accelerator here.","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"When JuMP is used, the accelerator settings can be passed in the usual way:","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"model = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"accelerator\" => with_options(AndersonAccelerator, mem = 15)))","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"Or using the set_optimizer_attribute() method:","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"model = JuMP.Model(COSMO.Optimizer);\nset_optimizer_attribute(model, \"accelerator\", with_options(AndersonAccelerator, mem = 15))","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"EditURL = \"../../../examples/sum_abs_k_eigenvalues.jl\"","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/#Minimizing-the-sum-of-the-k-largest-λ","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"","category":"section"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"We show how to find the sum of absolute value of the k largest eigenvalues of a symmetric matrix A in mathbbS^n. This problem can be solved as a semidefinite program. The primal and dual forms are stated in Alizadeh [1]:","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"beginarrayllll textmaximize textTr(AY) - textTr(AW) textminimize kz + Tr(U) + Tr(V) \ntextsubject to textTr(Y + W) = k textsubject to zI + V - A succeq 0 \n 0 preceq Y preceq I zI + U + A succeq 0 \n 0 preceq W preceq I U V succeq 0\nendarray","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"where Y W are the variables of the primal and U V are the variables of the dual problem.","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"using LinearAlgebra, JuMP, COSMO, Random\nrng = Random.MersenneTwister(212)\n\nn = 10\nA = 5 .* randn(rng, 10, 10)\nA = Symmetric(A, :U)","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"We are interested in minimizing the sum of absolute values of the k=3 largest eigenvalues. Let's formulate the problem in JuMP with COSMO as the backend solver:","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"k = 3\nmodel = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"verbose\" => true));\n@variable(model, Y[1:n, 1:n], PSD);\n@variable(model, W[1:n, 1:n], PSD);\n\n@objective(model, Max, tr(A * Y) - tr(A * W));\n@constraint(model, tr(Y + W) == k);\n@constraint(model, Symmetric(I - Y) in PSDCone());\n@constraint(model, Symmetric(I - W) in PSDCone());\nstatus = JuMP.optimize!(model)","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"opt_objective = JuMP.objective_value(model)","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"Now, we can check the solution by computing the sum of the absolute value of the 3-largest eigenvalues:","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"k_λ_abs = sum(sort(abs.(eigen(A).values), rev = true)[1:k])","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/#Solve-the-dual","page":"Minimizing the sum of the k-largest λ","title":"Solve the dual","text":"","category":"section"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"Alternatively, we can solve the dual problem:","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"model = JuMP.Model(with_optimizer(COSMO.Optimizer, verbose=true));\n@variable(model, V[1:n, 1:n], PSD);\n@variable(model, U[1:n, 1:n], PSD);\n@variable(model, z);\n\n@objective(model, Min, k * z + tr(V) + tr(U));\n@constraint(model, Symmetric(z .* diagm(0 => ones(n)) + V - A) in PSDCone());\n@constraint(model, Symmetric(z .* diagm(0 => ones(n)) + U + A) in PSDCone());\nstatus = JuMP.optimize!(model)","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"opt_objective = JuMP.objective_value(model)","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"This gives the same result.","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/#Problem-with-A-as-variable","page":"Minimizing the sum of the k-largest λ","title":"Problem with A as variable","text":"","category":"section"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"Above problems are mostly helpful for illustrative purpose. It is obviously easier to find the sum of the k-largest eigenvalues by simply computing the eigenvalues of A. However, above results become useful if finding A itself is part of the problem. For example, assume we want to find a valid matrix A under the constraints: C textvec(A) = b with the minimum sum of absolute values of the k-largest eigenvalues. We can then solve the equivalent problem:","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"beginarrayll textminimize kz + Tr(U) + Tr(V) \n textsubject to C textvec(A) = b \n zI + V - A succeq 0 \n zI + U + A succeq 0 \n U V succeq 0\nendarray","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/#References","page":"Minimizing the sum of the k-largest λ","title":"References","text":"","category":"section"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"[1] Alizadeh - Interior point methods in semidefinite programming with applications to combinatorial optimization (1995)","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"This page was generated using Literate.jl.","category":"page"},{"location":"api/#API-Reference","page":"API Reference","title":"API Reference","text":"","category":"section"},{"location":"api/#Model","page":"API Reference","title":"Model","text":"","category":"section"},{"location":"api/","page":"API Reference","title":"API Reference","text":"COSMO.Model\nCOSMO.assemble!\nCOSMO.set!\nCOSMO.empty_model!\nCOSMO.warm_start_primal!\nCOSMO.warm_start_slack!\nCOSMO.warm_start_dual!","category":"page"},{"location":"api/#COSMO.Model","page":"API Reference","title":"COSMO.Model","text":"Model{T <: AbstractFloat}()\n\nInitializes an empty COSMO model that can be filled with problem data using assemble!(model, P, q,constraints; [settings, x0, s0, y0]).\n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.assemble!","page":"API Reference","title":"COSMO.assemble!","text":"assemble!(model, P, q, constraint(s); [settings, x0, y0, s0])\n\nAssembles a COSMO.Model with a cost function defind by P and q, and a number of constraints.\n\nThe positive semidefinite matrix P and vector q are used to specify the cost function of the optimization problem:\n\nmin 1/2 x'Px + q'x\ns.t. Ax + b ∈ C\n\nconstraints is a COSMO.Constraint or an array of COSMO.Constraint objects that are used to describe the constraints on x.\n\n\n\nThe optional keyword argument settings can be used to pass custom solver settings:\n\ncustom_settings = COSMO.Settings(verbose = true);\nassemble!(model, P, q, constraints, settings = custom_settings)\n\n\n\nThe optional keyword arguments x0 and y0 can be used to provide the solver with warm starting values for the primal variable x and the dual variable y.\n\nx_0 = [1.0; 5.0; 3.0]\nCOSMO.assemble!(model, P, q, constraints, x0 = x_0)\n\n\n\n\n\n","category":"function"},{"location":"api/#COSMO.set!","page":"API Reference","title":"COSMO.set!","text":"set!(model, P, q, A, b, convex_sets, [settings])\n\nSets model data directly based on provided fields.\n\n\n\n\n\n","category":"function"},{"location":"api/#COSMO.empty_model!","page":"API Reference","title":"COSMO.empty_model!","text":"empty_model!(model)\n\nResets all the fields of model to that of a model created with COSMO.Model() (apart from the settings).\n\n\n\n\n\n","category":"function"},{"location":"api/#COSMO.warm_start_primal!","page":"API Reference","title":"COSMO.warm_start_primal!","text":"warm_start_primal!(model, x0, [ind])\n\nProvides the COSMO.Model with warm starting values for the primal variable x. ind can be used to warm start certain components of x.\n\n\n\n\n\n","category":"function"},{"location":"api/#COSMO.warm_start_slack!","page":"API Reference","title":"COSMO.warm_start_slack!","text":"warm_start_slack!(model, s0, [ind])\n\nProvides the COSMO.Model with warm starting values for the primal slack variable s. ind can be used to warm start certain components of s.\n\n\n\n\n\n","category":"function"},{"location":"api/#COSMO.warm_start_dual!","page":"API Reference","title":"COSMO.warm_start_dual!","text":"warm_start_dual!(model, y0, [ind])\n\nProvides the COSMO.Model with warm starting values for the dual variable y. ind can be used to warm start certain components of y.\n\n\n\n\n\n","category":"function"},{"location":"api/#Constraints","page":"API Reference","title":"Constraints","text":"","category":"section"},{"location":"api/","page":"API Reference","title":"API Reference","text":"COSMO.Constraint\nCOSMO.ZeroSet\nCOSMO.Nonnegatives\nCOSMO.Box\nCOSMO.SecondOrderCone\nCOSMO.PsdCone\nCOSMO.PsdConeTriangle\nCOSMO.ExponentialCone\nCOSMO.DualExponentialCone\nCOSMO.PowerCone\nCOSMO.DualPowerCone","category":"page"},{"location":"api/#COSMO.Constraint","page":"API Reference","title":"COSMO.Constraint","text":"Constraint{T <: AbstractFloat}(A, b, convex_set_type, dim = 0, indices = 0:0)\n\nCreates a COSMO constraint: Ax + b ∈ convex_set.\n\nBy default the following convex set types are supported: ZeroSet, Nonnegatives, SecondOrderCone, PsdCone, PsdConeTriangle.\n\nExamples\n\njulia> COSMO.Constraint([1 0;0 1], zeros(2), COSMO.Nonnegatives)\nConstraint\nSize of A: (2, 2)\nConvexSet: COSMO.Nonnegatives{Float64}\n\nFor convex sets that require their own data, it is possible to pass the pass the instantiated object directly rather than the type name.\n\nExamples\n\njulia> COSMO.Constraint([1 0;0 1], zeros(2), COSMO.Box([-1.;-1.],[1.;1.]))\nConstraint\nSize of A: (2, 2)\nConvexSet: COSMO.Box{Float64}\n\n\n\nThe optional arguments dim and indices can be used to specify A and b for subparts of variable x. If x has dimension dim = 4, then x[2] and x[3] can be constrained to the zero cone in the following way:\n\nExamples\n\njulia> c = COSMO.Constraint([1 0;0 1], zeros(2), COSMO.ZeroSet, 4, 2:3)\nConstraint\nSize of A: (2, 4)\nConvexSet: COSMO.ZeroSet{Float64}\n\nNotice that extra columns of A have been added automatically.\n\njulia>Matrix(c.A)\n2×4 Array{Float64,2}:\n0.0 1.0 0.0 0.0\n0.0 0.0 1.0 0.0\n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.ZeroSet","page":"API Reference","title":"COSMO.ZeroSet","text":"ZeroSet(dim)\n\nCreates the zero set 0 ^dim of dimension dim. If x ∈ ZeroSet then all entries of x are zero.\n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.Nonnegatives","page":"API Reference","title":"COSMO.Nonnegatives","text":"Nonnegatives(dim)\n\nCreates the nonnegative orthant x in mathbbR^dim x ge 0 of dimension dim.\n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.Box","page":"API Reference","title":"COSMO.Box","text":"Box(l, u)\n\nCreates a box or intervall with lower boundary vector l in mathbbR^m cup -infty^m and upper boundary vectoru in mathbbR^mcup +infty^m.\n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.SecondOrderCone","page":"API Reference","title":"COSMO.SecondOrderCone","text":"SecondOrderCone(dim)\n\nCreates the second-order cone (or Lorenz cone) (tx) in mathrmR^dim x _2 leq t .\n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.PsdCone","page":"API Reference","title":"COSMO.PsdCone","text":"PsdCone(dim)\n\nCreates the cone of symmetric positive semidefinite matrices mathcalS_+^dim. The entries of the matrix X are stored column-by-column in the vector x of dimension dim. Accordingly X in mathbbS_+ Rightarrow x in mathcalS_+^dim, where X = textmat(x).\n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.PsdConeTriangle","page":"API Reference","title":"COSMO.PsdConeTriangle","text":"PsdConeTriangle(dim)\n\nCreates the cone of symmetric positive semidefinite matrices. The entries of the upper-triangular part of matrix X are stored in the vector x of dimension dim. A r times r matrix has r(r+1)2 upper triangular elements and results in a vector of mathrmdim = r(r+1)2.\n\nExamples\n\nThe matrix\n\nbeginbmatrix x_1 x_2 x_4 x_2 x_3 x_5 x_4 x_5 x_6 endbmatrix\n\nis transformed to the vector x_1 x_2 x_3 x_4 x_5 x_6^top with corresponding constraint PsdConeTriangle(6).\n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.ExponentialCone","page":"API Reference","title":"COSMO.ExponentialCone","text":"ExponentialCone(MAX_ITERS = 100, EXP_TOL = 1e-8)\n\nCreates the exponential cone mathcalK_exp = (x y z) mid y geq 0 ye^xy z cup (xyz) mid x leq 0 y = 0 z geq 0 \n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.DualExponentialCone","page":"API Reference","title":"COSMO.DualExponentialCone","text":"DualExponentialCone(MAX_ITERS::Int = 100, EXP_TOL = 1e-8)\n\nCreates the dual exponential cone mathcalK^*_exp = (x y z) mid x 0 -xe^yx leq e^1 z cup (0yz) mid y geq 0 z geq 0 \n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.PowerCone","page":"API Reference","title":"COSMO.PowerCone","text":"PowerCone(alpha::Float64, MAX_ITERS::Int = 20, POW_TOL = 1e-8)\n\nCreates the 3-d power cone mathcalK_pow = (x y z) mid x^alpha y^(1-alpha) geq z x geq 0 y geq 0 with 0 alpha 1\n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.DualPowerCone","page":"API Reference","title":"COSMO.DualPowerCone","text":"DualPowerCone(alpha::Float64, MAX_ITERS::Int = 20, POW_TOL = 1e-8)\n\nCreates the 3-d dual power cone mathcalK^*_pow = (u v w) mid left( fracualpharight)^alpha left( fracv1-alpharight)^(1-alpha) geq w u geq 0 v geq 0 with 0 alpha 1\n\n\n\n\n\n","category":"type"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"EditURL = \"../arbitrary_precision.jl\"","category":"page"},{"location":"literate/build/arbitrary_precision/#Arbitrary-Precision","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"","category":"section"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"COSMO allows you to solve problems with arbitrary floating-point precision, e.g. by using BigFloat problem data. To do this, the desired floating point type has to be consistent across the model COSMO.Model{<: AbstractFloat}, the input data and the (optional) settings object COSMO.Settings{<: AbstractFloat}. As an example, assume we want to solve the following quadratic program:","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"beginarrayll textminimize 12 x^top P x + q^top x \ntextsubject to l leq A x leq u\nendarray","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"where P = beginbmatrix 4 1 1 2endbmatrix, q = 1 1^top, A = beginbmatrix 1 1 1 0 0 1endbmatrix and l= 10 0^top, u=1 07 07^top. We start by creating the model with the desired precision:","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"using COSMO, LinearAlgebra, SparseArrays\nmodel = COSMO.Model{BigFloat}()","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"Next, we define the problem data as BigFloat arrays and create the constraint:","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"q = BigFloat[1; 1.]\nP = sparse(BigFloat[4. 1; 1 2])\nA = BigFloat[1. 1; 1 0; 0 1]\nl = BigFloat[1.; 0; 0]\nu = BigFloat[1; 0.7; 0.7]\n\nconstraint = COSMO.Constraint(A, zeros(BigFloat, 3), COSMO.Box(l, u))","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"Notice that the constraint type parameter is dependent on the input data. The same is true for the constraint set Box. Next, we define the settings","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"settings = COSMO.Settings{BigFloat}(verbose = true, kkt_solver = QdldlKKTSolver)","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"and assemble and solve the problem:","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"assemble!(model, P, q, constraint, settings = settings)\nresult = COSMO.optimize!(model);\nnothing #hide","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"Moreover, notice that when no type parameter is specified, all objects default to Float64:","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"model = COSMO.Model()","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"Two limitations to arbitrary precision:","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"Since we call the LAPACK function syevr for eigenvalue decompositions, we currently only support solving problems with PSD constraints in Float32 and Float64.\nWe suggest to use the pure Julia QDLDL linear system solver (kkt_solver = QdldlKKTSolver) when working with arbitrary precision types as some of the other available solvers don't support all available precisions.","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"note: Note\n","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"If you want to use COSMO directly with MathOptInterface, you can use: COSMO.Optimizer{<: AbstractFloat} as your optimizer. Again, the problem data precision of your MathOptInterface-model has to agree with the optimizer's precision.","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"This page was generated using Literate.jl.","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"EditURL = \"../../../examples/maxcut.jl\"","category":"page"},{"location":"examples/maxcut/#Maximum-Cut-Problem","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"","category":"section"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"We are interested in solving an approximation to the maximum cut problem using semidefinite programming. Consider the graph G(VE) with weights w_ij shown below:","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"# create the weighted adjacency matrix of the graph and plot the graph\nusing LinearAlgebra, Plots, GraphRecipes\nn = 4;\nW = zeros(n, n);\nW[1, 2] = 1; W[1, 4] = 8;\nW[2, 3] = 2; W[2, 4] = 10;\nW[3, 4] = 6;\nW = Symmetric(W)\ngraphplot(W, names = 1:n, edgelabel = W, x = [0; 1; 1; 0], y = [1; 1; 0; 0], fontsize = 12, nodeshape =:circle)","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"The maximum cut problem tries to find a cut or a partition of the graph's vertices into two complementary sets S and barS such that the total weight of the edges between the two sets is maximized. For this small graph the problem is trivial. The optimal solution is S = 123 and barS=4. Formally, this problem can be written as a mixed-integer optimisation problem:","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"beginarrayll textmaximize frac12 sum_i j w_ij(1 - y_i y_j)\ntextsubject to y_i in -1 1 quad forall i in V\nendarray","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"where y_i = 1 indicates that v_i in S and y_i = -1 indicates that v_i in barS. This problem is of interest in the field of integrated circuit design, where one tries to minimize the number of cross-layer connections in a circuit under layout constraints.","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"For more complicated graphs this problem quickly becomes hard to solve to optimality and in fact is known to be NP-hard. For this example we are interested in the randomized approximation algorithm that relaxes the problem to an SDP and was devised in Goemans and Williamson (1995) [1] (see for more details).","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"The approach can be divided into three steps:","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"Solve a relaxed SDP to obtain Y^*\nRecover an approximate solution V via a Cholesky factorisation Y^* = V^top V\nRound the approximate solution using a random vector r from the unit sphere.","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"The authors showed that this approach guarantees a solution of at least 087856 times the optimal solution.","category":"page"},{"location":"examples/maxcut/#Solving-the-primal-SDP","page":"Maximum Cut Problem","title":"Solving the primal SDP","text":"","category":"section"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"Before we formulate the SDP, let's compute the Laplacian matrix L:","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"using LinearAlgebra, Random, SparseArrays, COSMO, JuMP\n\nrng = Random.MersenneTwister(1)\n\n# Compute the Laplacian matrix of the graph\nL = diagm(0 => W * ones(n)) - W;\nnothing #hide","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"Given the Laplacian matrix L, we are looking for the optimal Y^* that solves the following SDP:","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"beginarrayll textmaximize frac14 langle L Y rangle \ntextsubject to Y_ii = 1 quad textfor i = 1dotsn\n Y in mathbfS_+^n\nendarray","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"Notice that the solution Y^* can be viewed as a correlation matrix. Let's solve the problem using COSMO and JuMP.","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"model = JuMP.Model(COSMO.Optimizer);\n@variable(model, Y[1:n, 1:n], PSD);\n@objective(model, Max, 1 / 4 * dot(L, Y));\n@constraint(model, [i = 1:n], Y[i, i] == 1.);\nJuMP.optimize!(model)","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"Yopt = JuMP.value.(Y);\nobj_val = JuMP.objective_value(model)","category":"page"},{"location":"examples/maxcut/#Solving-the-dual-SDP","page":"Maximum Cut Problem","title":"Solving the dual SDP","text":"","category":"section"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"Notice that the decision matrix Y is generally dense (as correctly classified in the solver output above). Therefore, we won't be able to utilize COSMO's chordal decomposition features. However, assuming strong duality, it turns out that we can also solve the dual problem, which is given by:","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"beginarrayll textminimize sum_i gamma_i \ntextsubject to textdiag(gamma) - frac14 L = S \n S in mathbfS_+^n\nendarray","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"As you can see, the matrix S is constrained to have zeros in places specified by the graph (Laplacian). Therefore, COSMO can try to decompose this SDP (see solver output) and speed up its algorithm:","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"model_dual = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"complete_dual\" => true));\n@variable(model_dual, γ[1:n]);\n@objective(model_dual, Min, sum(γ));\n@constraint(model_dual, lmi, Symmetric(-1/4 * L + diagm(γ)) in JuMP.PSDCone());\nJuMP.optimize!(model_dual)\nobj_val = JuMP.objective_value(model_dual)","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"The primal variable Y^* can be recovered from the dual variable associated with the LMI-constraint:","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"Yopt = dual(lmi)","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"To get the correct positive semidefinite dual variable, we have to enable PSD completion of the dual variable in COSMO. Now, that we have a solution we can perform the remaining steps in the approximation algorithm.","category":"page"},{"location":"examples/maxcut/#Cholesky-factorisation-of-Y","page":"Maximum Cut Problem","title":"Cholesky factorisation of Y","text":"","category":"section"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"Compute the Cholesky factorisation of Y = V^top V to find the unit vectors v_1 dots v_n.","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"factor = cholesky(Yopt, Val(true); check = false);\nV = Matrix((factor.P * factor.L)');\n# normalize columns\nfor i in 1:n\n V[:, i] ./= norm(V[:, i]);\nend","category":"page"},{"location":"examples/maxcut/#Rounding-the-approximate-solution","page":"Maximum Cut Problem","title":"Rounding the approximate solution","text":"","category":"section"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"It remains to round the unit vectors v_i using a random vector r with each component drawn from mathcalU(0 1), to obtain the y_i's:","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"r = rand(rng, n)\nr /= norm(r, 2)\ny = ones(n)\nfor i in 1:n\n dot(r, V[:, i]) <= 0 && (y[i] = -1)\nend\ny","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"For larger graphs this rounding step could be repeated several times to improve the rounding. As expected S = 1 2 3 and barS= 4.","category":"page"},{"location":"examples/maxcut/#References","page":"Maximum Cut Problem","title":"References","text":"","category":"section"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"[1] Goemans and Williamson - Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programs (1995)","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"[2] Post-processing code steps 2 and 3 from this JuMP example","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"This page was generated using Literate.jl.","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"EditURL = \"../../../examples/svm_primal.jl\"","category":"page"},{"location":"examples/svm_primal/#Support-Vector-Machine","page":"Support Vector Machine","title":"Support Vector Machine","text":"","category":"section"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"We are showing how to solve a support vector machine problem with COSMO (and JuMP).","category":"page"},{"location":"examples/svm_primal/#Generating-the-Dataset","page":"Support Vector Machine","title":"Generating the Dataset","text":"","category":"section"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"We want to classify the points in this example dataset with m = 100 samples and n = 2 features:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"using Distributions: MvNormal\nusing Plots, LinearAlgebra, SparseArrays, Random, Test","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"# Generate dataset\nrng = Random.MersenneTwister(123);\nnum_samples = 100;\nXpos = rand(rng, MvNormal([1.5, 1.5], 1.25), div(num_samples, 2))';\nXneg = rand(rng, MvNormal([-1.5, -1.5], 1.25), div(num_samples, 2))';\nypos = ones(div(num_samples, 2));\nyneg = -ones(div(num_samples, 2));\nnothing #hide","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"# Plot dataset\nplot(Xpos[:, 1], Xpos[:, 2], color = :red, st=:scatter, markershape = :rect, label = \"positive\", xlabel = \"x1\", ylabel = \"x2\")\nplot!(Xneg[:, 1], Xneg[:, 2], color = :blue, st=:scatter, markershape = :circle, label = \"negative\")","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"with samples (x_1 x_2 ldots x_m) in mathbbR^2 and labels y_i in -11.","category":"page"},{"location":"examples/svm_primal/#Solving-SVM-as-a-QP","page":"Support Vector Machine","title":"Solving SVM as a QP","text":"","category":"section"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"We want to compute the weights w and bias term b of the (soft-margin) SVM classifier:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"beginarrayll\n textminimize w^2 + lambda sum_i=1^m textmax(0 1 - y_i(w^top x_i - b))\nendarray","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"where lambda is a hyperparameter. This problem can be solved as a quadratic program. We can rewrite above problem into an optimisation problem in primal form by introducing the auxiliary slack variables t_i:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"t_i = textmax(0 1 - y_i(w^T x_i - b)) quad t_i geq 0","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"This allows us to write the problems in standard QP format:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"beginarrayll\n textminimize w^2 + lambda sum_i=1^m t_i\n textsubject to y_i (w^top x_i - b) geq 1 - t_i quad textfor i = 1ldots m\n t_i geq 0 quad textfor i = 1ldots m\nendarray","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"Next, we will remove the bias term b by adding an initial feature x_0 = -1 to each sample (now: n = 3):","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"X = [-ones(num_samples) [Xpos; Xneg]];\ny = [ypos; yneg];\nm, n = size(X)","category":"page"},{"location":"examples/svm_primal/#Modelling-in-JuMP","page":"Support Vector Machine","title":"Modelling in JuMP","text":"","category":"section"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"We can model this problem using JuMP and then hand it to COSMO:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"using JuMP, COSMO","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"λ = 1.0; # hyperparameter\nmodel = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"verbose\" => true));\n\n\n@variable(model, w[1:n]);\n@variable(model, t[1:m] >= 0.);\n@objective(model, Min, w' * w + λ * ones(m)' * t);\n@constraint(model, diagm(0 => y) * X * w .+ t .- 1 .>= 0);\nstatus = JuMP.optimize!(model)","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"The optimal weights w = w_0 w_1 w_2^top (where w_0 = b) are:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"w_opt = JuMP.value.(w)","category":"page"},{"location":"examples/svm_primal/#Plotting-the-hyperplane","page":"Support Vector Machine","title":"Plotting the hyperplane","text":"","category":"section"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"The separating hyperplane is defined by w^top x - b = 0. To plot the hyperplane, we calculate x_2 over a range of x_1 values:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"x_2 = (-w_1 x_1 - w_0) w_2 text where w_0 = b","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"x1 = -4:0.1:4;\nx2 = (-w_opt[2] * x1 .- w_opt[1]) / w_opt[3]\nplot!(x1, x2, label = \"SVM separator\", legend = :topleft)","category":"page"},{"location":"examples/svm_primal/#Modelling-with-COSMO","page":"Support Vector Machine","title":"Modelling with COSMO","text":"","category":"section"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"The problem can also be solved by transforming it directly into COSMO's problem format. Define COSMO`s x-variable to be x=w t^top and choose P, q, accordingly:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"P = blockdiag(spdiagm(0 => ones(n)), spzeros(m, m));\nq = [zeros(n); 0.5 * λ * ones(m)];\nnothing #hide","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"Next we transform the first constraint y_i (w^top x_i - b) geq 1 - t_i quad textfor i = 1ldots m into COSMO's constraint format: Ax + b in mathcalK.","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"A1 = [(spdiagm(0 => y) * X) spdiagm(0 => ones(m))];\nb1 = -ones(m);\ncs1 = COSMO.Constraint(A1, b1, COSMO.Nonnegatives);\nnothing #hide","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"It remains to specify the constraint t_i geq 0 quad textfor i = 1ldots m:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"A2 = spdiagm(0 => ones(m));\nb2 = zeros(m);\ncs2 = COSMO.Constraint(A2, b2, COSMO.Nonnegatives, m+n, n+1:m+n);\nnothing #hide","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"Create, assemble and solve the COSMO.Model:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"model2 = COSMO.Model();\nassemble!(model2, P, q, [cs1; cs2]);\nresult2 = COSMO.optimize!(model2);\nw_opt2 = result2.x[1:3];\n@test norm(w_opt2 - w_opt, Inf) < 1e-3","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"This page was generated using Literate.jl.","category":"page"},{"location":"jump/#JuMP-Interface","page":"JuMP Interface","title":"JuMP Interface","text":"","category":"section"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"Our JuMP interface allows you to describe and modify your optimisation problem with JuMP and use COSMO as the backend solver. The interface is defined in /src/MOIWrapper.jl.","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"note: Note\nWe assume here that the latest JuMP release (~0.21.0) is used.","category":"page"},{"location":"jump/#Use-COSMO","page":"JuMP Interface","title":"Use COSMO","text":"","category":"section"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"To specify COSMO as the solver for your JuMP model, load the solver module with using COSMO and then pass a COSMO.Optimizer when initialising the JuMP model:","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"m = JuMP.Model(COSMO.Optimizer);","category":"page"},{"location":"jump/#Specify-Solver-Settings","page":"JuMP Interface","title":"Specify Solver Settings","text":"","category":"section"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"Solver-specific settings can be passed using the optimizer_with_attributes() function. For example, if you want to adjust the maximum number of iterations and turn on verbose printing use","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"m = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"max_iter\" => 5000, \"verbose\" => true));","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"Note that the attributes are passed as key-value pairs and the keys are strings. This is slightly different to using the native COSMO interface. Equivalently, one can also use:","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"m = JuMP.Model(COSMO.Optimizer);\nset_optimizer_attribute(m, \"max_iter\", 5000)\nset_optimizer_attribute(m, \"verbose\", true)","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"The full list of available settings can be found in the Settings section. All of them should be compatible with JuMP.","category":"page"},{"location":"jump/#Results","page":"JuMP Interface","title":"Results","text":"","category":"section"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"After solving the problem the result can be obtained using the standard JuMP commands. To see if the optimisation was successful use","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"JuMP.termination_status(m)\nJuMP.primal_status(m)","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"If a solution is available, the optimal objective value can be queried using","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"JuMP.objective_value(m)","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"and the value of a decision variable x can be obtained with","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"JuMP.value.(x)","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"To query the number of iterations of the algorithm use:","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"iter = MOI.get(m, COSMO.ADMMIterations())","category":"page"},{"location":"jump/#Feasibility-of-solution","page":"JuMP Interface","title":"Feasibility of solution","text":"","category":"section"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"In case JuMP.termination_status(m) is MOI.ITERATION_LIMIT, JuMP.primal_status(m) (resp. JuMP.dual_status(m)) should be checked to determine the feasibility status of the primal (resp. dual) solutions. A primal (resp. dual) solution is considered feasible (resported as MOI.FEASIBLE_POINT) if","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"r epsilon_textabs + sepsilon_textrel","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"where","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"r is the primal (resp. dual) residual r_prim = lVert Ax + s - b rVert_infty","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"(resp. r_dual = lVert P x + q - A^top * μ rVert_infty),","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"s is the primal (resp. dual) scaling of this residual","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"max_norm_prim = max(lVert Ax rVert_infty lVert s rVert_infty lVert b rVert_infty) (resp. max_norm_dual = max(lVert P x rVert_infty lVert q rVert_infty lVert A^top * μ rVert_infty)).","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"epsilon_textabs is the optimizer attribute eps_abs and\nepsilon_textrel is the optimizer attribute eps_rel.","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"A primal (resp. dual solution) is considered nearly feasible (resported as MOI.NEARLY_FEASIBLE_POINT) if it is not feasible but the same inequality is satisfied where the right-hand side is multiplied by the optimizer attribute nearly_ratio.","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"Otherwise, the solution is considered infeasible (resported as MOI.INFEASIBLE_POINT). Note that this does not mean that the problem is infeasible, it only means that the current primal (resp. dual) solution is infeasible.","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"Note that, in case JuMP.termination_status(m) is MOI.ITERATION_LIMIT, the feasibility of the solution is checked with the current optimizer attributes, eps_abs, eps_rel and nearly_ratio. This means that if they are changed after optimize!, the status may change! This can be used to check whether the solution is feasible up to a relaxed tolerance. For instance, the following could happen:","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"optimize!(m)\nprimal_status(m) # MOI.INFEASIBLE_POINT\n# This means that `r_prim >= nearly_ratio * (eps_abs * max_norm_prim * eps_rel)`\nset_optimizer_attribute(m, \"eps_abs\", 1e-4)\nprimal_status(m) # MOI.NEARLY_FEASIBLE_POINT\n# This means that `1 <= r_prim / (eps_abs * max_norm_prim * eps_rel) < nearly_ratio`","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"The values of r_prim, max_norm_prim, r_dual and max_norm_dual can also be accessed as the fields of the res_info struct that can be obtained as follows","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"res_info = MOI.get(m, COSMO.RawResult()).info","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"Then, the feasibility can either be checked manually as in","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"res_info.r_prim < eps_abs + res_info.max_norm_prim * eps_rel","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"or using one of the following:","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"COSMO.is_primal_feasible(res_info, eps_abs, eps_rel)\nCOSMO.is_primal_nearly_feasible(res_info, eps_abs, eps_rel)\nCOSMO.is_dual_feasible(res_info, eps_abs, eps_rel)\nCOSMO.is_dual_nearly_feasible(res_info, eps_abs, eps_rel)","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"For more information on how to use JuMP check the JuMP documentation.","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"EditURL = \"../portfolio_model_updates.jl\"","category":"page"},{"location":"literate/build/portfolio_model_updates/#Model-Updates","page":"Model Updates","title":"Model Updates","text":"","category":"section"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"When a pareto-optimal front in portfolio optimisation is computed for different risk-aversion parameters gamma, we are repeatedly solving the same model with only a slight change in the cost function. This allows us to reuse the KKT factorisation from previous solves and recompute the solution very efficiently, see Algorithm. For more information about solving portfolio optimsiation problems with COSMO see Portfolio Optimisation. We are planning to solve the following QP for different values of gamma:","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"beginarrayll textminimize x^top D x + y^top y - gamma^-1 mu^top x \ntextsubject to y = F^top x \n 1^top x = d + 1^top x^0 \n x geq 0\nendarray","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"As you can see gamma enters only in the linear part of the cost function, i.e. our vector q. Let's generate some example data:","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"using LinearAlgebra, SparseArrays, Random, COSMO\n\nrng = Random.MersenneTwister(1)\nk = 5; # number of factors\nn = k * 10; # number of assets\nD = spdiagm(0 => rand(rng, n) .* sqrt(k))\nF = sprandn(rng, n, k, 0.5); # factor loading matrix\nμ = (3 .+ 9. * rand(rng, n)) / 100. # expected returns between 3% - 12%\nd = 1 # we are starting from all cash\nx0 = zeros(n);\nnothing #hide","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"We want to solve this problem for the following risk-parameters:","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"gammas = [ 0.001, 0.01, 0.1, 0.5, 1., 3., 10, 100, 1000]","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"Let's solve the problem for the first parameter:","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"# x = (x, y)\nγ = gammas[1]\nP = blockdiag(D, spdiagm(0 => ones(k)))\nq = [-μ ./ (2 * γ); zeros(k)];\n\nAeq = [ transpose(F) -diagm(0 => ones(k));\n ones(1, n) zeros(1, k)]\nbeq = [zeros(k); -1.0]\nAineq = [spdiagm(0 => ones(n)) spzeros(n, k)]\n\ncs1 = COSMO.Constraint(Aeq, beq, COSMO.ZeroSet);\ncs2 = COSMO.Constraint(Aineq, zeros(n), COSMO.Nonnegatives);\ncosmo_settings = COSMO.Settings(verbose = false)\n\n# solve the problem once\nmodel = COSMO.Model();\nassemble!(model, P, q, [cs1; cs2], settings = cosmo_settings);\nresult = COSMO.optimize!(model);\nx_opt = view(result.x, 1:n);\ny_opt = view(result.x, n+1:n+k);\n\n# allocate some space for results\nrisks = zeros(length(gammas));\nreturns = zeros(length(gammas));\nreturns[1] = dot(μ, x_opt);\nrisks[1] = sqrt(dot(y_opt, y_opt));\nnothing #hide","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"After a successful solve, we can use the update!(model, q = nothing, b = nothing) function to update our model and re-solve the problem efficiently:","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"# solve the problem efficiently for the remaining γ using model updates\nfunction solve_repeatedly!(returns, risks, gammas, model, k, n, μ)\n for (i, γ) in enumerate(gammas[2:end])\n q_new = [-μ ./ (2 * γ); zeros(k)];\n\n # here we use the update! function to change the model\n update!(model, q = q_new)\n result = COSMO.optimize!(model);\n\n x_opt = view(result.x, 1:n)\n y_opt = view(result.x, n+1:n+k)\n returns[i + 1] = dot(μ, x_opt)\n risks[i + 1] = sqrt(dot(y_opt, y_opt))\n end\nend\nsolve_repeatedly!(returns, risks, gammas, model, k, n, μ);\nnothing #hide","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"We can now plot the risk-return trade-off curve:","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"using Plots\nPlots.plot(risks, returns, xlabel = \"Standard deviation (risk)\", ylabel = \"Expected return\", title = \"Risk-return trade-off for efficient portfolios\", legend = false)","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"This page was generated using Literate.jl.","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"EditURL = \"../../../examples/portfolio_optimisation.jl\"","category":"page"},{"location":"examples/portfolio_optimisation/#Portfolio-Optimisation","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"","category":"section"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"We consider a single-period Markowitz portfolio optimisation example. Assume that we have a portfolio with n assets at the beginning of time period t. Given some forecasts on risks and expected returns we try to find the optimal trade vector that rebalances the portfolio to achieve a good balance between expected risk (variance) x^top Sigma x and returns mu^top x. In it's most simple form we want to solve:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"beginarrayll textmaximize mu^top x - gamma (x^top Sigma x)\ntextsubject to 1^top x = d + 1^top x^0 \n x geq 0\nendarray","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"with variable x in mathbfR^n, mu forecasted (expected) returns, gamma 0 risk aversion parameter x^0_i represents the initial investment in asset i and d represents the cash reserve. Consequently, the equality constraint tells us that the sum of the new allocation vector x has to equal the initial allocation plus the cash reserve. Furthermore, the covariance matrix of our risk model is given by Sigma in mathbfS_+^n. Here we assume a factor risk model, i.e. we can write Sigma as Sigma = D + F F^top where D is diagonal and the factor matrix F has a lower rank k n. This approach allows us to reduce the number of nonzeros in the problem. Furthermore, note that we don't consider shortselling in this example. Let's generate some problem data:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"using LinearAlgebra, SparseArrays, Random, COSMO, JuMP, Test\n\n# generate the data\nrng = Random.MersenneTwister(1)\nk = 5; # number of factors\nn = k * 10; # number of assets\nD = spdiagm(0 => rand(rng, n) .* sqrt(k))\nF = sprandn(rng, n, k, 0.5); # factor loading matrix\nμ = (3 .+ 9. * rand(rng, n)) / 100. # expected returns between 3% - 12%\nγ = 1.0; # risk aversion parameter\nd = 1 # we are starting from all cash\nx0 = zeros(n);\nnothing #hide","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"We can now write the problem as a QP:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"beginarrayll textminimize x^top D x + y^top y - gamma^-1 mu^top x \ntextsubject to y = F^top x \n 1^top x = d + 1^top x^0 \n x geq 0\nendarray","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"Before considering other effects, let's create the model in JuMP and solve it using COSMO:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"model = JuMP.Model(COSMO.Optimizer);\n@variable(model, x[1:n]);\n@variable(model, y[1:k]);\n@objective(model, Min, x' * D * x + y' * y - 1/γ * μ' * x);\n@constraint(model, y .== F' * x);\n@constraint(model, sum(x) == d + sum(x0));\n@constraint(model, x .>= 0);\nJuMP.optimize!(model)","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"After solving the problem, we can calculate the expected return and risk sigma= sqrtx^* top Sigma x^*:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"x_opt = JuMP.value.(x);\ny_opt = JuMP.value.(y);\nexpected_return_basic = dot(μ, x_opt)","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"expected_risk_basic = sqrt(dot(y_opt, y_opt))","category":"page"},{"location":"examples/portfolio_optimisation/#Using-standard-deviation-in-the-model","page":"Portfolio Optimisation","title":"Using standard deviation in the model","text":"","category":"section"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"It is pointed out in [1] that above problem formulation can lead to numerical problems, e.g. if Sigma is not strictly positive semidefinite. Another option is to formulate the risk constraint in terms of the standard deviation M^top x where M M^top = D + F F^top and bound it using a second-order cone constraint:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"beginarrayll textminimize - mu^top x \ntextsubject to M^top x leq gamma \n 1^top x = d + 1^top x^0 \n x geq 0\nendarray","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"Mt = [D.^0.5; F']\nmodel = JuMP.Model(COSMO.Optimizer);\n@variable(model, x[1:n]);\n@objective(model, Min, - μ' * x);\n@constraint(model, [γ; Mt * x] in SecondOrderCone()); # ||M'x|| <= γ\n@constraint(model, sum(x) == d + sum(x0));\n@constraint(model, x .>= 0);\nJuMP.optimize!(model)","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"Note that the result is different from the example above because gamma scales the problem in a different way. Here it can be seen as an upper bound on the standard deviation of the portfolio.","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"x_opt = JuMP.value.(x);\nexpected_return = dot(μ, x_opt)","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"Let us verify that the bound holds:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"@test norm(Mt * x_opt) <= γ","category":"page"},{"location":"examples/portfolio_optimisation/#Pareto-optimal-front","page":"Portfolio Optimisation","title":"Pareto-optimal front","text":"","category":"section"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"The above portfolio optimisation approach yields the optimal expected return for a given level of risk. The result is obviously impacted by the risk aversion gamma parameter. To visualise the trade-off and present the investor with an efficient Pareto optimal portfolio for their risk appetite we can compute the optimal portfolio for many choices of gamma and plot the corresponding risk-return trade-off curve.","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"gammas = [ 0.001, 0.01, 0.1, 0.5, 1., 3., 10, 100, 1000]\nrisks = zeros(length(gammas))\nreturns = zeros(length(gammas))\nmodel = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"verbose\" => false));\n@variable(model, x[1:n]);\n@variable(model, y[1:k]);\n@objective(model, Min, x' * D * x + y' * y - 1/γ * μ' * x);\n@constraint(model, y .== F' * x);\n@constraint(model, sum(x) == d + sum(x0));\n@constraint(model, x .>= 0);\n\n# solve the same problem for different values of γ\nfor (k, gamma) in enumerate(gammas)\n coeff = - 1/gamma * μ\n JuMP.set_objective_coefficient.(model, x, coeff)\n JuMP.optimize!(model)\n local x_opt = JuMP.value.(x);\n local y_opt = JuMP.value.(y);\n returns[k] = dot(μ, x_opt)\n risks[k] = sqrt(dot(y_opt, y_opt))\nend","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"We can now plot the risk-return trade-off curve:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"using Plots\nPlots.plot(risks, returns, xlabel = \"Standard deviation (risk)\", ylabel = \"Expected return\", title = \"Risk-return trade-off for efficient portfolios\", legend = false)","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"note: Note\nWhen the model is updated in JuMP as above the JuMP.model is copied in full to COSMO. We are trying to improve the interface with respect to model updates in the future. Until then you can use Model Updates in COSMOs native interface.","category":"page"},{"location":"examples/portfolio_optimisation/#Transaction-costs","page":"Portfolio Optimisation","title":"Transaction costs","text":"","category":"section"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"In the model above we assume that trading the assets is free and does not impact the market. However, this is clearly not the case in reality. To make the example more realistic consider the following cost c_j associated with the trade δ_j = x_j - x_j^0:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"c_j(delta_j) = a_j delta_j + b_j delta_j^32","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"where the first term models the bid-ask spread and broker fees for asset j. The second term models the impact on the market that our trade has. This is obviously only a factor if the volume of our trade is significant. The constant b_j is a function of the total volume traded in the considered time periode and the price volatility of the asset and has to be estimated by the trader. To make this example simple we consider the same coefficients a and b for every asset. The delta_j^32 term can be easily modeled using a power cone constraint mathcalK_pow = (x y z) mid x^alpha y^(1-alpha) geq z x geq 0 y geq 0 0 leq alpha leq 1 . In fact this can be used to model any market impact function with exponent greater than 1. We can write the total transaction cost a^top s + b^top t where s_j bounds the absolute value of delta_j and t_j is used to bound the term x_j - x_j^0^32 leq t_j using a power cone formulation: (t_j 1 x_j - x_j^0) in mathcalK_pow(23).","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"a = 1e-3\nb = 1e-1\nγ = 1.0;\nmodel = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"eps_abs\" => 1e-5, \"eps_rel\" => 1e-5));\n@variable(model, x[1:n]);\n@variable(model, y[1:k]);\n@variable(model, t[1:n]);\n@variable(model, s[1:n]);\n@objective(model, Min, x' * D * x + y' * y - 1/γ * μ' * x);\n@constraint(model, y .== F' * x);\n@constraint(model, x .>= 0);\n\n# transaction costs\n@constraint(model, sum(x) + a * sum(s) + b * sum(t) == d + sum(x0) );\n@constraint(model, [i = 1:n], x[i] - x0[i] <= s[i]); # model the absolute value with slack variable s\n@constraint(model, [i = 1:n], x0[i] - x[i] <= s[i]);\n@constraint(model, [i = 1:n], [t[i], 1, x[i] - x0[i]] in MOI.PowerCone(2/3));\nJuMP.optimize!(model)","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"Let's look at the expected return and the total transaction cost:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"x_opt = JuMP.value.(x);\ny_opt = JuMP.value.(y);\ns_opt = JuMP.value.(s);\nt_opt = JuMP.value.(t);\nexpected_return = dot(μ, x_opt)","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"expected_risk = dot(y_opt, y_opt)","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"transaction_cost = a * sum(s_opt) + b * sum( t_opt)","category":"page"},{"location":"examples/portfolio_optimisation/#References","page":"Portfolio Optimisation","title":"References","text":"","category":"section"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"[1] Mosek Case Studies","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"This page was generated using Literate.jl.","category":"page"},{"location":"getting_started/#Getting-Started","page":"Getting Started","title":"Getting Started","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"This user guide describes the basic structures and functions to define an optimisation problem, to solve the problem and to analyse the result. If you want to use JuMP to describe the problem, see the JuMP Interface section.","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"COSMO solves optimisation problems in the following format:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"beginarrayll textminimize textstylefrac12x^top Px + q^top x textsubject to Ax + s = b s in mathcalK endarray","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"with decision variables x in mathbbR^n, s in mathbbR^m and data matrices P=P^top succeq 0, q in mathbbR^n, A in mathbbR^m times n, and b in mathbbR^m. The convex set mathcalK is a composition of convex sets and cones.","category":"page"},{"location":"getting_started/#Model","page":"Getting Started","title":"Model","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"The problem data, user settings and workspace variables are all stored in a Model. To get started define an empty model:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"model = COSMO.Model()","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"To initialize the model with an optimisation problem we need to define three more things:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"the objective function, i.e. the matrix P and the vector q in frac12x^top P x + q^top x\nan array of constraints\na Settings object that specifies how COSMO solves the problem (optional)","category":"page"},{"location":"getting_started/#Objective-Function","page":"Getting Started","title":"Objective Function","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"To set the objective function of your optimisation problem simply define the square positive semidefinite matrix P in mathrmR^ntimes n and the vector q in mathrmR^n. You might have to transform your optimisation problem into a solver compatible format for this step.","category":"page"},{"location":"getting_started/#Constraints","page":"Getting Started","title":"Constraints","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"The COSMO interface expects constraints to have the form A_i x + b_i in mathcalK_i, where mathcalK_i is one of the convex sets defined below:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Convex Set Description\nZeroSet The set 0 ^dim that contains the origin\nNonnegatives The nonnegative orthant x in mathbbR^dim x_i ge 0 forall i=1dotsmathrmdim \nBox(l, u) The hyperbox x in mathbbR^dim l leq x leq u with vectors l in mathbbR^dim cup -infty and u in mathbbR^dim cup +infty\nSecondOrderCone The second-order (Lorenz) cone (tx) in mathbbR^dim x_2 leq t \nPsdCone The vectorized positive semidefinite cone mathcalS_+^dim. x is the vector obtained by stacking the columns of the positive semidefinite matrix X, i.e. X in mathbbS^sqrtdim_+ rarr textvec(X) = x in mathcalS_+^dim\nPsdConeTriangle The vectorized positive semidefinite cone mathcalS_+^dim. x is the vector obtained by stacking the columns of the upper triangular part of the positive semidefinite matrix X, i.e. X in mathbbS^d_+ rarr textsvec(X) = x in mathcalS_+^dim where d=sqrt14 + 2 textdim - 12\nExponentialCone The exponential cone mathcalK_exp = (x y z) mid y geq 0 ye^xy z cup (xyz) mid x leq 0 y = 0 z geq 0 \nDualExponentialCone The dual exponential cone mathcalK^*_exp = (x y z) mid x 0 -xe^yx leq e^1 z cup (0yz) mid y geq 0 z geq 0 \nPowerCone(alpha) The 3d power cone mathcalK_pow = (x y z) mid x^alpha y^(1-alpha) geq z x geq 0 y geq 0 with 0 alpha 1\nDualPowerCone(alpha) The 3d dual power cone mathcalK^*_pow = (u v w) mid left( fracualpharight)^alpha left( fracv1-alpharight)^(1-alpha) geq w u geq 0 v geq 0 with 0 alpha 1","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"The constructor for a constraint expects a matrix A, a vector b and a convex_set.","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Lets consider a problem with a decision variable x in mathbbR^5. Suppose we want to create the two constraint x_2 + 5 geq 0 and x_3 - 3 geq 0. We can do this either by creating two constraints and adding them to an array:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":" constraint1 = COSMO.Constraint([0.0 1.0 0.0 0.0 0.0], 5.0, COSMO.Nonnegatives)\n constraint2 = COSMO.Constraint([0.0 0.0 1.0 0.0 0.0], -3.0, COSMO.Nonnegatives)\n constraints = [constraint1; constraint2]","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"The second option is to include both in one constraint:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"constraint1 = COSMO.Constraint([0.0 1.0 0.0 0.0 0.0; 0.0 0.0 1.0 0.0 0.0], [5.0; -3.0], COSMO.Nonnegatives)","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Another way to construct the constraint is to used the optional arguments dim, the dimension of x, and indices, the elements of x that appear in the constraint. When specifying these arguments, A and b only refer to the elements of x in indices:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"constraint1 = COSMO.Constraint([1.0 0.0; 0.0 1.0], [5.0; -3.0], COSMO.Nonnegatives, 5, 2:3)","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Consider as a second example the positive semidefinite constraint on a matrix X in mathbbS_+^3. Our decision variable is the vector x obtained by stacking the columns of X. We can specify the constraint on x in the following way:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"I_9 x + 0_9 in mathcalS_+^9","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"or in Julia:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"constraint1 = COSMO.Constraint(Matrix(1.0I, 9, 9), zeros(9), COSMO.PsdCone)","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Several constraints can be combined in an array:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"constraints = [constraint_1, constraint_2, ..., constraint_N]","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"It is usually enough to pass the convex_set as a type. However, some convex sets like Box, PowerCone and DualPowerCone require more information to be created. In that case you have to pass an object to the constructor, e.g.","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"l = -ones(2)\nu = ones(2)\nconstraint = COSMO.Constraint(Matrix(1.0I, 2, 2), zeros(2), COSMO.Box(l, u))","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"or in the case of a power Cone you specify the alpha:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"constraint = COSMO.Constraint(Matrix(1.0I, 3, 3), zeros(3), COSMO.PowerCone(0.6))","category":"page"},{"location":"getting_started/#Settings","page":"Getting Started","title":"Settings","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"The solver settings are stored in a Settings object and can be adjusted by the user. To create a Settings object just call the constructor:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"COSMO.Settings","category":"page"},{"location":"getting_started/#COSMO.Settings","page":"Getting Started","title":"COSMO.Settings","text":"COSMO.Settings{T}(; kwargs) where {T <: AbstractFloat}\n\nCreates a COSMO settings object that is used to pass user settings to the solver.\n\nArgument Description Values (default)\nrho ADMM rho step 0.1\nsigma ADMM sigma step 1e-6\nalpha Relaxation parameter 1.6\neps_abs Absolute residual tolerance 1e-5\neps_rel Relative residual tolerance 1e-5\nnearly_ratio Residual tolerance ratio between MOI.NEARLY_FEASIBLE_POINT and MOIFEASIBLE_POINT 100\neps_prim_inf Primal infeasibility tolerance 1e-5\neps_dual_inf Dual infeasibility tolerance 1e-5\nmax_iter Maximum number of iterations 5000\nverbose Verbose printing false\nverbose_timing Verbose timing false\nkkt_solver Linear System solver QdldlKKTSolver\ncheck_termination Check termination interval 25\ncheck_infeasibility Check infeasibility interval 40\nscaling Number of scaling iterations 10\nadaptive_rho Automatic adaptation of step size parameter true\nadaptiverhomax_adaptions Max number of rho adaptions typemax(Int64) (deactivated)\ndecompose Activate to decompose chordal psd constraints true\ncomplete_dual Activate to complete the dual variable after decomposition false\nmerge_strategy Choose a strategy for clique merging CliqueGraphMerge\ncompact_transformation Choose how a decomposed problem is transformed true\ntime_limit Set solver time limit in s 0 (deactivated)\naccelerator Acceleration scheme AndersonAccelerator{T, Type2{QRDecomp}, RestartedMemory, NoRegularizer}\naccelerator_activation Accelerator activation ImmediateActivation\nsafeguard Accelerator safeguarding true\nsafeguard_tol Safeguarding tolerance 2.0\n\n\n\n\n\n","category":"type"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"To adjust those values, either pass your preferred option and parameter as a key-value pair to the constructor or edit the corresponding field afterwards. For example if you want to enable verbose printing and increase the solver accuracy, you can type","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"settings = COSMO.Settings(verbose = true, eps_abs = 1e-5, eps_rel = 1e-5)\n# the following is equivalent\nsettings = COSMO.Settings()\nsettings.verbose = true\nsettings.eps_abs = 1e-5\nsettings.eps_rel = 1e-5","category":"page"},{"location":"getting_started/#Assembling-the-model","page":"Getting Started","title":"Assembling the model","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Once the objective function and an array of constraints have been defined, we can assemble the model with","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"COSMO.assemble!(model, P, q, constraints)","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"This simply sets the corresponding variables in the model and transforms the array of constraints into the problem format defined at the top of the page.","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"If you want to change the default settings, you can pass your settings object custom_settings to the assemble! function:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"COSMO.assemble!(model, P, q, constraints, settings = custom_settings)","category":"page"},{"location":"getting_started/#Warm-starting","page":"Getting Started","title":"Warm starting","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"One of the advantages of ADMM-based solvers is that they can be easily warm started. By providing starting values for the primal variable x and/or the dual variable y in the vicinity of their optimal values, the number of iterations to convergence can often be dramatically decreased.","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Consider the case where you have a decision variable x in mathbbR^3 and a dual variable y in mathbbR^2. Assume you expect their optimal values to be close to x_0 = (1 5 3) and y_0 = (1 2). You can pass these values when assembling the model.","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"x_0 = [1.0; 5.0; 3.0]\ny_0 = [1.0; 2.0]\nCOSMO.assemble!(model, P, q, constraints, x0 = x_0, y0 = y_0)","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Another option is to use","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"COSMO.assemble!(model, P, q, constraints)\nwarm_start_primal!(model, x_0)\nwarm_start_dual!(model, y_0)","category":"page"},{"location":"getting_started/#Solving","page":"Getting Started","title":"Solving","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"After the model has been assembled, we can solve the problem by typing","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"results = COSMO.optimize!(model)","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Once the solver algorithm terminates, it will return a Results object that gives information about the status of the solver. If successful, it contains the optimal objective value and optimal primal and dual variables. For more information see the following section.","category":"page"},{"location":"getting_started/#Results","page":"Getting Started","title":"Results","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"After attempting to solve the problem, COSMO will return a result object with the following fields:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"COSMO.Result\nCOSMO.ResultInfo","category":"page"},{"location":"getting_started/#COSMO.Result","page":"Getting Started","title":"COSMO.Result","text":"Result{T <: AbstractFloat}\n\nObject returned by the COSMO solver after calling optimize!(model). It has the following fields:\n\nFieldname Type Description\nx Vector{T} Primal variable\ny Vector{T} Dual variable\ns Vector{T} (Primal) set variable\nobj_val T Objective value\niter Int Total number of ADMM iterations (incl. safeguarding_iter)\nsafeguarding_iter Int Number of iterations due to safeguarding of accelerator\nstatus Symbol Solution status\ninfo COSMO.ResultInfo Struct with more information\ntimes COSMO.ResultTimes Struct with several measured times\n\n\n\n\n\n","category":"type"},{"location":"getting_started/#COSMO.ResultInfo","page":"Getting Started","title":"COSMO.ResultInfo","text":"ResultInfo{T <: AbstractFloat}\n\nObject that contains further information about the primal residual, the dual residuals and the rho updates.\n\n\n\n\n\n","category":"type"},{"location":"getting_started/#Status-Codes","page":"Getting Started","title":"Status Codes","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"COSMO will return one of the following statuses:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Status Code Description\n:Solved An optimal solution was found\n:Unsolved Default value\n:Max_iter_reached Solver reached iteration limit (set with Settings.max_iter)\n:Time_limit_reached Solver reached time limit (set with Settings.time_limit)\n:Primal_infeasible Problem is primal infeasible\n:Dual_infeasible Problem is dual infeasible","category":"page"},{"location":"getting_started/#Timings","page":"Getting Started","title":"Timings","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"If settings.verbose_timing is set to true, COSMO will report the following times in result.times:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"COSMO.ResultTimes","category":"page"},{"location":"getting_started/#COSMO.ResultTimes","page":"Getting Started","title":"COSMO.ResultTimes","text":"ResultTimes\n\nPart of the Result object returned by the solver. ResultTimes contains timing results for certain parts of the algorithm:\n\nTime Name Description\nsolver_time Total time used to solve the problem\nsetup_time Setup time = graph_time + init_factor_time + scaling_time\nscaling_time Time to scale the problem data\ngraph_time Time used to perform chordal decomposition\ninit_factor_time Time used for initial factorisation of the system of linear equations\nfactor_update_time Sum of times used to refactor the system of linear equations due to rho\niter_time Time spent in iteration loop\nproj_time Time spent in projection functions\npost_time Time used for post processing\nupdate_time Time spent in the update! function of the accelerator\naccelerate_time Time spent in the accelerate! function of the accelerator\n\nBy default COSMO only measures solver_time, setup_time and proj_time. To measure the other times set verbose_timing = true.\n\n\n\n\n\n","category":"type"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"It holds: solver_time = setup_time+ iter_time + factor_update_time + post_time,","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"setup_time = graph_time+ init_factor_time + scaling_time,","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"proj_time is a subset of iter_time.","category":"page"},{"location":"getting_started/#Updating-the-model","page":"Getting Started","title":"Updating the model","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"In some cases we want to solve a large number of similar models. COSMO allows you to update the model problem data vectors q and b after the first call of optimize!(). After changing the problem data, COSMO can reuse the factorisation step of the KKT matrix from the previous problem which can save a lot of time in the case of LPs and QPs.","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"COSMO.update!","category":"page"},{"location":"getting_started/#COSMO.update!","page":"Getting Started","title":"COSMO.update!","text":"update!(model, q = nothing, b = nothing)\n\nUpdates the model data for b or q. This can be done without refactoring the KKT matrix. The vectors will be appropriatly scaled.\n\n\n\n\n\n","category":"function"},{"location":"method/#Method","page":"Method","title":"Method","text":"","category":"section"},{"location":"method/","page":"Method","title":"Method","text":"This section describes COSMO's underlying ADMM algorithm and how the user can use the settings to adjust this algorithm. For a more detailed explanation take a look at the associated publication in Citing COSMO.","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"COSMO solves problems with quadratic objective function and a number of conic constraints in the following form:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"beginarrayll textminimize textstylefrac12x^top Px + q^top x textsubject to Ax + s = b s in mathcalK endarray","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"with primal decision variable x in mathbbR^n, primal slack variable s in mathbbR^m. The objective function is defined by positive semidefinite matrix P=P^top succeq 0 and vector q in mathbbR^n. The constraints are defined by matrix A in mathbbR^m times n, vector b in mathbbR^m and a non-empty, closed convex set mathcalK. The convex set itself can be a Cartesian product of convex sets in the form:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":" mathcalK = mathcalK_1^m_1 times mathcalK_2^m_2 times cdots times mathcalK_N^m_N","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"Accordingly, by an appropriate choice of convex sets one can represent any LP, QP, SOCP or SDP.","category":"page"},{"location":"method/#Dual-problem","page":"Method","title":"Dual problem","text":"","category":"section"},{"location":"method/","page":"Method","title":"Method","text":"The dual problem of the optimisation problem above is given by:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"beginarrayll\ntextmaximize -textstylefrac12x^top Px - b^top y - textsup_s in mathcalK(-y^top s ) \ntextsubject to Px + A^top y = -q\n y in (mathcalK^infty)^* \nendarray","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"with dual variable y in mathbbR^m.","category":"page"},{"location":"method/#Algorithm","page":"Method","title":"Algorithm","text":"","category":"section"},{"location":"method/","page":"Method","title":"Method","text":"The algorithm considers a slightly transformed problem. By introducing two dummy variables tildex = x and tildes = s we can rewrite the original problem:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"beginarrayll\ntextminimize textstylefrac12tildex^top P tildex + q^top tildex + mathcalI_Ax+s=b(tildextildes) + mathcalI_mathcalK(s)\ntextsubject to (tildextildes) = (xs)\nendarray","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"where indicator functions mathcalI were used to move the constraints into the objective function. The resulting problem is now in the right format to apply the alternating direction method of multipliers (ADMM). To apply ADMM we first find the augmented Lagrangian L:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":" L(xstildextildeslambday) = textstylefrac12tildex^top Ptildex + q^top tildex + mathcalI_Ax+s=b(tildextildes) + mathcalI_mathcalK(s) + fracsigma2 tildex - x + textstylefrac1sigma λ _2^2 + fracrho2 tildes - s + textstylefrac1rho y _2^2","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"Minimizing the Lagrangian in an alternating fashion with respect to the two variable pairs (tildextildes) and (xs) yields the following algorithm steps:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"beginaligned\n ( tildex^k+1tildes^k+1) rarr undersettildextildestextargmin Lleft( tildextildesx^ks^ky^k right)\n x^k+1 larr tildex^k+1\n s^k+1 larr undersetstextargmin fracrho2 tildes^k+1 - s+textstylefrac1rhoy^k _2^2 + I_mathcalK(s) \n y^k+1 larr y^k + rho left( tildes^k+1 -s^k+1 right)\nendaligned","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"By the construction of the ADMM method those iterates are converging to the global solution. These steps are executed in a loop until convergence. Two important parameters are the ADMM steps sizes rho (Settings.rho) and sigma (Settings.sigma) which can be adjusted via the solver settings.","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"The two most important steps of the algorithm happen in the first and third line. The evaluation of the first line turns out to be an equality constrained quadratic program. We get a solution for ( tildex^k+1tildes^k+1) at every iteration by solving the following linear system:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"beginaligned\nbeginbmatrix\nP + sigma I A^top A - frac1rhoI\n endbmatrixbeginbmatrixtildex^k+1 nu^k+1endbmatrix= beginbmatrix-q+sigma x^k b-s^k+frac1rhoy^kendbmatrix\ntildes^k+1 = s^k - frac1rholeft(nu^k+1 + y^kright)\nendaligned","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"Fortunately, the left hand matrix doesn't change, which is why COSMO only has to factor the matrix once at the start of the algorithm.","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"The second important step in the algorithm is the update equation for s^k+1 which can be interpreted as a projection onto the constraint set mathcalK:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"s^k+1 = Pi_mathcalKleft( tildes^k+1 + frac1rhoy^kright)","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"The computational cost of this projection is highly dependent on the constraints of the problem. While projections onto the zero set or the nonnegative orthant are inexpensive, projections onto the positive semidefinite cone of order N involve an eigen-decomposition. Since methods for eigen-decompositions have a complexity of mathcalO(N^3) the projection can become the computationally most expensive operation of the algorithm.","category":"page"},{"location":"method/#Scaling","page":"Method","title":"Scaling","text":"","category":"section"},{"location":"method/","page":"Method","title":"Method","text":"The convergence of ADMM-based algorithms depends on the relative scaling of the problem data. Especially to improve the convergence of badly scaled problems, COSMO tries to rescale the data in a preprocessing step.","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"We rescale the equality constraints with diagonal positive semidefinite matrices D and E. The scaled problem is given by:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"beginarrayll\ntextminimize textstylefrac12 hatx^top hatP hatx + hatq^top hatx\ntextsubject to hatA hatx + hats = hatb \n hats in EmathcalK\nendarray","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"with scaled problem data","category":"page"},{"location":"method/","page":"Method","title":"Method","text":" hatP=DPD quad hatq=Dq quadhatA=EAD quad hatb=Eb","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"and the scaled convex cone EmathcalK = Ev in mathbbR^m mid v in mathcalK . After solving the scaled problem the original solution is obtained by reversing the scaling:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":" x = Dhatx quad s = E^-1hats quad y = Ehaty","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"To obtain the scaling matrices D and E we use a modified Ruiz equilibration algorithm which involves a certain number of scaling iterations to equilibrate the column norms of the data matrices P and A. The number of these iterations can be adjusted by the user with scaling. To disable the scaling step set scaling = 0.","category":"page"},{"location":"method/#Termination-criteria","page":"Method","title":"Termination criteria","text":"","category":"section"},{"location":"method/","page":"Method","title":"Method","text":"The COSMO algorithm can terminate for four reasons:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"The maximum number of allowed iterations has been reached. The user can specify this value in the solver settings with max_iter.\nThe solver runtime reaches the time limit specified by the user (time_limit).\nCOSMO detects an infeasible problem.\nThe iterates fulfil the termination criteria for convergence.","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"COSMO uses the primal and dual residuals of the problem to determine if the algorithm has converged. The primal and dual residuals are given by:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"beginaligned\nr_p = Ax + s -b\nr_d = Px + q + A^T y\nendaligned","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"The solver terminates when the infty-norms of the residuals lie below a specified tolerance. COSMO uses the sum of an absolute and relative tolerance term:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"beginaligned\n r_p^k _infty leq epsilon_mathrmabs + epsilon_mathrmrel textmax left Ax^k _inftys^k_infty b_infty right\n r_d^k_infty leq epsilon_mathrmabs + epsilon_mathrmrel textmax leftPx^k_inftyq_infty A^top y^k_infty right\nendaligned","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"The absolute and relative tolerances epsilon_mathrmabsand epsilon_mathrmrel can be set by the user by specifying eps_abs and eps_rel. Furthermore, the user can adjust the number of iterations after which the convergence criteria are checked (check_termination).","category":"page"},{"location":"method/#Infeasibility-detection","page":"Method","title":"Infeasibility detection","text":"","category":"section"},{"location":"method/","page":"Method","title":"Method","text":"COSMO uses conditions based on separating hyperplanes to detect infeasible problems. The conditions for COSMO's problem format have been developed in [1]. Define the convex set mathcalC = mathcal-K + b then we can use the following infeasibility conditions:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"beginaligned\nmathcalP = leftx in mathbbR^n mid Px = 0 Ax in mathcalC^infty langle qx rangle 0 right \nmathcalD = lefty in mathbbR^m mid A^top y = 0 sigma_mathcalC(y) 0 right\nendaligned","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"The existence of some y in mathcalD is a certificate that the problem is primal infeasible, while the existence of some x in mathcalP is a certificate for dual infeasibility. COSMO regularly checks above conditions to detect infeasible problems. If the detection is successful, the solver terminates and returns the status codes :Primal_infeasible or :Dual_infeasible. COSMO checks the conditions every check_infeasibility iterations, which can be adjusted by the user.","category":"page"},{"location":"method/#References","page":"Method","title":"References","text":"","category":"section"},{"location":"method/","page":"Method","title":"Method","text":"[1] Banjac, G. et al. Infeasibility detection in the alternating direction method of multipliers for convex optimization. Preprint, 2017.","category":"page"},{"location":"examples/#Examples","page":"Examples","title":"Examples","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"Some example problems that are solved with COSMO can be found in the /examples folder.","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"In the first example the native solver interface is used to define and solve a Linear Program. In the second example JuMP is used to describe the problem and COSMO is set as the solver backend.","category":"page"},{"location":"examples/#Linear-Program","page":"Examples","title":"Linear Program","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"We want to solve the following linear program with decision variable x:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"beginarrayll textminimize c^top x\ntextsubject to A x leq b \n x geq 1 \n x_2 geq 5 \n x_1 + x_3 geq 4\nendarray","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"The problem can be solved with COSMO in the following way:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"using COSMO, LinearAlgebra, SparseArrays, Test\n\nc = [1; 2; 3; 4.]\nA = Matrix(1.0I, 4, 4)\nb = [10; 10; 10; 10]\nn = 4\n# -------------------\n# create constraints A * x + b in set\n# -------------------\n# Ax <= b\nc1 = COSMO.Constraint(-A, b, COSMO.Nonnegatives)\n# x >= 1\nc2 = COSMO.Constraint(Matrix(1.0I, n, n), -ones(n), COSMO.Nonnegatives)\n# x2 >= 5\nc3 = COSMO.Constraint(1, -5, COSMO.Nonnegatives, n, 2:2)\n# x1 + x3 >= 4\nc4 = COSMO.Constraint([1 0 1 0], -4, COSMO.Nonnegatives)\n\n# -------------------\n# define cost function\n# -------------------\nP = spzeros(4, 4)\nq = c\n\n# -------------------\n# assemble solver model\n# -------------------\nsettings = COSMO.Settings(max_iter=2500, verbose=true, eps_abs = 1e-4, eps_rel = 1e-5)\nmodel = COSMO.Model()\nassemble!(model, P, q, [c1; c2; c3; c4], settings = settings)\nres = COSMO.optimize!(model);\n\n@testset \"Linear Problem\" begin\n @test isapprox(res.x[1:4], [3; 5; 1; 1], atol=1e-2, norm = (x -> norm(x, Inf)))\n @test isapprox(res.obj_val, 20.0, atol=1e-2)\nend","category":"page"},{"location":"examples/#Closest-Correlation-Matrix","page":"Examples","title":"Closest Correlation Matrix","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"We consider the problem of finding the closest correlation matrix X to a given random matrix C. With closest correlation matrix we mean a positive semidefinite matrix with ones on the diagonal. The problem is given by:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"beginarrayll textminimize frac12X - C_F^2\ntextsubject to X_ii = 1 quad i=1dotsn \n X succeq 0\nendarray","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Notice how JuMP is used to describe the problem. COSMO is chosen as the backend solver using JuMP's optimizer_with_attributes() function.","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"using COSMO, JuMP, LinearAlgebra, SparseArrays, Test, Random\nrng = Random.MersenneTwister(12345);\n\n# create a random test matrix C\nn = 8\nC = -1 .+ rand(rng, n, n) .* 2;\nc = vec(C);\n\n# define problem in JuMP\nq = -vec(C);\nr = 0.5 * vec(C)' * vec(C);\nm = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"verbose\" => true, \"eps_abs\" => 1e-4));\n@variable(m, X[1:n, 1:n], PSD);\nx = vec(X);\n@objective(m, Min, 0.5 * x' * x + q' * x + r)\nfor i = 1:n\n @constraint(m, X[i, i] == 1.)\nend\n\n# solve and get results\nstatus = JuMP.optimize!(m)\nobj_val = JuMP.objective_value(m)\nX_sol = JuMP.value.(X)","category":"page"},{"location":"examples/#Logistic-Regression","page":"Examples","title":"Logistic Regression","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"Logistic regression problems can be solved using exponential cone constraints. An example on how to use COSMO to solve a logistic regression problem is presented in /examples/logistic_regression_regularization.ipynb.","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"EditURL = \"../../../examples/qp.jl\"","category":"page"},{"location":"examples/qp/#Quadratic-Program","page":"Quadratic Program","title":"Quadratic Program","text":"","category":"section"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"We want to solve the following quadratic program with decision variable x:","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"beginarrayll textminimize 12 x^top P x + q^top x \ntextsubject to l leq A x leq u\nendarray","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"The problem can be solved with COSMO in the following way. Start by defining the problem data","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"using COSMO, SparseArrays, LinearAlgebra, Test\n\nq = [1; 1.];\nP = sparse([4. 1; 1 2]);\nA = [1. 1; 1 0; 0 1];\nl = [1.; 0; 0];\nu = [1; 0.7; 0.7];\nnothing #hide","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"First, we decide to solve the problem with two one-sided constraints using COSMO.Nonnegatives as the convex set:","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"Aa = [-A; A]\nba = [u; -l]\nconstraint1 = COSMO.Constraint(Aa, ba, COSMO.Nonnegatives);\nnothing #hide","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"Next, we define the settings object, the model and then assemble everything:","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"settings = COSMO.Settings(verbose=true);\nmodel = COSMO.Model();\nassemble!(model, P, q, constraint1, settings = settings);\nres = COSMO.optimize!(model);\nnothing #hide","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"Alternatively, we can also use two-sided constraints with COSMO.Box:","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"constraint1 = COSMO.Constraint(A, zeros(3), COSMO.Box(l, u));\n\nmodel = COSMO.Model();\nassemble!(model, P, q, constraint1, settings = settings);\nres_box = COSMO.optimize!(model);\nnothing #hide","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"Let's check that the solution is correct:","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"@testset \"QP Problem\" begin\n @test norm(res.x[1:2] - [0.3; 0.7], Inf) < 1e-3\n @test norm(res_box.x[1:2] - [0.3; 0.7], Inf) < 1e-3\n @test abs(res.obj_val - 1.88) < 1e-3\n @test abs(res_box.obj_val - 1.88) < 1e-3\nend\nnothing","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"This page was generated using Literate.jl.","category":"page"},{"location":"lin_solver/#Linear-System-Solver","page":"Linear System Solver","title":"Linear System Solver","text":"","category":"section"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"One major step of COSMO's ADMM algorithm is solving a linear system of equations at each iteration. Fortunately, the left-hand matrix is only dependent on the problem data and therefore only needs to be factored once. Depending on the problem class this factorisation can be the computationally most expensive step of the algorithm (LPs, QPs). See the Method section for a more detailed description of the linear system.","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"COSMO allows you to specify the linear system solver that performs the factorisation and back-substitution. We also support indirect system solver which are useful for very large problems where a factorisation becomes inefficient. The table below shows the currently supported linear system solver:","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"Type Solver Description\nCholmodKKTSolver Cholmod Julia's default linear system solver (from SuiteSparse)\nQdldlKKTSolver QDLDL For more information QDLDL.jl\nPardisoDirectKKTSolver Pardiso (direct) Pardiso 6.0 direct solver\nPardisoIndirectKKTSolver Pardiso (indirect) Pardiso 6.0 indirect solver\nMKLPardisoKKTSolver Intel MKL Pardiso Pardiso optimised for Intel platforms\nCGIndirectKKTSolver IterativeSolvers.jl Conjugate Gradients on the reduced KKT linear system.\nMINRESIndirectKKTSolver IterativeSolvers.jl MINRES on the (full) KKT linear system.","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"note: Note\nTo use the Pardiso and Intel MKL Pardiso solver, you have to install the respective libraries and the corresponding Julia wrapper. For more information about installing these, visit the Pardiso.jl repository page. Likewise in order to use Indirect(Reduced)KKTSolver you have to install IterativeSolvers.jl (v0.9+) and LinearMaps.jl. We are using the Requires package for lazy loading of code related to Pardiso and IterativeSolvers. This means in order to use Pardiso / IterativeSolvers, you'll have to load these packages alongside COSMO, i.e. using Pardiso and using IterativeSolvers, LinearMaps.","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"COSMO uses the Cholmod linear system solver by default. You can specify a different solver in the settings by using the kkt_solver keyword and the respective type:","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"settings = COSMO.Settings(kkt_solver = CholmodKKTSolver)\n","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"COSMO also allows you to pass in solver-specific options with the with_options(solver_type, args...; kwargs...) syntax. For example, if you want to use Pardiso with verbose printing use the following syntax:","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"settings = COSMO.Settings(kkt_solver = with_options(PardisoDirectKKTSolver, msg_level_on = true))","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"Likewise, CGIndirectKKTSolver and MINRESIndirectKKTSolver are also parameterizable with with_options(solver_type, args...; kwargs...) and accept the following arguments:","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"Argument Description Values (default)\ntol_constant::T and tol_exponent::T Parameter that defines the solution tolerance for the iterative solvers across iterations. In particular, the solution tolerance at every iteration is defined as texttol_constant textiteration^texttol_exponent 1.0, 1.5","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"This also works if you want to use this configuration with JuMP:","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"model = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"kkt_solver\" => with_options(PardisoDirectKKTSolver, msg_level_on = true));\n","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"Or alternatively:","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"model = JuMP.Model(COSMO.Optimizer);\nset_optimizer_attribute(model, \"kkt_solver\", with_options(PardisoDirectKKTSolver, msg_level_on = true));","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"EditURL = \"../../../examples/closest_correlation_matrix.jl\"","category":"page"},{"location":"examples/closest_correlation_matrix/#Closest-Correlation-Matrix","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"","category":"section"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"We consider the problem of finding the closest correlation matrix X to a given random matrix C. With closest correlation matrix we mean a positive semidefinite matrix with ones on the diagonal. The problem is given by:","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"beginarrayll textminimize frac12X - C_F^2\ntextsubject to X_ii = 1 quad i=1dotsn \n X succeq 0\nendarray","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"Notice that we use JuMP to model the problem. COSMO is chosen as the backend solver. And COSMO-specific settings are passed using the optimizer_with_attributes() function.","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"using COSMO, JuMP, LinearAlgebra, SparseArrays, Test, Random","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"rng = Random.MersenneTwister(12345);\n# create a random test matrix C\nn = 8;\nC = -1 .+ rand(rng, n, n) .* 2;\nc = vec(C);\nnothing #hide","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"Define problem in JuMP:","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"q = -vec(C);\nr = 0.5 * vec(C)' * vec(C);\nm = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"verbose\" => true));\n@variable(m, X[1:n, 1:n], PSD);\nx = vec(X);\n@objective(m, Min, 0.5 * x' * x + q' * x + r);\nfor i = 1:n\n @constraint(m, X[i, i] == 1.);\nend","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"Solve the JuMP model with COSMO and query the solution X_sol:","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"status = JuMP.optimize!(m);\nobj_val = JuMP.objective_value(m);\nX_sol = JuMP.value.(X);\nnothing #hide","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"Double check result against known solution:","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"known_opt_val = 12.5406\nknown_solution = [\n 1.0 0.732562 -0.319491 -0.359985 -0.287543 -0.15578 0.0264044 -0.271438;\n 0.732562 1.0 0.0913246 -0.0386357 0.299199 -0.122733 0.126612 -0.187489;\n -0.319491 0.0913246 1.0 -0.0863377 0.432948 0.461783 -0.248641 -0.395299;\n -0.359985 -0.0386357 -0.0863377 1.0 0.503379 0.250601 0.141151 0.286088;\n -0.287543 0.299199 0.432948 0.503379 1.0 -0.0875199 0.137518 0.0262425;\n -0.15578 -0.122733 0.461783 0.250601 -0.0875199 1.0 -0.731556 0.0841783;\n 0.0264044 0.126612 -0.248641 0.141151 0.137518 -0.731556 1.0 -0.436274;\n -0.271438 -0.187489 -0.395299 0.286088 0.0262425 0.0841783 -0.436274 1.0 ];\n@test isapprox(obj_val, known_opt_val , atol=1e-3)","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"@test norm(X_sol - known_solution, Inf) < 1e-3","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"This page was generated using Literate.jl.","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"EditURL = \"../../../examples/lovasz_petersen.jl\"","category":"page"},{"location":"examples/lovasz_petersen/#Lovasz-Theta-Function","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"","category":"section"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"The Lovász theta function is an important concept in graph theory. Consider an undirected graph G = (VE) with vertex set V = 1dotsn and edge set E with E_ij = 1 if there exists an edge between the vertices i and j. A stable set (or independent set) is a subset of V such that the induced subgraph does not contain edges. The stability number alpha(G) of the graph is equal to the cardinality of the largest stable set. It is closely related to the Shannon capacity Theta(G) that models the amount of information that a noisy communication channel can carry if certain signal values can be confused with each other. Unfortunately, the determination of alpha(G) is an NP-hard problem and the computational complexity to determine Theta(G) remains unknown. The Lovász theta function vartheta(G) was introduced in [1], can be computed in polynomial time, and represents an upper bound on both the stability number and the Shannon capacity:","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"alpha(G) leq Theta(G) leq vartheta(G)","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"The value of vartheta(G) can be determined by computing the optimal value p^* of the following SDP:","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"beginarrayll textmaximize textTr(JX)\ntextsubject to textTr(X) = 1 \n X_ij = 0 quad (i j) in E \n X succeq 0\nendarray","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"with matrix variable X, matrix J in mathbfR^n times n of all ones and textTr() denoting the matrix trace. In this simple example we will compute the value of the Lovász theta function for the Petersen Graph (which is known to be 4).","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"using LinearAlgebra, COSMO, JuMP, Plots, GraphRecipes\n\n# let's define the Petersen graph\nn = 10\nE = zeros(n, n)\nE[1, 2] = E[1, 5] = E[1, 6] = 1.\nE[2, 3] = E[2, 7] = 1.\nE[3, 4] = E[3, 8] = 1.\nE[4, 5] = E[4, 9] = 1.\nE[5, 10] = 1.\nE[6, 8] = E[6, 9] = 1.\nE[7, 9] = E[7, 10] = 1.\nE[8, 10] = 1.\n\n# plot the graph\nri = 1.\nro = 2.\ncoordinates = []\nfor θ = 90:-72:-198\n push!(coordinates, [ro * cosd(θ), ro * sind(θ)])\nend\nfor θ = 90:-72:-198\n push!(coordinates, [ri * cosd(θ), ri * sind(θ)])\nend\ngraphplot(E, names = 1:n, x = getindex.(coordinates, 1), y = getindex.(coordinates, 2), fontsize = 10, nodesize = 1, nodeshape =:circle, curvature = 0.)","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"Let's solve the SDP with COSMO and JuMP:","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"model = JuMP.Model(COSMO.Optimizer);\n\n@variable(model, X[1:n, 1:n], PSD)\nx = vec(X)\n@objective(model, Max, sum(x))\n@constraint(model, tr(X)== 1.)\nfor j = 1:n\n for i = 1:j-1\n if E[i, j] == 1.\n @constraint(model, X[i, j] == 0)\n end\n end\nend\nstatus = JuMP.optimize!(model)","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"The optimal objective is given by:","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"JuMP.objective_value(model)","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"Which is the correct known value for the Petersen Graph.","category":"page"},{"location":"examples/lovasz_petersen/#References","page":"Lovász Theta Function","title":"References","text":"","category":"section"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"[1] Lovász - On the Shannon Capacity of a Graph, IEEE Transactions on Information Theory (1979)","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"This page was generated using Literate.jl.","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"EditURL = \"../../../examples/two_way_partitioning.jl\"","category":"page"},{"location":"examples/two_way_partitioning/#Relaxed-Two-Way-Partitioning-Problem","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"","category":"section"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"We consider the (nonconvex) two-way partitioning problem from Boyd and Vandenberghe (2004), p.219 [1]:","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"beginarrayll textminimize x^top W x \ntextsubject to x_i^2 = 1 quad i=1dots n\nendarray","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"with x in mathbfR^n and W in mathbfS^n. The problem can be interpreted as finding a partition of the points i = 1dots n into two sets, where the cost of two points in the same set is W_ij and -W_ij otherwise. Brute forcing the solution x^* takes 2^n attempts and becomes quickly intractable, e.g. for n geq 30. However, a lower-bound for this problem can be computed by solving the convex dual of the problem:","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"beginarrayll textmaximize -sum_i nu \ntextsubject to W + textdiag(nu) in mathbfS_+^n\nendarray","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"Solving this problem with optimal objective d^* yields a lower bound as least as good as a lower bound based on the minimum eigenvalue d^* geq lambda_textmin(W). Let's set up the problem for n = 20 and with a specially structured (banded) W:","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"using LinearAlgebra, Random, COSMO, JuMP, IterTools\n\nrng = Random.MersenneTwister(212313);\nn = 20;\n\nW = diagm(0 => randn(rng, n));\nW += diagm(1 => randn(rng, n-1));\nW = Symmetric(W)","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"As you can see, the matrix W imposes a structure on the LMI-constraint. Accordingly, COSMO will be able to use chordal decomposition to decompose the LMI constraint into multiple smaller constraints. This can make a significant difference in the performance of the algorithm for large n.","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"model = JuMP.Model(COSMO.Optimizer);\n@variable(model, ν[1:n]);\n@objective(model, Max, -ones(n)' * ν )\n@constraint(model, Symmetric(W + diagm(ν) ) in JuMP.PSDCone());\nJuMP.optimize!(model)","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"Looking at the solver output you can see how the PSD constraint was decomposed into 19 PSD constraints. Let's look at the lower bound:","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"JuMP.objective_value(model)","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"As n is small, we can verify our result by finding the optimal solution by trying out all possible combinations:","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"function brute_force_optimisation(W, n)\n opt_obj = Inf\n opt_x = Inf * ones(n)\n\n for xt in Iterators.product([[1.; -1.] for k = 1:n]...)\n x = [i for i in xt]\n obj_val = x' * W * x\n if obj_val < opt_obj\n opt_obj = obj_val\n opt_x = x\n end\n end\n return opt_obj, opt_x\nend\nopt_obj, opt_x = brute_force_optimisation(W, n)","category":"page"},{"location":"examples/two_way_partitioning/#References","page":"Relaxed Two-Way Partitioning Problem","title":"References","text":"","category":"section"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"[1] Boyd and Vandenberghe - Convex Optimization, Cambridge University Press (2004)","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"This page was generated using Literate.jl.","category":"page"},{"location":"","page":"Home","title":"Home","text":"COSMO.jl is a Julia implementation of the Conic Operator Splitting Method. The underlying ADMM-algorithm is well-suited for large convex conic problems. COSMO solves the following problem:","category":"page"},{"location":"","page":"Home","title":"Home","text":"beginarrayll textminimize textstylefrac12x^top Px + q^top x textsubject to Ax + s = b s in mathcalK\nendarray","category":"page"},{"location":"","page":"Home","title":"Home","text":"with decision variables x in mathbbR^n, s in mathbbR^m and data matrices P=P^top succeq 0, q in mathbbR^n, A in mathbbR^m times n, and b in mathbbR^m. The convex set mathcalK is a composition of convex sets and cones.","category":"page"},{"location":"#Features","page":"Home","title":"Features","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Versatile: COSMO solves linear programs, quadratic programs, second-order cone programs, semidefinite programs and problems involving exponential and power cones\nQuad SDPs: Positive semidefinite programs with quadratic objective functions are natively supported\nSafeguarded acceleration: robust and faster convergence to higher precision using COSMOAccelerators\nInfeasibility detection: Infeasible problems are detected without a homogeneous self-dual embedding of the problem\nJuMP / Convex.jl support: We provide an interface to MathOptInterface (MOI), which allows you to describe your problem in JuMP and Convex.jl.\nChordal decomposition: COSMO tries to decompose large structured PSD constraints into multiple smaller PSD constraints using chordal decomposition techniques. This often results in a significant speedup compared to solving the original problem.\nSmart clique merging: After an initial decomposition of a structured SDP, COSMO recombines overlapping cliques/blocks to speed up the algorithm.\nWarm starting: COSMO supports warm starting of the decision variables\nArbitrary precision types: You can solve problems with any floating point precision.\nOpen Source: Our code is available on GitHub and distributed under the Apache 2.0 Licence","category":"page"},{"location":"","page":"Home","title":"Home","text":"!!! Audience: User","category":"page"},{"location":"","page":"Home","title":"Home","text":"COSMO has both a native interface and can also be interfaced through JuMP. If you are interested in using COSMO to solve your optimization problem, we recommend using JuMP to define your problem. Install this package as described below, take a look at our [Examples](@ref) and the [JuMP documentation](https://jump.dev/JuMP.jl/stable/) for inspiration.","category":"page"},{"location":"","page":"Home","title":"Home","text":"!!! Audience: Researcher / Developer","category":"page"},{"location":"","page":"Home","title":"Home","text":"If you are interested in COSMO's algorithm, chordal decomposition or acceleration methods take a look at [Method](@ref), [Chordal Decomposition](@ref), and [Acceleration](@ref). If you want low-level control over the algorithm, it can make sense to start with COSMO's native interface.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"COSMO can be installed using the Julia package manager for Julia v1.6 and higher. Inside the Julia REPL, type ] to enter the Pkg REPL mode then run","category":"page"},{"location":"","page":"Home","title":"Home","text":"pkg> add COSMO","category":"page"},{"location":"","page":"Home","title":"Home","text":"If you want to install the latest version from master run","category":"page"},{"location":"","page":"Home","title":"Home","text":"pkg> add COSMO#master","category":"page"},{"location":"#Quick-Example","page":"Home","title":"Quick Example","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Consider the following 2x2 semidefinite program with decision variable X:","category":"page"},{"location":"","page":"Home","title":"Home","text":"beginarrayll textminimize texttr(CX)\ntextsubject to texttr(A X) = b \n X succeq 0\nendarray","category":"page"},{"location":"","page":"Home","title":"Home","text":"with problem data A, b and C:","category":"page"},{"location":"","page":"Home","title":"Home","text":"A = beginbmatrix 1 5 5 2endbmatrix\nC = beginbmatrix 1 2 2 2endbmatrix\nb = 4","category":"page"},{"location":"","page":"Home","title":"Home","text":"where tr denotes the trace of a matrix. We can solve this problem either using COSMO's interface:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using COSMO, LinearAlgebra\n\nC = [1. 2; 2 2]\nA = [1. 5; 5 2]\nb = 4.0;\n\nmodel = COSMO.Model();\n\n# define the cost function\nP = zeros(4, 4)\nq = vec(C)\n\n# define the constraints\n# A x = b\ncs1 = COSMO.Constraint(vec(A)', -b, COSMO.ZeroSet)\n# X in PSD cone\ncs2 = COSMO.Constraint(Matrix(1.0I, 4, 4), zeros(4), COSMO.PsdCone)\nconstraints = [cs1; cs2]\n\n# assemble and solve the model\nassemble!(model, P, q, constraints)\nresult = COSMO.optimize!(model);\n\nX_sol = reshape(result.x, 2, 2)\nobj_value = result.obj_val","category":"page"},{"location":"","page":"Home","title":"Home","text":"or we can describe the problem using JuMP and use COSMO as the backend solver:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using COSMO, JuMP, LinearAlgebra\n\nC = [1. 2; 2 2]\nA = [1. 5; 5 2]\nb = 4.0;\nm = JuMP.Model(COSMO.Optimizer);\n@variable(m, X[1:2, 1:2], PSD)\n@objective(m, Min, tr(C * X));\n@constraint(m, tr(A * X) == b);\nJuMP.optimize!(m);\n\nstatus = JuMP.termination_status(m)\nX_sol = JuMP.value.(X)\nobj_value = JuMP.objective_value(m)","category":"page"},{"location":"#Related-talks","page":"Home","title":"Related talks","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"A video of my first presentation of COSMO at JuMP-dev (Santiago) is available here: COSMO.jl at JuMP-dev\nA more detailed presentation on chordal decomposition and clique merging is available here: Chordal decomposition and clique merging\nI gave an overview on recent features of COSMO such as acceleration methods at JuliaCon 2021: What's new in COSMO?\nMore details on the acceleration methods were presented in my ECC 2022 talk Safeguarded Anderson acceleration for parametric nonexpansive operators","category":"page"},{"location":"#Credits","page":"Home","title":"Credits","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"The following people are involved in the development of COSMO:","category":"page"},{"location":"","page":"Home","title":"Home","text":"Michael Garstka (main development)\nNikitas Rontsis (algorithm performance)\nPaul Goulart (code architecture, maths and algorithms)\nMark Cannon (maths and algorithms)","category":"page"},{"location":"","page":"Home","title":"Home","text":"*all contributors are affiliated with the University of Oxford.","category":"page"},{"location":"","page":"Home","title":"Home","text":"If this project is useful for your work please consider","category":"page"},{"location":"","page":"Home","title":"Home","text":"Citing the relevant papers\nLeaving a star on the GitHub repository","category":"page"},{"location":"#Licence","page":"Home","title":"Licence","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"COSMO.jl is licensed under the Apache License 2.0. For more details click here.","category":"page"},{"location":"citing/#Citing-COSMO","page":"Citing COSMO","title":"Citing COSMO","text":"","category":"section"},{"location":"citing/","page":"Citing COSMO","title":"Citing COSMO","text":"If you find COSMO useful in your project, we kindly request that you cite the following paper:","category":"page"},{"location":"citing/","page":"Citing COSMO","title":"Citing COSMO","text":"@Article{Garstka_2021,\n author = {Michael Garstka and Mark Cannon and Paul Goulart},\n journal = {Journal of Optimization Theory and Applications},\n title = {{COSMO}: A Conic Operator Splitting Method for Convex Conic Problems},\n volume = {190},\n number = {3},\n pages = {779--810},\n year = {2021},\n publisher = {Springer},\n doi = {10.1007/s10957-021-01896-x},\n url = {https://doi.org/10.1007/s10957-021-01896-x}\n}","category":"page"},{"location":"citing/","page":"Citing COSMO","title":"Citing COSMO","text":"The article is available under Open Access here.","category":"page"},{"location":"citing/","page":"Citing COSMO","title":"Citing COSMO","text":"The following paper describes the algorithm used for our clique graph based merging strategy:","category":"page"},{"location":"citing/","page":"Citing COSMO","title":"Citing COSMO","text":"@InProceedings{Garstka_2020,\n author = {Michael Garstka and Mark Cannon and Paul Goulart},\n title = {A clique graph based merging strategy for decomposable {SDPs}},\n year = {2020},\n note = {21th IFAC World Congress},\n number = {2},\n pages = {7355-7361},\n volume = {53},\n doi = {10.1016/j.ifacol.2020.12.1255},\n issn = {2405-8963},\n journal = {IFAC-PapersOnLine},\n}","category":"page"},{"location":"citing/","page":"Citing COSMO","title":"Citing COSMO","text":"A preprint can be downloaded here.","category":"page"},{"location":"citing/","page":"Citing COSMO","title":"Citing COSMO","text":"The acceleration strategy is described in this paper:","category":"page"},{"location":"citing/","page":"Citing COSMO","title":"Citing COSMO","text":"@InProceedings{Garstka_2022,\n title={Safeguarded Anderson acceleration for parametric nonexpansive operators},\n author={Garstka, Michael and Cannon, Mark and Goulart, Paul},\n booktitle={2022 European Control Conference (ECC)},\n pages={435--440},\n year={2022},\n organization={IEEE}\n}","category":"page"},{"location":"citing/","page":"Citing COSMO","title":"Citing COSMO","text":"A preprint can be downloaded here.","category":"page"}] +[{"location":"logistic_regression/#Logistic-Regression","page":"Logistic Regression","title":"Logistic Regression","text":"","category":"section"},{"location":"logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"Logistic regression problems can be solved using exponential cone constraints. An example on how to use COSMO to solve a logistic regression problem is presented in /examples/logistic_regression_regularization.ipynb.","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"EditURL = \"../../../examples/logistic_regression.jl\"","category":"page"},{"location":"examples/logistic_regression/#Logistic-Regression","page":"Logistic Regression","title":"Logistic Regression","text":"","category":"section"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"The presented example is adapted from the Machine Learning - course by Andrew Ng.","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"In this example we use logistic regression to estimate the parameters theta_i of a logistic model in a classification problem. We will first transform the logistic regression problem into an exponential cone optimisation problem. We will then solve the optimisation problem with COSMO and determine the model parameters and the decision boundary.","category":"page"},{"location":"examples/logistic_regression/#Visualizing-the-data","page":"Logistic Regression","title":"Visualizing the data","text":"","category":"section"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"Before we start, let's load and take a look at the example data from examples/chip_data.txt:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"using LinearAlgebra, SparseArrays, COSMO, JuMP\nusing Plots\n\n# load example data\nf = open(joinpath(@__DIR__, \"chip_data.txt\"))\nlines = readlines(f)\nclose(f)\nn_data = length(lines)\nn_half = div(n_data, 2)\nx1 = zeros(n_data)\nx2 = zeros(n_data)\ny = zeros(Float64, n_data)\nfor (i, l) in enumerate(lines)\n s = split(l, \",\")\n x1[i] = parse(Float64, s[1])\n x2[i] = parse(Float64, s[2])\n y[i] = parse(Float64, s[3])\nend\n\n\n# visualize data\nplot(x1[1:n_half-1], x2[1:n_half-1], color = :blue, st=:scatter, markershape = :cross, aspect_ratio=:equal, label = \"Accepted\", xlabel = \"x1 - Microchip Test Score 1\", ylabel = \"x2 - Microchip Test Score 2\")\nplot!(x1[n_half:end], x2[n_half:end], color = :red, st=:scatter, markershape = :circle, label = \"Rejected\")","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"The plot shows two test scores of n microchip samples from a fabrication plant and whether the chip passed the quality check. Based on this data we would like to build a logistic model that takes into account the test scores and helps us predict the likelihood of a chip being accepted.","category":"page"},{"location":"examples/logistic_regression/#Defining-the-logistic-model","page":"Logistic Regression","title":"Defining the logistic model","text":"","category":"section"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"The logistic regression hypothesis is given by","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"h_theta(x) = g(theta^top x)","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"where g is the sigmoid function:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"g(theta^top x) = frac11+exp(-theta^top x)","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"The vector x represents the independent variables and theta represents the model parameters. For our samples we set the dependent variable y =1 if the chip was accepted and y = 0 otherwise.","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"The function h_theta(x) can be interpreted as the probability of the outcome being true rather than false. We want to find the parameters theta that maximize the log-likelihood over all (independently Bernoulli distributed) observations","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"J(theta) = sum_i y_i = 1 log h_theta(x_i) + sum_i y_i = 0 log (1-h_theta(x_i))","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"Consequently, we want to solve the following optimization problem:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"textminimize quad -J(theta) + mu theta _2","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"where we added a regularization term with parameter mu to prevent overfitting.","category":"page"},{"location":"examples/logistic_regression/#Feature-mapping","page":"Logistic Regression","title":"Feature mapping","text":"","category":"section"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"As our dataset only has two independent variables (the test scores) our model y = theta_0 + theta_1 x_1 + theta_2 x_2 will have the form of a straight line. Looking at the plot one can see that a line will not perform well in separating the samples. Therefore, we will create more features based on each data point by mapping the original features (x_1, x_2) into all polynomial terms of x_1 and x_2 up to the 6th power:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"textmap_feature(x_1x_2) = 1 x_1 x_2 x_1^2 x_1x_2 x_2^2 x_1^3 dots x_1x_2^5 x_2^6 ","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"This will create 28 features for each sample.","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"function map_feature(x1, x2)\n deg = 6\n x_new = ones(length(x1))\n for i = 1:deg, j = 0:i\n x_new = hcat(x_new, x1.^(i-j) .* x2.^j)\n end\n return x_new\nend\n\nX = map_feature(x1, x2);\nsize(X)","category":"page"},{"location":"examples/logistic_regression/#Transformation-into-a-conic-optimisation-problem","page":"Logistic Regression","title":"Transformation into a conic optimisation problem","text":"","category":"section"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"We can rewrite above likelihood maximisation problem as a conic optimisation problem with exponential-cone-, second-order-cone-, equality-, and inequality constraints:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"beginarrayll\ntextminimize sum_i^n epsilon_i + mu v\ntextsubject to theta _2 leq v\n log(1 + exp(-theta^top x_i)) leq epsilon_i quad textif y_i = 1 \n log(1 + exp(theta^top x_i)) leq epsilon_i quadtext otherwise\nendarray","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"Implementing the constraint log(1 + exp(z)) leq epsilon for each of the n samples requires two exponential cone constraints, one inequality constraint and two equality constraints. To see this, take the exponential on both sides and then divide by exp(epsilon) to get:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"exp(-epsilon) + exp(z - epsilon) leq 1","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"This constraint is equivalent to:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"beginarrayll\n(z - epsilon s_1 t_1) in K_textexp \n(-epsilon s_2 t_2) in K_textexp \nt_1 + t_2 leq 1\ns_1 = s_2 = 1\nendarray","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"where we defined the exponential cone as:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"K_textexp = (r s t) mid s 0 s exp(rs) leq t cup r leq 0 s = 0 t geq 0 ","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"Based on this transformation our optimisation problem will have 5n + n_theta + 1 variables, 1 SOCP constraint, 2n exponential cone constraints, n inequality constraints and 2n equality constraints. Let's model the problem with JuMP and COSMO:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"n_theta = size(X, 2)\nn = n_data\nμ = 1.\n\nm = JuMP.Model(COSMO.Optimizer)\n@variable(m, v)\n@variable(m, θ[1:n_theta])\n@variable(m, e[1:n])\n@variable(m, t1[1:n])\n@variable(m, t2[1:n])\n@variable(m, s1[1:n])\n@variable(m, s2[1:n])\n\n@objective(m, Min, μ * v + sum(e))\n@constraint(m, [v; θ] in MOI.SecondOrderCone(n_theta + 1))\n\n# create the constraints for each sample points\nfor i = 1:n\n yi = y[i]\n x = X[i, :]\n yi == 1. ? (a = -1) : (a = 1)\n @constraint(m, [a * dot(θ, x) - e[i]; s1[i]; t1[i] ] in MOI.ExponentialCone())\n @constraint(m, [-e[i]; s2[i]; t2[i]] in MOI.ExponentialCone())\n @constraint(m, t1[i] + t2[i] <= 1)\n @constraint(m, s1[i] == 1)\n @constraint(m, s2[i] == 1)\nend\nJuMP.optimize!(m)","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"theta = value.(θ)","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"Once we have solved the optimisation problem and obtained the parameter vector theta, we can plot the decision boundary. This can be done by evaluating our model over a grid of points (uv) and then plotting the contour line where the function g returns a probability of p=05.","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"# First we evaluate our model over a grid of points z = θ' x\nu = collect(range(-1., stop = 1.5, length = 50))\nv = collect(range(-1., stop = 1.5, length = 50))\nz = zeros(length(u), length(v));\nfor i = 1:length(u), j = 1:length(v)\n z[i, j] = dot(map_feature(u[i], v[j]), theta);\nend","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"To add the decision boundary we have to plot the line indicating 50 probability of acceptance, i.e. g(theta^top x) = g(z) = 05 which we get at z=0.","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"plot(x1[1:n_half-1], x2[1:n_half-1], color = :blue, st = :scatter, markershape = :cross, aspect_ratio=:equal, label = \"Accepted\", xlabel = \"x1 - Microchip Test Score 1\", ylabel = \"x2 - Microchip Test Score 2\")\nplot!(x1[n_half:end], x2[n_half:end], color = :red, st = :scatter, markershape = :circle, label = \"Rejected\")\ncontour!(u, v, z', levels = [0.], c = :black, linewidth = 2)","category":"page"},{"location":"examples/logistic_regression/#Solving-the-optimisation-problem-directly-with-COSMO","page":"Logistic Regression","title":"Solving the optimisation problem directly with COSMO","text":"","category":"section"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"We can solve the problem directly in COSMO by using its modeling interface. The problem will have nn = 5 n + n_theta + 1 variables. Let us define the cost function frac12x^top P x + q^top x:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"nn = 5 * n + n_theta + 1\nP = spzeros(nn, nn)\nq = zeros(nn)\nq[1] = μ\nfor i = 1:n\n q[1 + n_theta + (i - 1) * 5 + 1] = 1.\nend","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"Next we define a function that creates the COSMO.Constraints for a given sample:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"# the order of the variables\n# v, thetas, [e1 t11 t12 s11 s12] [e2 t21 t22 s21 s22] ...\n# for each sample create two exponential cone constraints,\n# 1 nonnegatives constraint, 2 zeroset constraints\nfunction add_log_regression_constraints!(constraint_list, x, y, n, sample_num)\n num_thetas = length(x)\n # 1st exponential cone constraint (zi - ei, s1, t1) in Kexp\n c_start = 1 + num_thetas + (sample_num - 1) * 5 + 1\n A = spzeros(3, n)\n A[1, c_start] = -1.\n y == 1. ? (a = -1) : (a = 1)\n for k = 1:num_thetas\n A[1, 2 + k - 1] = a * x[k]\n end\n A[2, c_start + 3] = 1.\n A[3, c_start + 1] = 1.\n b = zeros(3)\n push!(constraint_list, COSMO.Constraint(A, b, COSMO.ExponentialCone))\n\n # 2nd exponential cone constraint (-e, s2, t2)\n A = spzeros(3, n)\n A[1, c_start] = -1.\n A[2, c_start + 4] = 1.\n A[3, c_start + 2] = 1.\n b = zeros(3)\n push!(constraint_list, COSMO.Constraint(A, b, COSMO.ExponentialCone))\n\n # Nonnegatives constraint t1 + t2 <= 1\n A = spzeros(1, n)\n A[1, c_start + 1] = -1.\n A[1, c_start + 2] = -1.\n b = [1.]\n push!(constraint_list, COSMO.Constraint(A, b, COSMO.Nonnegatives))\n\n # ZeroSet constraint s1 == 1, s2 == 1\n A = spzeros(2, n)\n A[1, c_start + 3] = 1.\n A[2, c_start + 4] = 1.\n b = -1 * ones(2)\n push!(constraint_list, COSMO.Constraint(A, b, COSMO.ZeroSet))\nend\nnothing","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"Now we can use this function to loop over the sample points and add the constraints to our constraint list:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"constraint_list = Array{COSMO.Constraint{Float64}}(undef, 0)\nfor i = 1:n\n add_log_regression_constraints!(constraint_list, X[i, :], y[i], nn, i )\nend","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"It remains to add a second order cone constraint for the regularisation term: theta _2 leq v","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"push!(constraint_list, COSMO.Constraint(Matrix(1.0I, n_theta + 1, n_theta + 1), zeros(n_theta + 1), COSMO.SecondOrderCone, nn, 1:n_theta+1));\nnothing #hide","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"We can now create, assemble, and solve our COSMO.Model:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"model = COSMO.Model()\nassemble!(model, P, q, constraint_list, settings = COSMO.Settings(verbose=true))\nres = COSMO.optimize!(model);\nnothing #hide","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"Let us double check that we get the same theta as in the previous section:","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"using Test\ntheta_cosmo = res.x[2:2+n_theta-1]\n@test norm(theta_cosmo - theta) < 1e-4","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"","category":"page"},{"location":"examples/logistic_regression/","page":"Logistic Regression","title":"Logistic Regression","text":"This page was generated using Literate.jl.","category":"page"},{"location":"contributing/#Contributing","page":"Contributing","title":"Contributing","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"This package is currently in maintenance mode. ","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"We are trying to keep it compatible with new releases of JuMP/MOI. Contributions for bug fixes or new features are always welcome. Enhancement ideas are tagged as such in the Issues section. If you are unsure how to develop, continue with the section below.","category":"page"},{"location":"contributing/#How-to-develop","page":"Contributing","title":"How-to develop","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Let's assume you found a bug that you want to fix or develop a new feature. You have already downloaded the latest julia version as well as git. Follow these steps to make changes, test the changes, and open your first PR. This assumes that you don't yet have push access to the repository and need to fork.","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Create a new fork in Github by clicking on Fork at the top right of the repository page and name it e.g. migarstka/COSMO.jl. (If you have repository push rights, you don't need to fork and can clone the repository directly.)\nClone the forked repository to your machine with:","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"$ git clone git@github.com:migarstka/COSMO.jl.git","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Navigate into the new folder and create a new branch:","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"$ git checkout -b mg/bug_fix","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Make your code changes by modifying the files in the repository.\nAs a first sanity check that nothing obvious is broken, you can run one of the examples in the examples/ folder. Open julia and navigate into the git repository. Next, we want to activate and instantiate the repositories virtual environment. In the julia REPL type ], then to activate the current environment (i.e. current Project.toml file):","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"(@v1.8) pkg> activate .","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Next, instantiate the environment, i.e. install the solver's dependencies defined in Project.toml:","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"(COSMO) pkg> instantiate","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Since we instantiated the local envrionment, COSMO now refers to the local version that includes your changes. You can solve an example problem using the modified code:","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"julia> include(\"examples/qp.jl\")","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Let's assume you have made some changes and want to check whether the bug is fixed or the new feature broke anything. In both cases you should add some new tests for the change to /test/UnitTest and /test/run_cosmo_tests.jl. If you want to run existing tests follow the steps below.","category":"page"},{"location":"contributing/#How-to-run-tests","page":"Contributing","title":"How-to run tests","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"You can run all package-native tests with the main test file:","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"julia> include(\"test/run_cosmo_tests.jl\")","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"The full test suite also includes several MathOptInterface test problems that you can run with:","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"julia> include(\"test/runtests.jl\")","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"These take longer to complete, so I recommend using test/run_cosmo_tests.jl during iterative development and then switch to test/runtests.jl once the previous tests pass.","category":"page"},{"location":"contributing/#Open-a-PR","page":"Contributing","title":"Open a PR","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Once all tests pass, you have added additional tests for your changes, and used docstrings to describe any new functionality in the code, you can open a PR. Commit your changes with a detailed commit message, and push your branch to the (fork) remote.","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"$ git add .\n$ git commit \n[Write commit message]\n$ git push","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Then navigate to https://github.com/oxfordcontrol/COSMO.jl/compare and create a new PR with base repository: oxfordcontrol/COSMO.jl base: master and head repository: [your-fork-repo] base:mg/bug_fix. Ask me or other collaborators to review it, before it can get merged.","category":"page"},{"location":"contributing/#Making-a-new-release","page":"Contributing","title":"Making a new release","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Changes to master are periodically bundled in a new release version of the package. If you have permissions, you can make a new release by following these steps:","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Ensure all tests pass (locally and in Github Actions)\nLocally run bumpversion patch (or bumpversion minor or bumpversion major depending on semver convention) to increment the version number across the repository (most importantly in Project.toml).\nCheck and commit the changes and use the commit message Bump version to X. \nPush changes to remote with git push.\nLet the julia package registry know about the new release by commenting @JuliaRegistrator register on the bump commit (like here).\nThis will open a PR in the JuliaRegistries repository and a new release will typically be approved within hours.\nUpdate the CHANGELOG and describe the changes bundled in the new release.","category":"page"},{"location":"contributing/#Updating-COSMO's-documentation","page":"Contributing","title":"Updating COSMO's documentation","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"The code for COSMO's documentation resides in the /docs folder of the repository and is created with the Documenter.jl package. There are two main versions of this documentation. Stable is based on the committed changes of the latest tagged release. Dev is based on the latest commit of master.","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"The documentation has its own environment defined inside /docs (see docs/Project.toml file). To edit the documentation, edit the files in /docs. It's advised to run the documentation generation process locally do ensure the layout is as expected and that Literate code examples are built without errors. To generate the documentation locally, follow these steps:","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Clone the repository and make your changes (as described in the section above).\nNavigate into the /docs folder and start julia.\nActivate and instantiate the (docs) environment, after ], type:","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"(@v1.8) pkg> activate .","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"(docs) pkg> instantiate","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Build the documentation locally into the /build folder with","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"include(\"make.jl\")","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"(The first build will take several minutes. Subsequent changes will take ~30s.)","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"After the new documentation has been built, you can serve the files in /build in your browser, e.g. using a python webserver. Navigate into /build and run:","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"python3 -m http.server --bind localhost","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"View the local documentation in your browser of choice at http://127.0.0.1:8000/\nIf you are satisfied with the results, open a PR for the changes as described in the section above.","category":"page"},{"location":"contributing/#Code-Style-Guide","page":"Contributing","title":"Code Style Guide","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"The code in this repository follows the naming and style conventions of Julia Base with a few modifications. This style guide is heavily \"inspired\" by the guides of John Myles White and JuMP.","category":"page"},{"location":"contributing/#Formatting","page":"Contributing","title":"Formatting","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Use one tab when indenting a new block (except module)\nUse spaces between operators, except for ^, ', and :\nUse single space after commas and semicolons\nDon't use spaces around parentheses, or braces","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Bad: f(x,y) = [5*sin(x+y);y'] Good: f(x, y) = [5 * sin(x + y); y']","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Use spacing with keyword arguments","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Bad: foo(x::Integer=1) Good: foo(x::Integer = 1)","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Don't parenthesize conditions","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Bad: if (a == b) Good: if a == b","category":"page"},{"location":"contributing/#Naming","page":"Contributing","title":"Naming","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Modules and Type names use capitilization and camel case, e.g. module LinearAlgebra, struct ConvexSets.\nFunctions are lowercase and use underscores to seperate words, e.g. has_key(x), is_valid(y).\nNormal variables are lowercase and use underscores like functions, e.g. convex_set\nConstants are uppercase, e.g. const MY_CONSTANT\nAlways append ! to names of functions that modify their arguments.\nFunction arguments that are mutated come first. Otherwise follow the rules layed out in Julia Base Argument ordering\nFiles are named like functions, e.g. my_new_file.jl","category":"page"},{"location":"contributing/#Syntax","page":"Contributing","title":"Syntax","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Use 1.0 instead of 1.","category":"page"},{"location":"contributing/#Git(hub)-specific-conventions","page":"Contributing","title":"Git(hub)-specific conventions","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Branch names should be prepended with the initials of the creator and a forward slash, e.g. mg/newIdea instead of newIdea\nCommit messages should have the following format:","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"<#IssueId> Short (72 chars or less) summary\n\nMore detailed explanatory text. Wrap it to 72 characters. The blank\nline separating the summary from the body is critical.\n\nImperative style for the commit message: \"Fix bug\" and not \"Fixed\nbug\" or \"Fixes bug.\"\n\nThe issue id can be ommitted if the commit does not related to a specific open issue","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"EditURL = \"../custom_cone.jl\"","category":"page"},{"location":"literate/build/custom_cone/#Custom-Cone-Constraint","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"","category":"section"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"This example demonstrate how the user can extend COSMO with his own custom convex cones and use them in constraints. To make things easy, we will implement the simple cone of Nonpositives, i.e. mathbbR_-^n = x in mathbbR^n mid x_i leq 0.","category":"page"},{"location":"literate/build/custom_cone/#Cone-definition-and-projection-function","page":"Custom Cone Constraint","title":"Cone definition and projection function","text":"","category":"section"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"We start by creating a concrete subtype of our cone interface-type: COSMO.AbstractConvexCone{T}, where T is a parameter for the floating-point precision. The only field that is strictly required for our cone definition is dim, which is used by the solver to store information about the cone dimension. However, you can also use the object to store other information, or allocate the workspace necessary for the projection step.","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"using COSMO, LinearAlgebra, SparseArrays, Test\n\n# define new cone type\nstruct Nonpositives{T} <: COSMO.AbstractConvexCone{T}\n dim::Int64\nend","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"Next, we define a projection method for Nonpositives{T} that takes a vector x and projects it onto our custom cone. For the cone of nonpositive vectors the projection simply sets all positive elements of x to zero:","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"function COSMO.project!(x::AbstractVector{T}, C::Nonpositives{T}) where {T <: AbstractFloat}\n x .= min.(x, zero(T))\nend","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"This is all that is required to get a basic constraint working. Let's test our new cone/constraint by solving the following LP:","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"beginarrayll textmaximize x_1 + x_2 + x_3\ntextsubject to x_1 leq 3 \n x_2 leq 2 \n x_1 + x_3 = 5\nendarray","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"We define our decision vector x = (x_1 x_2 x_3) and setup the problem:","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"n = 3\nP = spzeros(n, n)\nq = -ones(n)\n\n# A1 * x + b1 <= 0\nA1 = diagm(0 => ones(2))\nb1 = [-3.; -2.]\nc1 = COSMO.Constraint(A1, b1, Nonpositives, n, 1:2); #here we use our new cone\n\n# x_1 + x_3 == 5\nA2 = [1. 0 1.]\nb2 = [-5.]\nc2 = COSMO.Constraint(A2, b2, COSMO.ZeroSet);\n\n# assemple and solve\nmodel = COSMO.Model();\nassemble!(model, P, q, [c1; c2]);\nres = COSMO.optimize!(model);\nnothing #hide","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"res.x","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"obj_val = -dot(q, res.x)","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"The problem was solved using constraints involving our new cone and yields the expected result. Next, we want to add the ability to detect infeasible problems.","category":"page"},{"location":"literate/build/custom_cone/#Support-infeasibility-detection","page":"Custom Cone Constraint","title":"Support infeasibility detection","text":"","category":"section"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"If no further information about the new cone is provided, the infeasibility detection is disabled. However, by defining the two additional methods in_dual and in_pol_recc we can support infeasibility detection. More information on how infeasible problems are detected in COSMO can be found here [1]. We will have to add the ability for the solver to check whether a vector is in the dual cone mathcalK^* of our new cone mathcalK and whether a vector is in the recession cone of the polar cone mathcalK^circ^infty of our cone. The first function is used to check for primal infeasibility, while the second function is used to check for dual infeasibility. Luckily, for the Nonpositives-cone this is straightforward:","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"mathbbR_-^n^* = mathbbR_-^n quad mathbbR_-^n^circ^infty = mathbbR_+^n","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"function COSMO.in_dual(x::AbstractVector{T}, C::Nonpositives{T}, tol::T) where {T <: AbstractFloat}\n\treturn !any( x-> (x > -tol), x)\nend\n\nfunction COSMO.in_pol_recc(x::AbstractVector{T}, C::Nonpositives{T}, tol::T) where {T <: AbstractFloat}\n\treturn !any( x-> (x < tol), x)\nend","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"Notice that for numerical reasons, we give the check-functions a bit of slack using the tolerance parameter tol. To test the infeasibility detection, we will attempt to use our new cone to solve the clearly dual infeasible problem:","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"beginarrayll textminimize x\ntextsubject to xleq 3\nendarray","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"n = 1\nP = spzeros(n, n)\nq = [1.]\n\n# x <= 3 <=> x - 3 in Nonpositives\nAi = [1.]\nbi = [-3.]\nci = COSMO.Constraint(Ai, bi, Nonpositives);\n\nmodel = COSMO.Model();\nassemble!(model, P, q, [ci]);\nres = COSMO.optimize!(model);\nnothing #hide","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"@test res.status == :Dual_infeasible","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"which is what we would expect.","category":"page"},{"location":"literate/build/custom_cone/#Using-new-cone-with-JuMP","page":"Custom Cone Constraint","title":"Using new cone with JuMP","text":"","category":"section"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"The nice composibility of Julia-code allows us to use our new cone definition even when the problem is modelled with JuMP. For this to work, we define a concrete subtype of MOI.AbstractSet, that JuMP can use to build a constraint.","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"using MathOptInterface, JuMP\nimport Base.copy\nconst MOI = MathOptInterface\n\nstruct NonPos <: MOI.AbstractVectorSet\n\tdimension::Int\nend\nBase.copy(set::NonPos) = set #this is needed for MOI","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"Next, we tell COSMO that whenever JuMP wants to solve a problem with a NonPos-cone it should translate it into a Nonpositives constraint and we also tell MOI that we now support constraints of this new mysterious cone NonPos.","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"function COSMO.processSet!(b::Vector{T}, rows::UnitRange{Int}, cs, s::NonPos) where {T <: AbstractFloat}\n push!(cs, Nonpositives{T}(length(rows)))\n nothing\nend\n#tell MOI, that COSMO supports constraints with this new cone\nMOI.supports_constraint(optimizer::COSMO.Optimizer, ::Type{<:Union{MOI.VectorOfVariables, MOI.VectorAffineFunction{Float64}}}, ::Type{NonPos}) = true","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"We should now be able to model the LP from above in JuMP and solve it internally using our Nonpositives cone and projection function.","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"model = JuMP.Model(COSMO.Optimizer);\n@variable(model, x[1:3]);\n@objective(model, Max, x[1] + x[2] + x[3]);\n@constraint(model, A1 * x[1:2] .+ b1 in NonPos(2));\n@constraint(model, x[1] + x[3] == 5);\nJuMP.optimize!(model)","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"x_opt = JuMP.value.(x)","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"You can see in the solver output that indeed a set Nonpositives of dim: 2 was used.","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"The discussed cone of Nonpositives is of course trivial, but the ability to define new cones and constraints for COSMO can be very powerful to model complex problems.","category":"page"},{"location":"literate/build/custom_cone/#References","page":"Custom Cone Constraint","title":"References","text":"","category":"section"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"[1] Garstka et al. - COSMO: A conic operator splitting method for convex conic problems (2020)","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"","category":"page"},{"location":"literate/build/custom_cone/","page":"Custom Cone Constraint","title":"Custom Cone Constraint","text":"This page was generated using Literate.jl.","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"EditURL = \"../../../examples/lp.jl\"","category":"page"},{"location":"examples/lp/#Linear-Program","page":"Linear Program","title":"Linear Program","text":"","category":"section"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"We want to solve the following linear program with decision variable x:","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"beginarrayll textminimize c^top x\ntextsubject to A x leq b \n x geq 1 \n x_2 geq 5 \n x_1 + x_3 geq 4\nendarray","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"The problem can be solved with COSMO in the following way:","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"using COSMO, LinearAlgebra, SparseArrays, Test","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"##Define problem data:\nc = [1; 2; 3; 4.];\nA = Matrix(1.0I, 4, 4);\nb = [10.; 10; 10; 10];\nn = 4;\nnothing #hide","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"Create the constraints Ax + b in mathcalK:","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"# Ax <= b\nc1 = COSMO.Constraint(-A, b, COSMO.Nonnegatives);\n# x >= 1\nc2 = COSMO.Constraint(Matrix(1.0I, n, n), -ones(n), COSMO.Nonnegatives);\n# x2 >= 5\nc3 = COSMO.Constraint(1, -5, COSMO.Nonnegatives, n, 2:2);\n# x1 + x3 >= 4\nc4 = COSMO.Constraint([1 0 1 0], -4, COSMO.Nonnegatives);\nnothing #hide","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"Define matrix P and vector q for the objective function:","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"P = spzeros(4, 4);\nq = c;\nnothing #hide","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"Create, assemble and solve the model:","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"settings = COSMO.Settings(verbose=true, eps_abs = 1e-4, eps_rel = 1e-5);\nmodel = COSMO.Model();\nassemble!(model, P, q, [c1; c2; c3; c4], settings = settings);\nres = COSMO.optimize!(model)","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"Compare the result to the known solution:","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"@test isapprox(res.x[1:4], [3; 5; 1; 1], atol=1e-2, norm = (x -> norm(x, Inf)))","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"@test isapprox(res.obj_val, 20.0, atol=1e-2)","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"","category":"page"},{"location":"examples/lp/","page":"Linear Program","title":"Linear Program","text":"This page was generated using Literate.jl.","category":"page"},{"location":"performance/#Performance-Tips","page":"Performance Tips","title":"Performance Tips","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"There are a number of ways to improve the performance of the solver given a particular problem. If you are not satisfied with the performance there are a number of things you have to determine first. Is the solver slow because","category":"page"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"it's the first time you ran it in the current Julia session or\nis it because the solver needs a lot of iterations (convergence) or\nis each iteration or the initial factorisation slow (computational performance)?","category":"page"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"Let's see how each point can be addressed.","category":"page"},{"location":"performance/#First-run","page":"Performance Tips","title":"First run","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"Whenever a new Julia session is started, the first run will trigger a compilation of all functions based on their arguments used in your script. This will slow the first execution of COSMO down. After that Julia will call the fast compiled functions. To get around this, you can either keep your current Julia session open and discard the first run. Alternatively, if your problem is very large, you could solve a small version of your problem first to trigger the compilation. Another option is to use PackageCompiler to save compiled functions into a sysimage that can be loaded at startup.","category":"page"},{"location":"performance/#Solver-Timings","page":"Performance Tips","title":"Solver Timings","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"It is often instructive to look at the detailed solver timing results for your problem. This can reveal where most of the time is spent. To achieve this, run COSMO with the setting verbose_timing = true. After solving the problem with result = COSMO.optimize!(model) you can look at result.times for a breakdown of the times spent in different parts of the algorithm, see Timings for more details. Especially take a look at the ratio of factorisation time and iteration time. If you use JuMP to solve the problem, you can take a look at the timings with backend(model).optimizer.model.optimizer.results.times.","category":"page"},{"location":"performance/#Convergence","page":"Performance Tips","title":"Convergence","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"It is possible that COSMO converges slowly, i.e. needs a large number of iterations, for your problem given its default parameters.","category":"page"},{"location":"performance/#Parameters","page":"Performance Tips","title":"Parameters","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"You could try changing any of the following parameters:","category":"page"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"rho: The initial algorithm step parameter has a large influence on the convergence. Try different values between 1e-5 and 10.\nadaptive_rho = false: You can try to disable the automatic rho adaption and use different rho values.\nadaptive_rho_interval: This specifies after how many iterations COSMO tries to adapt the rho parameter. You can also set adaptive_rho_interval = 0 which adapts the rho parameter after the time spent iterating passes 40% of the factorisation time. This is currently the default in OSQP and works well with QPs.\nalpha = 1.0: This disables the over-relaxation that is used in the algorithm. We recommend values between 1.0 - 1.6.\nscaling = 0: This disables the problem scaling.\neps_abs and eps_rel: Check the impact of modifying the stopping accuracies.","category":"page"},{"location":"performance/#Use-warm-starting","page":"Performance Tips","title":"Use warm starting","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"The number of iterations can be dramatically decreased by providing a good initial guess for x, s and y. Examples where warm starting is commonly used are model predictive control and portfolio backtests, see Warm starting.","category":"page"},{"location":"performance/#Computational-performance","page":"Performance Tips","title":"Computational performance","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"If the convergence of the algorithm is not an issue, there are still a number of steps you can take to make COSMO faster.","category":"page"},{"location":"performance/#Intel-MKL-BLAS/LAPACK","page":"Performance Tips","title":"Intel MKL BLAS/LAPACK","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"We experienced significant performance improvements on Intel CPUs if Julia is compiled with MKL BLAS. This is because Julia's linear algebra function will use Intel MKL BLAS and LAPACK functions that are optimised for Intel hardware. The effect is especially significant for SDPs because most of the time is spent in the LAPACK function syevr. If you are running Julia on Intel hardware, an easy way to compile Julia with MKL is to add and build the MKL package, see MKL.jl. To verify your current BLAS vendor you can use julia> LinearAlgebra.BLAS.vendor().","category":"page"},{"location":"performance/#Linear-system-solver","page":"Performance Tips","title":"Linear system solver","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"COSMO uses QDLDL.jl as the default linear system solver. In our experience this seems to be a competitive choice until about 1e5 - 1e6 nonzeros in the constraint matrix. After that it is worth trying one of the indirect system solvers, such as CG or MINRES. Furthermore, we also recommend trying Pardiso (or MKLPardiso) for problems of that dimension. More details can be found here: Linear System Solver.","category":"page"},{"location":"performance/#Custom-cones","page":"Performance Tips","title":"Custom cones","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"In some cases the computations can be speed-up if certain constraints in the problem allow the implementation of a fast projection function. We allow the user to define their own custom convex cone with a corresponding projection function. The custom cone has to be defined as struct CustomCone{T} <: COSMO.AbstractConvexSet{T}. Furthermore, the user has to define a function that projects an input vector x onto the custom cone, i.e. function COSMO.project!(x::AbstractVector{T}, C::CustomCone{T}) where {T} ... end.","category":"page"},{"location":"performance/#Multithreading","page":"Performance Tips","title":"Multithreading","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"COSMO allows the execution of the projection step for multiple constraints in parallel using Julia's multithreading features. This is currently not enabled in the tagged release because of stability issues in earlier Julia versions. To use multithreading checkout the branch with_multi_threading, which we keep in sync with the latest tagged release. This can be installed via Julia's package manager with pkg> add COSMO#with_multi_threading. Afterwards, before starting Julia, set export JULIA_NUM_THREADS=[NUMBER_PHYSICAL_CORES_HERE]. In Julia you can verify the number of threads with julia> Threads.nthreads().","category":"page"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"Notice that the extra overhead for multithreading can slow the solver down if the problem is small. However, we noticed significant performance improvements if the problem contained multiple positive semidefinite constraints or when one large constraint was decomposed. In that case it also helps to restrict the number of BLAS threads per Julia thread with julia> BLAS.set_num_threads(1) to prevent oversubscription of the available cores.","category":"page"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"Multithreading can also be used in the factorisation step if the Pardiso or MKLPardiso solver are selected. This is only advisable for constraint matrices with more than 1e5 nonzeros.","category":"page"},{"location":"performance/#Chordal-decomposition-and-Clique-merging","page":"Performance Tips","title":"Chordal decomposition and Clique merging","text":"","category":"section"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"When solving large structured and sparse SDPs significant performance improvements are achievable if the problem is passed to COSMO in the right way. This means the solver has to be able to infer the structure of the positive semidefinite variable from the constraint. See the section on Chordal Decomposition for more details. In some cases the primal SDP doesn't allow decomposition but the dual SDP does, consider the Maximum Cut Problem and the Relaxed Two-Way Partitioning Problem for examples.","category":"page"},{"location":"performance/","page":"Performance Tips","title":"Performance Tips","text":"If the problem is decomposable it is also worth experimenting with different clique merging strategies to see how they impact the performance. More details can be found here: Clique merging.","category":"page"},{"location":"decomposition/#Chordal-Decomposition","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"","category":"section"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"For very large sparse semidefinite programs (SDPs) it is often helpful to analyse the sparsity structure of the PSD constraint(s). If the equality constraints impose a sparsity structure on the matrix variable, one PSD constraint on a large matrix variable can be decomposed into several smaller constraints. This results usually in a significant speedup and reduction in solve time.","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"The following example gives a short overview on chordal decomposition and clique merging. For more details, take a look at our paper on clique merging or watch the corresponding presentation that I gave on the topic.","category":"page"},{"location":"decomposition/#Example-problem","page":"Chordal Decomposition","title":"Example problem","text":"","category":"section"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Let's consider the following SDP in standard dual form:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"beginarrayll\n textminimize c^top x \n textsubject to displaystyle sum_i=1^m A_i x_i + S = B\n S in mathbbS^n_+\nendarray","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"with problem data matrices A_1 ldots A_m B in mathbbS^n, vector variable x in mathbbR^n, and matrix variable S in mathbbS^n_+.","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Let's look at the following example problem with m=2 and n=9:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"A1 = [-4.0 0.0 -2.0 0.0 0.0 -1.0 0.0 0.0 0.0; 0.0 -3.0 -1.0 0.0 0.0 0.0 0.0 0.0 0.0; -2.0 -1.0 -2.0 0.0 0.0 5.0 4.0 -4.0 0.0; 0.0 0.0 0.0 -4.0 -5.0 0.0 0.0 3.0 0.0; 0.0 0.0 0.0 -5.0 4.0 0.0 0.0 2.0 0.0; -1.0 0.0 5.0 0.0 0.0 5.0 -4.0 -4.0 -5.0; 0.0 0.0 4.0 0.0 0.0 -4.0 -1.0 -1.0 -3.0; 0.0 0.0 -4.0 3.0 2.0 -4.0 -1.0 2.0 -2.0; 0.0 0.0 0.0 0.0 0.0 -5.0 -3.0 -2.0 -3.0];\n\nA2 = [-5.0 0.0 3.0 0.0 0.0 -2.0 0.0 0.0 0.0; 0.0 -3.0 -5.0 0.0 0.0 0.0 0.0 0.0 0.0; 3.0 -5.0 3.0 0.0 0.0 5.0 -4.0 -5.0 0.0; 0.0 0.0 0.0 3.0 2.0 0.0 0.0 -2.0 0.0; 0.0 0.0 0.0 2.0 4.0 0.0 0.0 -3.0 0.0; -2.0 0.0 5.0 0.0 0.0 1.0 -5.0 -2.0 -4.0; 0.0 0.0 -4.0 0.0 0.0 -5.0 -2.0 -3.0 3.0; 0.0 0.0 -5.0 -2.0 -3.0 -2.0 -3.0 5.0 3.0; 0.0 0.0 0.0 0.0 0.0 -4.0 3.0 3.0 -4.0];\n\nB = [-0.11477375644968069 0.0 6.739182490600791 0.0 0.0 -1.2185593245043502 0.0 0.0 0.0; 0.0 1.2827680528587497 -5.136452036888789 0.0 0.0 0.0 0.0 0.0 0.0; 6.739182490600791 -5.136452036888789 7.344770673489607 0.0 0.0 -0.2224400187044442 -10.505300166831221 -1.2627361794562273 0.0; 0.0 0.0 0.0 10.327710040060499 8.91534585379813 0.0 0.0 -6.525873789637007 0.0; 0.0 0.0 0.0 8.91534585379813 0.8370459338528677 0.0 0.0 -6.210900615408826 0.0; -1.2185593245043502 0.0 -0.2224400187044442 0.0 0.0 -3.8185953011245024 -0.994033914192722 2.8156077981712997 1.4524716674219218; 0.0 0.0 -10.505300166831221 0.0 0.0 -0.994033914192722 0.029162208619863517 -2.8123790276830745 7.663416446183705; 0.0 0.0 -1.2627361794562273 -6.525873789637007 -6.210900615408826 2.8156077981712997 -2.8123790276830745 4.71893305728242 6.322431630550857; 0.0 0.0 0.0 0.0 0.0 1.4524716674219218 7.663416446183705 6.322431630550857 0.5026094532322212];\n\nc = [-0.21052661285686525, -1.263324575834677];","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Note that the data matrices A_1dots A_m B all have the same sparsity pattern (common zeros in certain entries). Take a look at A_1:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"A1 = [-4.0 0.0 -2.0 0.0 0.0 -1.0 0.0 0.0 0.0; 0.0 -3.0 -1.0 0.0 0.0 0.0 0.0 0.0 0.0; -2.0 -1.0 -2.0 0.0 0.0 5.0 4.0 -4.0 0.0; 0.0 0.0 0.0 -4.0 -5.0 0.0 0.0 3.0 0.0; 0.0 0.0 0.0 -5.0 4.0 0.0 0.0 2.0 0.0; -1.0 0.0 5.0 0.0 0.0 5.0 -4.0 -4.0 -5.0; 0.0 0.0 4.0 0.0 0.0 -4.0 -1.0 -1.0 -3.0; 0.0 0.0 -4.0 3.0 2.0 -4.0 -1.0 2.0 -2.0; 0.0 0.0 0.0 0.0 0.0 -5.0 -3.0 -2.0 -3.0] # hide\nA1","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Since all the data matrices have zeros in the same places, the equality constraint of the problem tells us that our matrix variable S will also have zeros in these places. The aggregated pattern matrix of S is shown in the following figure, where a dot represents a nonzero entry:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"(Image: )","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Since the matrices are symmetric we only show the lower triangle. We can represent the aggregated sparsity pattern of the problem using a graph G(V E), where the vertex set V is given by the column indices of the matrix and we introduce an edge (ij) in E for every nonzero matrix element S_ij. The graph for the example problem is shown in the right figure. In order to decompose the problem, we have to find the cliques, i.e. completely connected subgraphs, of the graph G. These cliques represent the dense subblocks of non-zero entries in the matrix (see the colored entries in the figure). In order for the theory to work, we also have to require G to be a chordal graph. For the purpose of this example, we just assume that G is chordal and keep in mind that we can always make a graph chordal by adding more edges.","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"COSMO finds the cliques of the graph automatically, if a constraint of type COSMO.PsdCone or COSMO.PsdConeTriangle is present in the problem and additional equality constraints impose a structure on them. For the example problem COSMO finds the following cliques: mathcalC_1=136 mathcalC_2=23 mathcalC_3=3678 mathcalC_4=458 and mathcalC_5=6789.","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Let's denote the set of cliques mathcalB=mathcalC_1ldotsmathcalC_p. To represent the relationship between different cliques, e.g. in terms of overlapping entries, it is helpful to represent them either as a clique tree mathcalT(mathcalB mathcalE) (left) or a clique graph mathcalG(mathcalB xi) (right), shown in the following figures:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"(Image: ) (Image: )","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Once the cliques have been found, we can use the following theorem to decompose the problem (and speed up the solver significantly).","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Theorem (Agler's Theorem)","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Let G(VE) be a chordal graph with a set of maximal cliques mathcalC_1ldotsmathcalC_p . Then S in mathbbS^n_+(E0) if and only if there exist matrices S_ell in S^mathcalC_ell for ell = 1ldotsp such that","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"S = displaystyle sum_ell = 1^p T_ell^top S_ell T_ell","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"where T_ell is an entry-selector matrix that maps the subblocks S_ell into the correct entries of the original matrix S.","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"In other words, instead of solving the original problem and choosing S to be positive semidefinite, we only have to choose the subblocks S_ell of the matrix S to be positive semidefinite and make sure that the individual blocks \"add up\" to S. Thus, we can solve the equivalent problem:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"beginarrayll\n textminimize c^top x \n textsubject to displaystyle sum_i=1^m A_i x + sum_ell = 1^p T_l^top S_ell T_ell = B\n S_ell in mathbbS^mathcalC_ell_+ l = 1 ldots p\nendarray","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Compare this again with the problem at the top of the page. We have replaced the positive semidefinite constraint on the matrix variable S by p constraints on the smaller subblocks S_ell. Why is this useful?","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"When solving an SDP, the major linear algebra operation is a projection onto the constraint set, see Method. This projection is done at each iteration of the algorithm and for positive semidefinite constraints involves an eigenvalue decomposition of the matrix iterate. Unfortunately, an eigenvalue decomposition of a matrix of dimension n has a complexity of roughly mathcalO(n^3).","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"For our example this means that we now project one 2times 2 block, two 3 times 3 blocks and two 4 times 4 blocks instead of one 9 times 9 block. For small problems, this doesn't seem to make much of a difference, but consider of a problem with 1000 11 times 11 blocks along the diagonal and overlapping by one entry. Taking into account that the computational cost scales cubicly, we can improve the performance of the solver by projecting 1000 11times 11 blocks instead of projecting one giant 10001times 10001 block at each iteration.","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Let's go back to our example and solve it with COSMO and JuMP:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"model = JuMP.Model(with_optimizer(COSMO.Optimizer, decompose = true, merge_strategy = COSMO.NoMerge));\n@variable(model, x[1:2]);\n@objective(model, Min, c' * x )\n@constraint(model, Symmetric(B - A1 .* x[1] - A2 .* x[2] ) in JuMP.PSDCone());\n\nJuMP.optimize!(model)","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Notice that we set the solver settings decompose = true to allow COSMO to decompose the problem. Furthermore, we set the clique merging strategy to COSMO.NoMerge. This just means that after we have found the cliques we don't attempt to merge some of them (more about clique merging below). We get the following output from the solver:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"------------------------------------------------------------------\n COSMO v0.5.0 - A Quadratic Objective Conic Solver\n Michael Garstka\n University of Oxford, 2017 - 2019\n------------------------------------------------------------------\n\nProblem: x ∈ R^{13},\n constraints: A ∈ R^{35x13} (70 nnz),\n matrix size to factor: 48x48 (188 nnz)\nSets: PsdConeTriangle of dim: 10\n PsdConeTriangle of dim: 10\n PsdConeTriangle of dim: 6\n PsdConeTriangle of dim: 6\n PsdConeTriangle of dim: 3\nDecomp: Num of original PSD cones: 1\n Num decomposable PSD cones: 1\n Num PSD cones after decomposition: 5\n Merge Strategy: NoMerge\nSettings: ϵ_abs = 1.0e-04, ϵ_rel = 1.0e-04,\n ϵ_prim_inf = 1.0e-06, ϵ_dual_inf = 1.0e-04,\n ρ = 0.1, σ = 1.0e-6, α = 1.6,\n max_iter = 2500,\n scaling iter = 10 (on),\n check termination every 40 iter,\n check infeasibility every 40 iter,\n KKT system solver: QDLDL\nSetup Time: 0.14ms\n\nIter: Objective: Primal Res: Dual Res: Rho:\n40 -1.4134e+00 1.2320e-05 3.0409e-05 1.0000e-01\n\n------------------------------------------------------------------\n>>> Results\nStatus: Solved\nIterations: 40\nOptimal objective: -1.4134\nRuntime: 0.002s (1.93ms)","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Under sets we can indeed see that COSMO solved a problem with five PSD constraints corresponding to the cliques mathcalC_1ldots mathcalC_5 discovered in the sparsity pattern. (Note that the dimension printed is the number of entries in the upper triangle of the matrix block.)","category":"page"},{"location":"decomposition/#Clique-merging","page":"Chordal Decomposition","title":"Clique merging","text":"","category":"section"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"After we have found the cliques of the sparsity pattern, we are allowed to merge some of them back together. For the graph of the sparsity pattern this just means adding more edges or treating some structural zeros as numerical zeros. The main reason to merge two cliques is that they might overlap a lot and therefore it is not advantageous to treat them as two different blocks. Consider the two extreme cases below:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"(Image: ) (Image: )","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"In the left figure we have the ideal case that all the blocks overlap in just one entry. A full decomposition would leave us with a large number of small blocks. The sparsity pattern in the right figure has two large blocks overlapping almost entirely. In this case it would be disadvantageous to decompose the blocks. Instead, we would do the initial decomposition, realize the large overlap, and then merge the two blocks back together. For sparsity patterns that arise from real applications the case is not always as clear and we have to use more sophisticated strategies to decide which blocks to merge.","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"COSMO currently provides three different strategies that can be selected by the user:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"COSMO.NoMerge\nCOSMO.ParentChildMerge\nCOSMO.CliqueGraphMerge","category":"page"},{"location":"decomposition/#COSMO.NoMerge","page":"Chordal Decomposition","title":"COSMO.NoMerge","text":"NoMerge <: AbstractMergeStrategy\n\nA strategy that does not merge cliques.\n\n\n\n\n\n","category":"type"},{"location":"decomposition/#COSMO.ParentChildMerge","page":"Chordal Decomposition","title":"COSMO.ParentChildMerge","text":"ParentChildMerge(t_fill = 8, t_size = 8) <: AbstractTreeBasedMerge\n\nThe merge strategy suggested in Sun and Andersen - Decomposition in conic optimization with partially separable structure (2014). The initial clique tree is traversed in topological order and a clique mathcalC_ell is greedily merged to its parent clique mathcalC_par(ell) if at least one of the two conditions are met\n\n( mathcalC_par(ell) - eta_ell) (mathcalC_ell - eta_ell) leq t_textfill (fill-in condition)\nmax left nu_ell nu_par(ell) right leq t_textsize (supernode size condition)\n\n\n\n\n\n","category":"type"},{"location":"decomposition/#COSMO.CliqueGraphMerge","page":"Chordal Decomposition","title":"COSMO.CliqueGraphMerge","text":"CliqueGraphMerge(edge_weight::AbstractEdgeWeight = ComplexityWeight()) <: AbstractGraphBasedMerge\n\nThe (default) merge strategy based on the reduced clique graph mathcalG(mathcalB xi), for a set of cliques mathcalB = mathcalC_1 dots mathcalC_p where the edge set xi is obtained by taking the edges of the union of clique trees.\n\nMoreover, given an edge weighting function e(mathcalC_imathcalC_j) = w_ij, we compute a weight for each edge that quantifies the computational savings of merging the two cliques. After the initial weights are computed, we merge cliques in a loop:\n\nwhile clique graph contains positive weights:\n\nselect two permissible cliques with the highest weight w_ij\nmerge cliques rightarrow update clique graph\nrecompute weights for updated clique graph\n\nCustom edge weighting functions can be used by defining your own CustomEdgeWeight <: AbstractEdgeWeight and a corresponding edge_metric method. By default, the ComplexityWeight <: AbstractEdgeWeight is used which computes the weight based on the cardinalities of the cliques: e(mathcalC_imathcalC_j) = mathcalC_i^3 + mathcalC_j^3 - mathcalC_i cup mathcalC_j^3.\n\nSee also: Garstka, Cannon, Goulart - A clique graph based merging strategy for decomposable SDPs (2019)\n\n\n\n\n\n","category":"type"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"In our example problem we have two cliques mathcalC_3 = 3678 and mathcalC_5 = 6789 that overlap in three entries. Let's solve the problem again and choose the default clique merging strategy merge_strategy = COSMO.CliqueGraphMerge:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"model = JuMP.Model(with_optimizer(COSMO.Optimizer, decompose = true, merge_strategy = COSMO.CliqueGraphMerge));\n@variable(model, x[1:2]);\n@objective(model, Min, c' * x )\n@constraint(model, Symmetric(B - A1 .* x[1] - A2 .* x[2] ) in JuMP.PSDCone());\n\nJuMP.optimize!(model)","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"------------------------------------------------------------------\n COSMO v0.5.0 - A Quadratic Objective Conic Solver\n Michael Garstka\n University of Oxford, 2017 - 2019\n------------------------------------------------------------------\n\nProblem: x ∈ R^{7},\n constraints: A ∈ R^{30x7} (58 nnz),\n matrix size to factor: 37x37 (153 nnz)\nSets: PsdConeTriangle of dim: 15\n PsdConeTriangle of dim: 6\n PsdConeTriangle of dim: 6\n PsdConeTriangle of dim: 3\nDecomp: Num of original PSD cones: 1\n Num decomposable PSD cones: 1\n Num PSD cones after decomposition: 4\n Merge Strategy: CliqueGraphMerge\nSettings: ϵ_abs = 1.0e-04, ϵ_rel = 1.0e-04,\n ϵ_prim_inf = 1.0e-06, ϵ_dual_inf = 1.0e-04,\n ρ = 0.1, σ = 1.0e-6, α = 1.6,\n max_iter = 2500,\n scaling iter = 10 (on),\n check termination every 40 iter,\n check infeasibility every 40 iter,\n KKT system solver: QDLDL\nSetup Time: 0.26ms\n\nIter: Objective: Primal Res: Dual Res: Rho:\n40 -1.4134e+00 1.7924e-05 1.0416e-04 1.0000e-01\n\n------------------------------------------------------------------\n>>> Results\nStatus: Solved\nIterations: 40\nOptimal objective: -1.4134\nRuntime: 0.003s (2.68ms)","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Unsurprisingly, we can see in the output that COSMO solved a problem with four PSD constraints. One of them is of dimension 15, i.e. a 5times 5 block, which correspond to the merged clique mathcalC_3 cup mathcalC_5 = 36789 .","category":"page"},{"location":"decomposition/#Completing-the-dual-variable","page":"Chordal Decomposition","title":"Completing the dual variable","text":"","category":"section"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"After a decomposed problem is solved, we can recover the solution to the original problem by assembling the matrix variable S from its subblocks S_ell:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"S = displaystyle sum_ell = 1^p T_ell^top S_ell T_ell","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"Following Agler's Theorem, S will be a positive semidefinite matrix. However, this is not true for the corresponding dual variable matrix Y. The dual variable returned after solving the decomposed problem will be in the space of PSD completable matrices Y in mathbbS_+^n(E). This means that the entries in Y corresponding to the blocks S_ell (black dots) have been chosen correctly. The numerical values for all the other entries (corresponding to the zeros in S and denoted with a red dot) have to be chosen in the right way to make Y positive semidefinite.","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"(Image: )","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"For more information about PSD matrix completion and the completion algorithm used in COSMO take a look at Vandenberghe and Andersen - Chordal Graphs and Semidefinite Optimization (Ch.10). To configure COSMO to complete the dual variable after solving the problem you have to set the complete_dual option:","category":"page"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"model = JuMP.Model(with_optimizer(COSMO.Optimizer, complete_dual = true));","category":"page"},{"location":"decomposition/#Example-Code","page":"Chordal Decomposition","title":"Example Code","text":"","category":"section"},{"location":"decomposition/","page":"Chordal Decomposition","title":"Chordal Decomposition","text":"The code used for this example can be found in /examples/chordal_decomposition.jl.","category":"page"},{"location":"acceleration/#Acceleration","page":"Acceleration","title":"Acceleration","text":"","category":"section"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"By default COSMO's ADMM algorithm is wrapped in a safeguarded acceleration method to achieve faster convergence to higher precision. COSMO uses accelerators from the COSMOAccelerators.jl package.","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"By default, the solver uses the accelerator type AndersonAccelerator{T, Type2{QRDecomp}, RestartedMemory, NoRegularizer}. This is the classic type-II Anderson acceleration method where the least squares subproblem is solved using an updated QR method. Moreover, the method is restarted, i.e. the history of iterates is deleted, after mem steps and no regularisation for the least-squares method is used.","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"In addition, the method is safeguarded (safeguard = true), i.e. the residual-norm of the accelerated point can not deviate too much from the current point. Otherwise, the point is discarded and the ADMM algorithm performs a normal step instead.","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"The acceleration method can be altered as usual via the solver settings and the accelerator keyword. To deactivate acceleration pass an EmptyAccelerator:","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"settings = COSMO.Settings(accelerator = EmptyAccelerator)","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"To use the default accelerator but with a different memory size (number of stored iterates) use:","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"settings = COSMO.Settings(accelerator = with_options(AndersonAccelerator, mem = 15))","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"To turn the safeguarding off use:","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"settings = COSMO.Settings(safeguard = false)","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"To use an Anderson Accelerator of Type-I with a rolling-memory (oldest iterate replaced by newest) approach, use:","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"settings = COSMO.Settings(accelerator = AndersonAccelerator{Float64, Type1, RollingMemory, NoRegularizer})","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"For more fine-grained control look at the implementation of the accelerator here.","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"When JuMP is used, the accelerator settings can be passed in the usual way:","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"model = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"accelerator\" => with_options(AndersonAccelerator, mem = 15)))","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"Or using the set_optimizer_attribute() method:","category":"page"},{"location":"acceleration/","page":"Acceleration","title":"Acceleration","text":"model = JuMP.Model(COSMO.Optimizer);\nset_optimizer_attribute(model, \"accelerator\", with_options(AndersonAccelerator, mem = 15))","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"EditURL = \"../../../examples/sum_abs_k_eigenvalues.jl\"","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/#Minimizing-the-sum-of-the-k-largest-λ","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"","category":"section"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"We show how to find the sum of absolute value of the k largest eigenvalues of a symmetric matrix A in mathbbS^n. This problem can be solved as a semidefinite program. The primal and dual forms are stated in Alizadeh [1]:","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"beginarrayllll textmaximize textTr(AY) - textTr(AW) textminimize kz + Tr(U) + Tr(V) \ntextsubject to textTr(Y + W) = k textsubject to zI + V - A succeq 0 \n 0 preceq Y preceq I zI + U + A succeq 0 \n 0 preceq W preceq I U V succeq 0\nendarray","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"where Y W are the variables of the primal and U V are the variables of the dual problem.","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"using LinearAlgebra, JuMP, COSMO, Random\nrng = Random.MersenneTwister(212)\n\nn = 10\nA = 5 .* randn(rng, 10, 10)\nA = Symmetric(A, :U)","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"We are interested in minimizing the sum of absolute values of the k=3 largest eigenvalues. Let's formulate the problem in JuMP with COSMO as the backend solver:","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"k = 3\nmodel = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"verbose\" => true));\n@variable(model, Y[1:n, 1:n], PSD);\n@variable(model, W[1:n, 1:n], PSD);\n\n@objective(model, Max, tr(A * Y) - tr(A * W));\n@constraint(model, tr(Y + W) == k);\n@constraint(model, Symmetric(I - Y) in PSDCone());\n@constraint(model, Symmetric(I - W) in PSDCone());\nstatus = JuMP.optimize!(model)","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"opt_objective = JuMP.objective_value(model)","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"Now, we can check the solution by computing the sum of the absolute value of the 3-largest eigenvalues:","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"k_λ_abs = sum(sort(abs.(eigen(A).values), rev = true)[1:k])","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/#Solve-the-dual","page":"Minimizing the sum of the k-largest λ","title":"Solve the dual","text":"","category":"section"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"Alternatively, we can solve the dual problem:","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"model = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"verbose\" => true));\n@variable(model, V[1:n, 1:n], PSD);\n@variable(model, U[1:n, 1:n], PSD);\n@variable(model, z);\n\n@objective(model, Min, k * z + tr(V) + tr(U));\n@constraint(model, Symmetric(z .* diagm(0 => ones(n)) + V - A) in PSDCone());\n@constraint(model, Symmetric(z .* diagm(0 => ones(n)) + U + A) in PSDCone());\nstatus = JuMP.optimize!(model)","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"opt_objective = JuMP.objective_value(model)","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"This gives the same result.","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/#Problem-with-A-as-variable","page":"Minimizing the sum of the k-largest λ","title":"Problem with A as variable","text":"","category":"section"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"Above problems are mostly helpful for illustrative purpose. It is obviously easier to find the sum of the k-largest eigenvalues by simply computing the eigenvalues of A. However, above results become useful if finding A itself is part of the problem. For example, assume we want to find a valid matrix A under the constraints: C textvec(A) = b with the minimum sum of absolute values of the k-largest eigenvalues. We can then solve the equivalent problem:","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"beginarrayll textminimize kz + Tr(U) + Tr(V) \n textsubject to C textvec(A) = b \n zI + V - A succeq 0 \n zI + U + A succeq 0 \n U V succeq 0\nendarray","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/#References","page":"Minimizing the sum of the k-largest λ","title":"References","text":"","category":"section"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"[1] Alizadeh - Interior point methods in semidefinite programming with applications to combinatorial optimization (1995)","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"","category":"page"},{"location":"examples/sum_abs_k_eigenvalues/","page":"Minimizing the sum of the k-largest λ","title":"Minimizing the sum of the k-largest λ","text":"This page was generated using Literate.jl.","category":"page"},{"location":"api/#API-Reference","page":"API Reference","title":"API Reference","text":"","category":"section"},{"location":"api/#Model","page":"API Reference","title":"Model","text":"","category":"section"},{"location":"api/","page":"API Reference","title":"API Reference","text":"COSMO.Model\nCOSMO.assemble!\nCOSMO.set!\nCOSMO.empty_model!\nCOSMO.warm_start_primal!\nCOSMO.warm_start_slack!\nCOSMO.warm_start_dual!","category":"page"},{"location":"api/#COSMO.Model","page":"API Reference","title":"COSMO.Model","text":"Model{T <: AbstractFloat}()\n\nInitializes an empty COSMO model that can be filled with problem data using assemble!(model, P, q,constraints; [settings, x0, s0, y0]).\n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.assemble!","page":"API Reference","title":"COSMO.assemble!","text":"assemble!(model, P, q, constraint(s); [settings, x0, y0, s0])\n\nAssembles a COSMO.Model with a cost function defind by P and q, and a number of constraints.\n\nThe positive semidefinite matrix P and vector q are used to specify the cost function of the optimization problem:\n\nmin 1/2 x'Px + q'x\ns.t. Ax + b ∈ C\n\nconstraints is a COSMO.Constraint or an array of COSMO.Constraint objects that are used to describe the constraints on x.\n\n\n\nThe optional keyword argument settings can be used to pass custom solver settings:\n\ncustom_settings = COSMO.Settings(verbose = true);\nassemble!(model, P, q, constraints, settings = custom_settings)\n\n\n\nThe optional keyword arguments x0 and y0 can be used to provide the solver with warm starting values for the primal variable x and the dual variable y.\n\nx_0 = [1.0; 5.0; 3.0]\nCOSMO.assemble!(model, P, q, constraints, x0 = x_0)\n\n\n\n\n\n","category":"function"},{"location":"api/#COSMO.set!","page":"API Reference","title":"COSMO.set!","text":"set!(model, P, q, A, b, convex_sets, [settings])\n\nSets model data directly based on provided fields.\n\n\n\n\n\n","category":"function"},{"location":"api/#COSMO.empty_model!","page":"API Reference","title":"COSMO.empty_model!","text":"empty_model!(model)\n\nResets all the fields of model to that of a model created with COSMO.Model() (apart from the settings).\n\n\n\n\n\n","category":"function"},{"location":"api/#COSMO.warm_start_primal!","page":"API Reference","title":"COSMO.warm_start_primal!","text":"warm_start_primal!(model, x0, [ind])\n\nProvides the COSMO.Model with warm starting values for the primal variable x. ind can be used to warm start certain components of x.\n\n\n\n\n\n","category":"function"},{"location":"api/#COSMO.warm_start_slack!","page":"API Reference","title":"COSMO.warm_start_slack!","text":"warm_start_slack!(model, s0, [ind])\n\nProvides the COSMO.Model with warm starting values for the primal slack variable s. ind can be used to warm start certain components of s.\n\n\n\n\n\n","category":"function"},{"location":"api/#COSMO.warm_start_dual!","page":"API Reference","title":"COSMO.warm_start_dual!","text":"warm_start_dual!(model, y0, [ind])\n\nProvides the COSMO.Model with warm starting values for the dual variable y. ind can be used to warm start certain components of y.\n\n\n\n\n\n","category":"function"},{"location":"api/#Constraints","page":"API Reference","title":"Constraints","text":"","category":"section"},{"location":"api/","page":"API Reference","title":"API Reference","text":"COSMO.Constraint\nCOSMO.ZeroSet\nCOSMO.Nonnegatives\nCOSMO.Box\nCOSMO.SecondOrderCone\nCOSMO.PsdCone\nCOSMO.PsdConeTriangle\nCOSMO.ExponentialCone\nCOSMO.DualExponentialCone\nCOSMO.PowerCone\nCOSMO.DualPowerCone","category":"page"},{"location":"api/#COSMO.Constraint","page":"API Reference","title":"COSMO.Constraint","text":"Constraint{T <: AbstractFloat}(A, b, convex_set_type, dim = 0, indices = 0:0)\n\nCreates a COSMO constraint: Ax + b ∈ convex_set.\n\nBy default the following convex set types are supported: ZeroSet, Nonnegatives, SecondOrderCone, PsdCone, PsdConeTriangle.\n\nExamples\n\njulia> COSMO.Constraint([1 0;0 1], zeros(2), COSMO.Nonnegatives)\nConstraint\nSize of A: (2, 2)\nConvexSet: COSMO.Nonnegatives{Float64}\n\nFor convex sets that require their own data, it is possible to pass the pass the instantiated object directly rather than the type name.\n\nExamples\n\njulia> COSMO.Constraint([1 0;0 1], zeros(2), COSMO.Box([-1.;-1.],[1.;1.]))\nConstraint\nSize of A: (2, 2)\nConvexSet: COSMO.Box{Float64}\n\n\n\nThe optional arguments dim and indices can be used to specify A and b for subparts of variable x. If x has dimension dim = 4, then x[2] and x[3] can be constrained to the zero cone in the following way:\n\nExamples\n\njulia> c = COSMO.Constraint([1 0;0 1], zeros(2), COSMO.ZeroSet, 4, 2:3)\nConstraint\nSize of A: (2, 4)\nConvexSet: COSMO.ZeroSet{Float64}\n\nNotice that extra columns of A have been added automatically.\n\njulia>Matrix(c.A)\n2×4 Array{Float64,2}:\n0.0 1.0 0.0 0.0\n0.0 0.0 1.0 0.0\n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.ZeroSet","page":"API Reference","title":"COSMO.ZeroSet","text":"ZeroSet(dim)\n\nCreates the zero set 0 ^dim of dimension dim. If x ∈ ZeroSet then all entries of x are zero.\n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.Nonnegatives","page":"API Reference","title":"COSMO.Nonnegatives","text":"Nonnegatives(dim)\n\nCreates the nonnegative orthant x in mathbbR^dim x ge 0 of dimension dim.\n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.Box","page":"API Reference","title":"COSMO.Box","text":"Box(l, u)\n\nCreates a box or intervall with lower boundary vector l in mathbbR^m cup -infty^m and upper boundary vectoru in mathbbR^mcup +infty^m.\n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.SecondOrderCone","page":"API Reference","title":"COSMO.SecondOrderCone","text":"SecondOrderCone(dim)\n\nCreates the second-order cone (or Lorenz cone) (tx) in mathrmR^dim x _2 leq t .\n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.PsdCone","page":"API Reference","title":"COSMO.PsdCone","text":"PsdCone(dim)\n\nCreates the cone of symmetric positive semidefinite matrices mathcalS_+^dim. The entries of the matrix X are stored column-by-column in the vector x of dimension dim. Accordingly X in mathbbS_+ Rightarrow x in mathcalS_+^dim, where X = textmat(x).\n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.PsdConeTriangle","page":"API Reference","title":"COSMO.PsdConeTriangle","text":"PsdConeTriangle(dim)\n\nCreates the cone of symmetric positive semidefinite matrices. The entries of the upper-triangular part of matrix X are stored in the vector x of dimension dim. A r times r matrix has r(r+1)2 upper triangular elements and results in a vector of mathrmdim = r(r+1)2.\n\nExamples\n\nThe matrix\n\nbeginbmatrix x_1 x_2 x_4 x_2 x_3 x_5 x_4 x_5 x_6 endbmatrix\n\nis transformed to the vector x_1 x_2 x_3 x_4 x_5 x_6^top with corresponding constraint PsdConeTriangle(6).\n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.ExponentialCone","page":"API Reference","title":"COSMO.ExponentialCone","text":"ExponentialCone(MAX_ITERS = 100, EXP_TOL = 1e-8)\n\nCreates the exponential cone mathcalK_exp = (x y z) mid y geq 0 ye^xy z cup (xyz) mid x leq 0 y = 0 z geq 0 \n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.DualExponentialCone","page":"API Reference","title":"COSMO.DualExponentialCone","text":"DualExponentialCone(MAX_ITERS::Int = 100, EXP_TOL = 1e-8)\n\nCreates the dual exponential cone mathcalK^*_exp = (x y z) mid x 0 -xe^yx leq e^1 z cup (0yz) mid y geq 0 z geq 0 \n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.PowerCone","page":"API Reference","title":"COSMO.PowerCone","text":"PowerCone(alpha::Float64, MAX_ITERS::Int = 20, POW_TOL = 1e-8)\n\nCreates the 3-d power cone mathcalK_pow = (x y z) mid x^alpha y^(1-alpha) geq z x geq 0 y geq 0 with 0 alpha 1\n\n\n\n\n\n","category":"type"},{"location":"api/#COSMO.DualPowerCone","page":"API Reference","title":"COSMO.DualPowerCone","text":"DualPowerCone(alpha::Float64, MAX_ITERS::Int = 20, POW_TOL = 1e-8)\n\nCreates the 3-d dual power cone mathcalK^*_pow = (u v w) mid left( fracualpharight)^alpha left( fracv1-alpharight)^(1-alpha) geq w u geq 0 v geq 0 with 0 alpha 1\n\n\n\n\n\n","category":"type"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"EditURL = \"../arbitrary_precision.jl\"","category":"page"},{"location":"literate/build/arbitrary_precision/#Arbitrary-Precision","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"","category":"section"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"COSMO allows you to solve problems with arbitrary floating-point precision, e.g. by using BigFloat problem data. To do this, the desired floating point type has to be consistent across the model COSMO.Model{<: AbstractFloat}, the input data and the (optional) settings object COSMO.Settings{<: AbstractFloat}. As an example, assume we want to solve the following quadratic program:","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"beginarrayll textminimize 12 x^top P x + q^top x \ntextsubject to l leq A x leq u\nendarray","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"where P = beginbmatrix 4 1 1 2endbmatrix, q = 1 1^top, A = beginbmatrix 1 1 1 0 0 1endbmatrix and l= 10 0^top, u=1 07 07^top. We start by creating the model with the desired precision:","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"using COSMO, LinearAlgebra, SparseArrays\nmodel = COSMO.Model{BigFloat}()","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"Next, we define the problem data as BigFloat arrays and create the constraint:","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"q = BigFloat[1; 1.]\nP = sparse(BigFloat[4. 1; 1 2])\nA = BigFloat[1. 1; 1 0; 0 1]\nl = BigFloat[1.; 0; 0]\nu = BigFloat[1; 0.7; 0.7]\n\nconstraint = COSMO.Constraint(A, zeros(BigFloat, 3), COSMO.Box(l, u))","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"Notice that the constraint type parameter is dependent on the input data. The same is true for the constraint set Box. Next, we define the settings","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"settings = COSMO.Settings{BigFloat}(verbose = true, kkt_solver = QdldlKKTSolver)","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"and assemble and solve the problem:","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"assemble!(model, P, q, constraint, settings = settings)\nresult = COSMO.optimize!(model);\nnothing #hide","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"Moreover, notice that when no type parameter is specified, all objects default to Float64:","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"model = COSMO.Model()","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"Two limitations to arbitrary precision:","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"Since we call the LAPACK function syevr for eigenvalue decompositions, we currently only support solving problems with PSD constraints in Float32 and Float64.\nWe suggest to use the pure Julia QDLDL linear system solver (kkt_solver = QdldlKKTSolver) when working with arbitrary precision types as some of the other available solvers don't support all available precisions.","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"note: Note\n","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"If you want to use COSMO directly with MathOptInterface, you can use: COSMO.Optimizer{<: AbstractFloat} as your optimizer. Again, the problem data precision of your MathOptInterface-model has to agree with the optimizer's precision.","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"","category":"page"},{"location":"literate/build/arbitrary_precision/","page":"Arbitrary Precision","title":"Arbitrary Precision","text":"This page was generated using Literate.jl.","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"EditURL = \"../../../examples/maxcut.jl\"","category":"page"},{"location":"examples/maxcut/#Maximum-Cut-Problem","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"","category":"section"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"We are interested in solving an approximation to the maximum cut problem using semidefinite programming. Consider the graph G(VE) with weights w_ij shown below:","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"# create the weighted adjacency matrix of the graph and plot the graph\nusing LinearAlgebra, Plots, GraphRecipes\nn = 4;\nW = zeros(n, n);\nW[1, 2] = 1; W[1, 4] = 8;\nW[2, 3] = 2; W[2, 4] = 10;\nW[3, 4] = 6;\nW = Symmetric(W)\ngraphplot(W, names = 1:n, edgelabel = W, x = [0; 1; 1; 0], y = [1; 1; 0; 0], fontsize = 12, nodeshape =:circle)","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"The maximum cut problem tries to find a cut or a partition of the graph's vertices into two complementary sets S and barS such that the total weight of the edges between the two sets is maximized. For this small graph the problem is trivial. The optimal solution is S = 123 and barS=4. Formally, this problem can be written as a mixed-integer optimisation problem:","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"beginarrayll textmaximize frac12 sum_i j w_ij(1 - y_i y_j)\ntextsubject to y_i in -1 1 quad forall i in V\nendarray","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"where y_i = 1 indicates that v_i in S and y_i = -1 indicates that v_i in barS. This problem is of interest in the field of integrated circuit design, where one tries to minimize the number of cross-layer connections in a circuit under layout constraints.","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"For more complicated graphs this problem quickly becomes hard to solve to optimality and in fact is known to be NP-hard. For this example we are interested in the randomized approximation algorithm that relaxes the problem to an SDP and was devised in Goemans and Williamson (1995) [1] (see for more details).","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"The approach can be divided into three steps:","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"Solve a relaxed SDP to obtain Y^*\nRecover an approximate solution V via a Cholesky factorisation Y^* = V^top V\nRound the approximate solution using a random vector r from the unit sphere.","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"The authors showed that this approach guarantees a solution of at least 087856 times the optimal solution.","category":"page"},{"location":"examples/maxcut/#Solving-the-primal-SDP","page":"Maximum Cut Problem","title":"Solving the primal SDP","text":"","category":"section"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"Before we formulate the SDP, let's compute the Laplacian matrix L:","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"using LinearAlgebra, Random, SparseArrays, COSMO, JuMP\n\nrng = Random.MersenneTwister(1)\n\n# Compute the Laplacian matrix of the graph\nL = diagm(0 => W * ones(n)) - W;\nnothing #hide","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"Given the Laplacian matrix L, we are looking for the optimal Y^* that solves the following SDP:","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"beginarrayll textmaximize frac14 langle L Y rangle \ntextsubject to Y_ii = 1 quad textfor i = 1dotsn\n Y in mathbfS_+^n\nendarray","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"Notice that the solution Y^* can be viewed as a correlation matrix. Let's solve the problem using COSMO and JuMP.","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"model = JuMP.Model(COSMO.Optimizer);\n@variable(model, Y[1:n, 1:n], PSD);\n@objective(model, Max, 1 / 4 * dot(L, Y));\n@constraint(model, [i = 1:n], Y[i, i] == 1.);\nJuMP.optimize!(model)","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"Yopt = JuMP.value.(Y);\nobj_val = JuMP.objective_value(model)","category":"page"},{"location":"examples/maxcut/#Solving-the-dual-SDP","page":"Maximum Cut Problem","title":"Solving the dual SDP","text":"","category":"section"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"Notice that the decision matrix Y is generally dense (as correctly classified in the solver output above). Therefore, we won't be able to utilize COSMO's chordal decomposition features. However, assuming strong duality, it turns out that we can also solve the dual problem, which is given by:","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"beginarrayll textminimize sum_i gamma_i \ntextsubject to textdiag(gamma) - frac14 L = S \n S in mathbfS_+^n\nendarray","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"As you can see, the matrix S is constrained to have zeros in places specified by the graph (Laplacian). Therefore, COSMO can try to decompose this SDP (see solver output) and speed up its algorithm:","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"model_dual = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"complete_dual\" => true));\n@variable(model_dual, γ[1:n]);\n@objective(model_dual, Min, sum(γ));\n@constraint(model_dual, lmi, Symmetric(-1/4 * L + diagm(γ)) in JuMP.PSDCone());\nJuMP.optimize!(model_dual)\nobj_val = JuMP.objective_value(model_dual)","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"The primal variable Y^* can be recovered from the dual variable associated with the LMI-constraint:","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"Yopt = dual(lmi)","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"To get the correct positive semidefinite dual variable, we have to enable PSD completion of the dual variable in COSMO. Now, that we have a solution we can perform the remaining steps in the approximation algorithm.","category":"page"},{"location":"examples/maxcut/#Cholesky-factorisation-of-Y","page":"Maximum Cut Problem","title":"Cholesky factorisation of Y","text":"","category":"section"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"Compute the Cholesky factorisation of Y = V^top V to find the unit vectors v_1 dots v_n.","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"factor = cholesky(Yopt, Val(true); check = false);\nV = Matrix((factor.P * factor.L)');\n# normalize columns\nfor i in 1:n\n V[:, i] ./= norm(V[:, i]);\nend","category":"page"},{"location":"examples/maxcut/#Rounding-the-approximate-solution","page":"Maximum Cut Problem","title":"Rounding the approximate solution","text":"","category":"section"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"It remains to round the unit vectors v_i using a random vector r with each component drawn from mathcalU(0 1), to obtain the y_i's:","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"r = rand(rng, n)\nr /= norm(r, 2)\ny = ones(n)\nfor i in 1:n\n dot(r, V[:, i]) <= 0 && (y[i] = -1)\nend\ny","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"For larger graphs this rounding step could be repeated several times to improve the rounding. As expected S = 1 2 3 and barS= 4.","category":"page"},{"location":"examples/maxcut/#References","page":"Maximum Cut Problem","title":"References","text":"","category":"section"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"[1] Goemans and Williamson - Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programs (1995)","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"[2] Post-processing code steps 2 and 3 from this JuMP example","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"","category":"page"},{"location":"examples/maxcut/","page":"Maximum Cut Problem","title":"Maximum Cut Problem","text":"This page was generated using Literate.jl.","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"EditURL = \"../../../examples/svm_primal.jl\"","category":"page"},{"location":"examples/svm_primal/#Support-Vector-Machine","page":"Support Vector Machine","title":"Support Vector Machine","text":"","category":"section"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"We are showing how to solve a support vector machine problem with COSMO (and JuMP).","category":"page"},{"location":"examples/svm_primal/#Generating-the-Dataset","page":"Support Vector Machine","title":"Generating the Dataset","text":"","category":"section"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"We want to classify the points in this example dataset with m = 100 samples and n = 2 features:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"using Distributions: MvNormal\nusing Plots, LinearAlgebra, SparseArrays, Random, Test","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"# Generate dataset\nrng = Random.MersenneTwister(123);\nnum_samples = 100;\nXpos = rand(rng, MvNormal([1.5, 1.5], 1.25), div(num_samples, 2))';\nXneg = rand(rng, MvNormal([-1.5, -1.5], 1.25), div(num_samples, 2))';\nypos = ones(div(num_samples, 2));\nyneg = -ones(div(num_samples, 2));\nnothing #hide","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"# Plot dataset\nplot(Xpos[:, 1], Xpos[:, 2], color = :red, st=:scatter, markershape = :rect, label = \"positive\", xlabel = \"x1\", ylabel = \"x2\")\nplot!(Xneg[:, 1], Xneg[:, 2], color = :blue, st=:scatter, markershape = :circle, label = \"negative\")","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"with samples (x_1 x_2 ldots x_m) in mathbbR^2 and labels y_i in -11.","category":"page"},{"location":"examples/svm_primal/#Solving-SVM-as-a-QP","page":"Support Vector Machine","title":"Solving SVM as a QP","text":"","category":"section"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"We want to compute the weights w and bias term b of the (soft-margin) SVM classifier:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"beginarrayll\n textminimize w^2 + lambda sum_i=1^m textmax(0 1 - y_i(w^top x_i - b))\nendarray","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"where lambda is a hyperparameter. This problem can be solved as a quadratic program. We can rewrite above problem into an optimisation problem in primal form by introducing the auxiliary slack variables t_i:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"t_i = textmax(0 1 - y_i(w^T x_i - b)) quad t_i geq 0","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"This allows us to write the problems in standard QP format:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"beginarrayll\n textminimize w^2 + lambda sum_i=1^m t_i\n textsubject to y_i (w^top x_i - b) geq 1 - t_i quad textfor i = 1ldots m\n t_i geq 0 quad textfor i = 1ldots m\nendarray","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"Next, we will remove the bias term b by adding an initial feature x_0 = -1 to each sample (now: n = 3):","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"X = [-ones(num_samples) [Xpos; Xneg]];\ny = [ypos; yneg];\nm, n = size(X)","category":"page"},{"location":"examples/svm_primal/#Modelling-in-JuMP","page":"Support Vector Machine","title":"Modelling in JuMP","text":"","category":"section"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"We can model this problem using JuMP and then hand it to COSMO:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"using JuMP, COSMO","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"λ = 1.0; # hyperparameter\nmodel = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"verbose\" => true));\n\n\n@variable(model, w[1:n]);\n@variable(model, t[1:m] >= 0.);\n@objective(model, Min, w' * w + λ * ones(m)' * t);\n@constraint(model, diagm(0 => y) * X * w .+ t .- 1 .>= 0);\nstatus = JuMP.optimize!(model)","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"The optimal weights w = w_0 w_1 w_2^top (where w_0 = b) are:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"w_opt = JuMP.value.(w)","category":"page"},{"location":"examples/svm_primal/#Plotting-the-hyperplane","page":"Support Vector Machine","title":"Plotting the hyperplane","text":"","category":"section"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"The separating hyperplane is defined by w^top x - b = 0. To plot the hyperplane, we calculate x_2 over a range of x_1 values:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"x_2 = (-w_1 x_1 - w_0) w_2 text where w_0 = b","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"x1 = -4:0.1:4;\nx2 = (-w_opt[2] * x1 .- w_opt[1]) / w_opt[3]\nplot!(x1, x2, label = \"SVM separator\", legend = :topleft)","category":"page"},{"location":"examples/svm_primal/#Modelling-with-COSMO","page":"Support Vector Machine","title":"Modelling with COSMO","text":"","category":"section"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"The problem can also be solved by transforming it directly into COSMO's problem format. Define COSMO`s x-variable to be x=w t^top and choose P, q, accordingly:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"P = blockdiag(spdiagm(0 => ones(n)), spzeros(m, m));\nq = [zeros(n); 0.5 * λ * ones(m)];\nnothing #hide","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"Next we transform the first constraint y_i (w^top x_i - b) geq 1 - t_i quad textfor i = 1ldots m into COSMO's constraint format: Ax + b in mathcalK.","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"A1 = [(spdiagm(0 => y) * X) spdiagm(0 => ones(m))];\nb1 = -ones(m);\ncs1 = COSMO.Constraint(A1, b1, COSMO.Nonnegatives);\nnothing #hide","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"It remains to specify the constraint t_i geq 0 quad textfor i = 1ldots m:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"A2 = spdiagm(0 => ones(m));\nb2 = zeros(m);\ncs2 = COSMO.Constraint(A2, b2, COSMO.Nonnegatives, m+n, n+1:m+n);\nnothing #hide","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"Create, assemble and solve the COSMO.Model:","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"model2 = COSMO.Model();\nassemble!(model2, P, q, [cs1; cs2]);\nresult2 = COSMO.optimize!(model2);\nw_opt2 = result2.x[1:3];\n@test norm(w_opt2 - w_opt, Inf) < 1e-3","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"","category":"page"},{"location":"examples/svm_primal/","page":"Support Vector Machine","title":"Support Vector Machine","text":"This page was generated using Literate.jl.","category":"page"},{"location":"jump/#JuMP-Interface","page":"JuMP Interface","title":"JuMP Interface","text":"","category":"section"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"Our JuMP interface allows you to describe and modify your optimisation problem with JuMP and use COSMO as the backend solver. The interface is defined in /src/MOIWrapper.jl.","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"note: Note\nWe assume here that the latest JuMP release (~0.21.0) is used.","category":"page"},{"location":"jump/#Use-COSMO","page":"JuMP Interface","title":"Use COSMO","text":"","category":"section"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"To specify COSMO as the solver for your JuMP model, load the solver module with using COSMO and then pass a COSMO.Optimizer when initialising the JuMP model:","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"m = JuMP.Model(COSMO.Optimizer);","category":"page"},{"location":"jump/#Specify-Solver-Settings","page":"JuMP Interface","title":"Specify Solver Settings","text":"","category":"section"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"Solver-specific settings can be passed using the optimizer_with_attributes() function. For example, if you want to adjust the maximum number of iterations and turn on verbose printing use","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"m = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"max_iter\" => 5000, \"verbose\" => true));","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"Note that the attributes are passed as key-value pairs and the keys are strings. This is slightly different to using the native COSMO interface. Equivalently, one can also use:","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"m = JuMP.Model(COSMO.Optimizer);\nset_optimizer_attribute(m, \"max_iter\", 5000)\nset_optimizer_attribute(m, \"verbose\", true)","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"The full list of available settings can be found in the Settings section. All of them should be compatible with JuMP.","category":"page"},{"location":"jump/#Results","page":"JuMP Interface","title":"Results","text":"","category":"section"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"After solving the problem the result can be obtained using the standard JuMP commands. To see if the optimisation was successful use","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"JuMP.termination_status(m)\nJuMP.primal_status(m)","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"If a solution is available, the optimal objective value can be queried using","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"JuMP.objective_value(m)","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"and the value of a decision variable x can be obtained with","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"JuMP.value.(x)","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"To query the number of iterations of the algorithm use:","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"iter = MOI.get(m, COSMO.ADMMIterations())","category":"page"},{"location":"jump/#Feasibility-of-solution","page":"JuMP Interface","title":"Feasibility of solution","text":"","category":"section"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"In case JuMP.termination_status(m) is MOI.ITERATION_LIMIT, JuMP.primal_status(m) (resp. JuMP.dual_status(m)) should be checked to determine the feasibility status of the primal (resp. dual) solutions. A primal (resp. dual) solution is considered feasible (resported as MOI.FEASIBLE_POINT) if","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"r epsilon_textabs + sepsilon_textrel","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"where","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"r is the primal (resp. dual) residual r_prim = lVert Ax + s - b rVert_infty","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"(resp. r_dual = lVert P x + q - A^top * μ rVert_infty),","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"s is the primal (resp. dual) scaling of this residual","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"max_norm_prim = max(lVert Ax rVert_infty lVert s rVert_infty lVert b rVert_infty) (resp. max_norm_dual = max(lVert P x rVert_infty lVert q rVert_infty lVert A^top * μ rVert_infty)).","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"epsilon_textabs is the optimizer attribute eps_abs and\nepsilon_textrel is the optimizer attribute eps_rel.","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"A primal (resp. dual solution) is considered nearly feasible (resported as MOI.NEARLY_FEASIBLE_POINT) if it is not feasible but the same inequality is satisfied where the right-hand side is multiplied by the optimizer attribute nearly_ratio.","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"Otherwise, the solution is considered infeasible (resported as MOI.INFEASIBLE_POINT). Note that this does not mean that the problem is infeasible, it only means that the current primal (resp. dual) solution is infeasible.","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"Note that, in case JuMP.termination_status(m) is MOI.ITERATION_LIMIT, the feasibility of the solution is checked with the current optimizer attributes, eps_abs, eps_rel and nearly_ratio. This means that if they are changed after optimize!, the status may change! This can be used to check whether the solution is feasible up to a relaxed tolerance. For instance, the following could happen:","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"optimize!(m)\nprimal_status(m) # MOI.INFEASIBLE_POINT\n# This means that `r_prim >= nearly_ratio * (eps_abs * max_norm_prim * eps_rel)`\nset_optimizer_attribute(m, \"eps_abs\", 1e-4)\nprimal_status(m) # MOI.NEARLY_FEASIBLE_POINT\n# This means that `1 <= r_prim / (eps_abs * max_norm_prim * eps_rel) < nearly_ratio`","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"The values of r_prim, max_norm_prim, r_dual and max_norm_dual can also be accessed as the fields of the res_info struct that can be obtained as follows","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"res_info = MOI.get(m, COSMO.RawResult()).info","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"Then, the feasibility can either be checked manually as in","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"res_info.r_prim < eps_abs + res_info.max_norm_prim * eps_rel","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"or using one of the following:","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"COSMO.is_primal_feasible(res_info, eps_abs, eps_rel)\nCOSMO.is_primal_nearly_feasible(res_info, eps_abs, eps_rel)\nCOSMO.is_dual_feasible(res_info, eps_abs, eps_rel)\nCOSMO.is_dual_nearly_feasible(res_info, eps_abs, eps_rel)","category":"page"},{"location":"jump/","page":"JuMP Interface","title":"JuMP Interface","text":"For more information on how to use JuMP check the JuMP documentation.","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"EditURL = \"../portfolio_model_updates.jl\"","category":"page"},{"location":"literate/build/portfolio_model_updates/#Model-Updates","page":"Model Updates","title":"Model Updates","text":"","category":"section"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"When a pareto-optimal front in portfolio optimisation is computed for different risk-aversion parameters gamma, we are repeatedly solving the same model with only a slight change in the cost function. This allows us to reuse the KKT factorisation from previous solves and recompute the solution very efficiently, see Algorithm. For more information about solving portfolio optimsiation problems with COSMO see Portfolio Optimisation. We are planning to solve the following QP for different values of gamma:","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"beginarrayll textminimize x^top D x + y^top y - gamma^-1 mu^top x \ntextsubject to y = F^top x \n 1^top x = d + 1^top x^0 \n x geq 0\nendarray","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"As you can see gamma enters only in the linear part of the cost function, i.e. our vector q. Let's generate some example data:","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"using LinearAlgebra, SparseArrays, Random, COSMO\n\nrng = Random.MersenneTwister(1)\nk = 5; # number of factors\nn = k * 10; # number of assets\nD = spdiagm(0 => rand(rng, n) .* sqrt(k))\nF = sprandn(rng, n, k, 0.5); # factor loading matrix\nμ = (3 .+ 9. * rand(rng, n)) / 100. # expected returns between 3% - 12%\nd = 1 # we are starting from all cash\nx0 = zeros(n);\nnothing #hide","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"We want to solve this problem for the following risk-parameters:","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"gammas = [ 0.001, 0.01, 0.1, 0.5, 1., 3., 10, 100, 1000]","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"Let's solve the problem for the first parameter:","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"# x = (x, y)\nγ = gammas[1]\nP = blockdiag(D, spdiagm(0 => ones(k)))\nq = [-μ ./ (2 * γ); zeros(k)];\n\nAeq = [ transpose(F) -diagm(0 => ones(k));\n ones(1, n) zeros(1, k)]\nbeq = [zeros(k); -1.0]\nAineq = [spdiagm(0 => ones(n)) spzeros(n, k)]\n\ncs1 = COSMO.Constraint(Aeq, beq, COSMO.ZeroSet);\ncs2 = COSMO.Constraint(Aineq, zeros(n), COSMO.Nonnegatives);\ncosmo_settings = COSMO.Settings(verbose = false)\n\n# solve the problem once\nmodel = COSMO.Model();\nassemble!(model, P, q, [cs1; cs2], settings = cosmo_settings);\nresult = COSMO.optimize!(model);\nx_opt = view(result.x, 1:n);\ny_opt = view(result.x, n+1:n+k);\n\n# allocate some space for results\nrisks = zeros(length(gammas));\nreturns = zeros(length(gammas));\nreturns[1] = dot(μ, x_opt);\nrisks[1] = sqrt(dot(y_opt, y_opt));\nnothing #hide","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"After a successful solve, we can use the update!(model, q = nothing, b = nothing) function to update our model and re-solve the problem efficiently:","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"# solve the problem efficiently for the remaining γ using model updates\nfunction solve_repeatedly!(returns, risks, gammas, model, k, n, μ)\n for (i, γ) in enumerate(gammas[2:end])\n q_new = [-μ ./ (2 * γ); zeros(k)];\n\n # here we use the update! function to change the model\n update!(model, q = q_new)\n result = COSMO.optimize!(model);\n\n x_opt = view(result.x, 1:n)\n y_opt = view(result.x, n+1:n+k)\n returns[i + 1] = dot(μ, x_opt)\n risks[i + 1] = sqrt(dot(y_opt, y_opt))\n end\nend\nsolve_repeatedly!(returns, risks, gammas, model, k, n, μ);\nnothing #hide","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"We can now plot the risk-return trade-off curve:","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"using Plots\nPlots.plot(risks, returns, xlabel = \"Standard deviation (risk)\", ylabel = \"Expected return\", title = \"Risk-return trade-off for efficient portfolios\", legend = false)","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"","category":"page"},{"location":"literate/build/portfolio_model_updates/","page":"Model Updates","title":"Model Updates","text":"This page was generated using Literate.jl.","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"EditURL = \"../../../examples/portfolio_optimisation.jl\"","category":"page"},{"location":"examples/portfolio_optimisation/#Portfolio-Optimisation","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"","category":"section"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"We consider a single-period Markowitz portfolio optimisation example. Assume that we have a portfolio with n assets at the beginning of time period t. Given some forecasts on risks and expected returns we try to find the optimal trade vector that rebalances the portfolio to achieve a good balance between expected risk (variance) x^top Sigma x and returns mu^top x. In it's most simple form we want to solve:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"beginarrayll textmaximize mu^top x - gamma (x^top Sigma x)\ntextsubject to 1^top x = d + 1^top x^0 \n x geq 0\nendarray","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"with variable x in mathbfR^n, mu forecasted (expected) returns, gamma 0 risk aversion parameter x^0_i represents the initial investment in asset i and d represents the cash reserve. Consequently, the equality constraint tells us that the sum of the new allocation vector x has to equal the initial allocation plus the cash reserve. Furthermore, the covariance matrix of our risk model is given by Sigma in mathbfS_+^n. Here we assume a factor risk model, i.e. we can write Sigma as Sigma = D + F F^top where D is diagonal and the factor matrix F has a lower rank k n. This approach allows us to reduce the number of nonzeros in the problem. Furthermore, note that we don't consider shortselling in this example. Let's generate some problem data:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"using LinearAlgebra, SparseArrays, Random, COSMO, JuMP, Test\n\n# generate the data\nrng = Random.MersenneTwister(1)\nk = 5; # number of factors\nn = k * 10; # number of assets\nD = spdiagm(0 => rand(rng, n) .* sqrt(k))\nF = sprandn(rng, n, k, 0.5); # factor loading matrix\nμ = (3 .+ 9. * rand(rng, n)) / 100. # expected returns between 3% - 12%\nγ = 1.0; # risk aversion parameter\nd = 1 # we are starting from all cash\nx0 = zeros(n);\nnothing #hide","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"We can now write the problem as a QP:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"beginarrayll textminimize x^top D x + y^top y - gamma^-1 mu^top x \ntextsubject to y = F^top x \n 1^top x = d + 1^top x^0 \n x geq 0\nendarray","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"Before considering other effects, let's create the model in JuMP and solve it using COSMO:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"model = JuMP.Model(COSMO.Optimizer);\n@variable(model, x[1:n]);\n@variable(model, y[1:k]);\n@objective(model, Min, x' * D * x + y' * y - 1/γ * μ' * x);\n@constraint(model, y .== F' * x);\n@constraint(model, sum(x) == d + sum(x0));\n@constraint(model, x .>= 0);\nJuMP.optimize!(model)","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"After solving the problem, we can calculate the expected return and risk sigma= sqrtx^* top Sigma x^*:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"x_opt = JuMP.value.(x);\ny_opt = JuMP.value.(y);\nexpected_return_basic = dot(μ, x_opt)","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"expected_risk_basic = sqrt(dot(y_opt, y_opt))","category":"page"},{"location":"examples/portfolio_optimisation/#Using-standard-deviation-in-the-model","page":"Portfolio Optimisation","title":"Using standard deviation in the model","text":"","category":"section"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"It is pointed out in [1] that above problem formulation can lead to numerical problems, e.g. if Sigma is not strictly positive semidefinite. Another option is to formulate the risk constraint in terms of the standard deviation M^top x where M M^top = D + F F^top and bound it using a second-order cone constraint:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"beginarrayll textminimize - mu^top x \ntextsubject to M^top x leq gamma \n 1^top x = d + 1^top x^0 \n x geq 0\nendarray","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"Mt = [D.^0.5; F']\nmodel = JuMP.Model(COSMO.Optimizer);\n@variable(model, x[1:n]);\n@objective(model, Min, - μ' * x);\n@constraint(model, [γ; Mt * x] in SecondOrderCone()); # ||M'x|| <= γ\n@constraint(model, sum(x) == d + sum(x0));\n@constraint(model, x .>= 0);\nJuMP.optimize!(model)","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"Note that the result is different from the example above because gamma scales the problem in a different way. Here it can be seen as an upper bound on the standard deviation of the portfolio.","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"x_opt = JuMP.value.(x);\nexpected_return = dot(μ, x_opt)","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"Let us verify that the bound holds:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"@test norm(Mt * x_opt) <= γ","category":"page"},{"location":"examples/portfolio_optimisation/#Pareto-optimal-front","page":"Portfolio Optimisation","title":"Pareto-optimal front","text":"","category":"section"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"The above portfolio optimisation approach yields the optimal expected return for a given level of risk. The result is obviously impacted by the risk aversion gamma parameter. To visualise the trade-off and present the investor with an efficient Pareto optimal portfolio for their risk appetite we can compute the optimal portfolio for many choices of gamma and plot the corresponding risk-return trade-off curve.","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"gammas = [ 0.001, 0.01, 0.1, 0.5, 1., 3., 10, 100, 1000]\nrisks = zeros(length(gammas))\nreturns = zeros(length(gammas))\nmodel = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"verbose\" => false));\n@variable(model, x[1:n]);\n@variable(model, y[1:k]);\n@objective(model, Min, x' * D * x + y' * y - 1/γ * μ' * x);\n@constraint(model, y .== F' * x);\n@constraint(model, sum(x) == d + sum(x0));\n@constraint(model, x .>= 0);\n\n# solve the same problem for different values of γ\nfor (k, gamma) in enumerate(gammas)\n coeff = - 1/gamma * μ\n JuMP.set_objective_coefficient.(model, x, coeff)\n JuMP.optimize!(model)\n local x_opt = JuMP.value.(x);\n local y_opt = JuMP.value.(y);\n returns[k] = dot(μ, x_opt)\n risks[k] = sqrt(dot(y_opt, y_opt))\nend","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"We can now plot the risk-return trade-off curve:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"using Plots\nPlots.plot(risks, returns, xlabel = \"Standard deviation (risk)\", ylabel = \"Expected return\", title = \"Risk-return trade-off for efficient portfolios\", legend = false)","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"note: Note\nWhen the model is updated in JuMP as above the JuMP.model is copied in full to COSMO. We are trying to improve the interface with respect to model updates in the future. Until then you can use Model Updates in COSMOs native interface.","category":"page"},{"location":"examples/portfolio_optimisation/#Transaction-costs","page":"Portfolio Optimisation","title":"Transaction costs","text":"","category":"section"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"In the model above we assume that trading the assets is free and does not impact the market. However, this is clearly not the case in reality. To make the example more realistic consider the following cost c_j associated with the trade δ_j = x_j - x_j^0:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"c_j(delta_j) = a_j delta_j + b_j delta_j^32","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"where the first term models the bid-ask spread and broker fees for asset j. The second term models the impact on the market that our trade has. This is obviously only a factor if the volume of our trade is significant. The constant b_j is a function of the total volume traded in the considered time periode and the price volatility of the asset and has to be estimated by the trader. To make this example simple we consider the same coefficients a and b for every asset. The delta_j^32 term can be easily modeled using a power cone constraint mathcalK_pow = (x y z) mid x^alpha y^(1-alpha) geq z x geq 0 y geq 0 0 leq alpha leq 1 . In fact this can be used to model any market impact function with exponent greater than 1. We can write the total transaction cost a^top s + b^top t where s_j bounds the absolute value of delta_j and t_j is used to bound the term x_j - x_j^0^32 leq t_j using a power cone formulation: (t_j 1 x_j - x_j^0) in mathcalK_pow(23).","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"a = 1e-3\nb = 1e-1\nγ = 1.0;\nmodel = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"eps_abs\" => 1e-5, \"eps_rel\" => 1e-5));\n@variable(model, x[1:n]);\n@variable(model, y[1:k]);\n@variable(model, t[1:n]);\n@variable(model, s[1:n]);\n@objective(model, Min, x' * D * x + y' * y - 1/γ * μ' * x);\n@constraint(model, y .== F' * x);\n@constraint(model, x .>= 0);\n\n# transaction costs\n@constraint(model, sum(x) + a * sum(s) + b * sum(t) == d + sum(x0) );\n@constraint(model, [i = 1:n], x[i] - x0[i] <= s[i]); # model the absolute value with slack variable s\n@constraint(model, [i = 1:n], x0[i] - x[i] <= s[i]);\n@constraint(model, [i = 1:n], [t[i], 1, x[i] - x0[i]] in MOI.PowerCone(2/3));\nJuMP.optimize!(model)","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"Let's look at the expected return and the total transaction cost:","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"x_opt = JuMP.value.(x);\ny_opt = JuMP.value.(y);\ns_opt = JuMP.value.(s);\nt_opt = JuMP.value.(t);\nexpected_return = dot(μ, x_opt)","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"expected_risk = dot(y_opt, y_opt)","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"transaction_cost = a * sum(s_opt) + b * sum( t_opt)","category":"page"},{"location":"examples/portfolio_optimisation/#References","page":"Portfolio Optimisation","title":"References","text":"","category":"section"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"[1] Mosek Case Studies","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"","category":"page"},{"location":"examples/portfolio_optimisation/","page":"Portfolio Optimisation","title":"Portfolio Optimisation","text":"This page was generated using Literate.jl.","category":"page"},{"location":"getting_started/#Getting-Started","page":"Getting Started","title":"Getting Started","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"This user guide describes the basic structures and functions to define an optimisation problem, to solve the problem and to analyse the result. If you want to use JuMP to describe the problem, see the JuMP Interface section.","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"COSMO solves optimisation problems in the following format:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"beginarrayll textminimize textstylefrac12x^top Px + q^top x textsubject to Ax + s = b s in mathcalK endarray","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"with decision variables x in mathbbR^n, s in mathbbR^m and data matrices P=P^top succeq 0, q in mathbbR^n, A in mathbbR^m times n, and b in mathbbR^m. The convex set mathcalK is a composition of convex sets and cones.","category":"page"},{"location":"getting_started/#Model","page":"Getting Started","title":"Model","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"The problem data, user settings and workspace variables are all stored in a Model. To get started define an empty model:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"model = COSMO.Model()","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"To initialize the model with an optimisation problem we need to define three more things:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"the objective function, i.e. the matrix P and the vector q in frac12x^top P x + q^top x\nan array of constraints\na Settings object that specifies how COSMO solves the problem (optional)","category":"page"},{"location":"getting_started/#Objective-Function","page":"Getting Started","title":"Objective Function","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"To set the objective function of your optimisation problem simply define the square positive semidefinite matrix P in mathrmR^ntimes n and the vector q in mathrmR^n. You might have to transform your optimisation problem into a solver compatible format for this step.","category":"page"},{"location":"getting_started/#Constraints","page":"Getting Started","title":"Constraints","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"The COSMO interface expects constraints to have the form A_i x + b_i in mathcalK_i, where mathcalK_i is one of the convex sets defined below:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Convex Set Description\nZeroSet The set 0 ^dim that contains the origin\nNonnegatives The nonnegative orthant x in mathbbR^dim x_i ge 0 forall i=1dotsmathrmdim \nBox(l, u) The hyperbox x in mathbbR^dim l leq x leq u with vectors l in mathbbR^dim cup -infty and u in mathbbR^dim cup +infty\nSecondOrderCone The second-order (Lorenz) cone (tx) in mathbbR^dim x_2 leq t \nPsdCone The vectorized positive semidefinite cone mathcalS_+^dim. x is the vector obtained by stacking the columns of the positive semidefinite matrix X, i.e. X in mathbbS^sqrtdim_+ rarr textvec(X) = x in mathcalS_+^dim\nPsdConeTriangle The vectorized positive semidefinite cone mathcalS_+^dim. x is the vector obtained by stacking the columns of the upper triangular part of the positive semidefinite matrix X, i.e. X in mathbbS^d_+ rarr textsvec(X) = x in mathcalS_+^dim where d=sqrt14 + 2 textdim - 12\nExponentialCone The exponential cone mathcalK_exp = (x y z) mid y geq 0 ye^xy z cup (xyz) mid x leq 0 y = 0 z geq 0 \nDualExponentialCone The dual exponential cone mathcalK^*_exp = (x y z) mid x 0 -xe^yx leq e^1 z cup (0yz) mid y geq 0 z geq 0 \nPowerCone(alpha) The 3d power cone mathcalK_pow = (x y z) mid x^alpha y^(1-alpha) geq z x geq 0 y geq 0 with 0 alpha 1\nDualPowerCone(alpha) The 3d dual power cone mathcalK^*_pow = (u v w) mid left( fracualpharight)^alpha left( fracv1-alpharight)^(1-alpha) geq w u geq 0 v geq 0 with 0 alpha 1","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"The constructor for a constraint expects a matrix A, a vector b and a convex_set.","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Lets consider a problem with a decision variable x in mathbbR^5. Suppose we want to create the two constraint x_2 + 5 geq 0 and x_3 - 3 geq 0. We can do this either by creating two constraints and adding them to an array:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":" constraint1 = COSMO.Constraint([0.0 1.0 0.0 0.0 0.0], 5.0, COSMO.Nonnegatives)\n constraint2 = COSMO.Constraint([0.0 0.0 1.0 0.0 0.0], -3.0, COSMO.Nonnegatives)\n constraints = [constraint1; constraint2]","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"The second option is to include both in one constraint:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"constraint1 = COSMO.Constraint([0.0 1.0 0.0 0.0 0.0; 0.0 0.0 1.0 0.0 0.0], [5.0; -3.0], COSMO.Nonnegatives)","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Another way to construct the constraint is to used the optional arguments dim, the dimension of x, and indices, the elements of x that appear in the constraint. When specifying these arguments, A and b only refer to the elements of x in indices:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"constraint1 = COSMO.Constraint([1.0 0.0; 0.0 1.0], [5.0; -3.0], COSMO.Nonnegatives, 5, 2:3)","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Consider as a second example the positive semidefinite constraint on a matrix X in mathbbS_+^3. Our decision variable is the vector x obtained by stacking the columns of X. We can specify the constraint on x in the following way:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"I_9 x + 0_9 in mathcalS_+^9","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"or in Julia:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"constraint1 = COSMO.Constraint(Matrix(1.0I, 9, 9), zeros(9), COSMO.PsdCone)","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Several constraints can be combined in an array:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"constraints = [constraint_1, constraint_2, ..., constraint_N]","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"It is usually enough to pass the convex_set as a type. However, some convex sets like Box, PowerCone and DualPowerCone require more information to be created. In that case you have to pass an object to the constructor, e.g.","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"l = -ones(2)\nu = ones(2)\nconstraint = COSMO.Constraint(Matrix(1.0I, 2, 2), zeros(2), COSMO.Box(l, u))","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"or in the case of a power Cone you specify the alpha:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"constraint = COSMO.Constraint(Matrix(1.0I, 3, 3), zeros(3), COSMO.PowerCone(0.6))","category":"page"},{"location":"getting_started/#Settings","page":"Getting Started","title":"Settings","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"The solver settings are stored in a Settings object and can be adjusted by the user. To create a Settings object just call the constructor:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"COSMO.Settings","category":"page"},{"location":"getting_started/#COSMO.Settings","page":"Getting Started","title":"COSMO.Settings","text":"COSMO.Settings{T}(; kwargs) where {T <: AbstractFloat}\n\nCreates a COSMO settings object that is used to pass user settings to the solver.\n\nArgument Description Values (default)\nrho ADMM rho step 0.1\nsigma ADMM sigma step 1e-6\nalpha Relaxation parameter 1.6\neps_abs Absolute residual tolerance 1e-5\neps_rel Relative residual tolerance 1e-5\nnearly_ratio Residual tolerance ratio between MOI.NEARLY_FEASIBLE_POINT and MOIFEASIBLE_POINT 100\neps_prim_inf Primal infeasibility tolerance 1e-5\neps_dual_inf Dual infeasibility tolerance 1e-5\nmax_iter Maximum number of iterations 5000\nverbose Verbose printing false\nverbose_timing Verbose timing false\nkkt_solver Linear System solver QdldlKKTSolver\ncheck_termination Check termination interval 25\ncheck_infeasibility Check infeasibility interval 40\nscaling Number of scaling iterations 10\nadaptive_rho Automatic adaptation of step size parameter true\nadaptiverhomax_adaptions Max number of rho adaptions typemax(Int64) (deactivated)\ndecompose Activate to decompose chordal psd constraints true\ncomplete_dual Activate to complete the dual variable after decomposition false\nmerge_strategy Choose a strategy for clique merging CliqueGraphMerge\ncompact_transformation Choose how a decomposed problem is transformed true\ntime_limit Set solver time limit in s 0 (deactivated)\naccelerator Acceleration scheme AndersonAccelerator{T, Type2{QRDecomp}, RestartedMemory, NoRegularizer}\naccelerator_activation Accelerator activation ImmediateActivation\nsafeguard Accelerator safeguarding true\nsafeguard_tol Safeguarding tolerance 2.0\n\n\n\n\n\n","category":"type"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"To adjust those values, either pass your preferred option and parameter as a key-value pair to the constructor or edit the corresponding field afterwards. For example if you want to enable verbose printing and increase the solver accuracy, you can type","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"settings = COSMO.Settings(verbose = true, eps_abs = 1e-5, eps_rel = 1e-5)\n# the following is equivalent\nsettings = COSMO.Settings()\nsettings.verbose = true\nsettings.eps_abs = 1e-5\nsettings.eps_rel = 1e-5","category":"page"},{"location":"getting_started/#Assembling-the-model","page":"Getting Started","title":"Assembling the model","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Once the objective function and an array of constraints have been defined, we can assemble the model with","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"COSMO.assemble!(model, P, q, constraints)","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"This simply sets the corresponding variables in the model and transforms the array of constraints into the problem format defined at the top of the page.","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"If you want to change the default settings, you can pass your settings object custom_settings to the assemble! function:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"COSMO.assemble!(model, P, q, constraints, settings = custom_settings)","category":"page"},{"location":"getting_started/#Warm-starting","page":"Getting Started","title":"Warm starting","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"One of the advantages of ADMM-based solvers is that they can be easily warm started. By providing starting values for the primal variable x and/or the dual variable y in the vicinity of their optimal values, the number of iterations to convergence can often be dramatically decreased.","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Consider the case where you have a decision variable x in mathbbR^3 and a dual variable y in mathbbR^2. Assume you expect their optimal values to be close to x_0 = (1 5 3) and y_0 = (1 2). You can pass these values when assembling the model.","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"x_0 = [1.0; 5.0; 3.0]\ny_0 = [1.0; 2.0]\nCOSMO.assemble!(model, P, q, constraints, x0 = x_0, y0 = y_0)","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Another option is to use","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"COSMO.assemble!(model, P, q, constraints)\nwarm_start_primal!(model, x_0)\nwarm_start_dual!(model, y_0)","category":"page"},{"location":"getting_started/#Solving","page":"Getting Started","title":"Solving","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"After the model has been assembled, we can solve the problem by typing","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"results = COSMO.optimize!(model)","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Once the solver algorithm terminates, it will return a Results object that gives information about the status of the solver. If successful, it contains the optimal objective value and optimal primal and dual variables. For more information see the following section.","category":"page"},{"location":"getting_started/#Results","page":"Getting Started","title":"Results","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"After attempting to solve the problem, COSMO will return a result object with the following fields:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"COSMO.Result\nCOSMO.ResultInfo","category":"page"},{"location":"getting_started/#COSMO.Result","page":"Getting Started","title":"COSMO.Result","text":"Result{T <: AbstractFloat}\n\nObject returned by the COSMO solver after calling optimize!(model). It has the following fields:\n\nFieldname Type Description\nx Vector{T} Primal variable\ny Vector{T} Dual variable\ns Vector{T} (Primal) set variable\nobj_val T Objective value\niter Int Total number of ADMM iterations (incl. safeguarding_iter)\nsafeguarding_iter Int Number of iterations due to safeguarding of accelerator\nstatus Symbol Solution status\ninfo COSMO.ResultInfo Struct with more information\ntimes COSMO.ResultTimes Struct with several measured times\n\n\n\n\n\n","category":"type"},{"location":"getting_started/#COSMO.ResultInfo","page":"Getting Started","title":"COSMO.ResultInfo","text":"ResultInfo{T <: AbstractFloat}\n\nObject that contains further information about the primal residual, the dual residuals and the rho updates.\n\n\n\n\n\n","category":"type"},{"location":"getting_started/#Status-Codes","page":"Getting Started","title":"Status Codes","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"COSMO will return one of the following statuses:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"Status Code Description\n:Solved An optimal solution was found\n:Unsolved Default value\n:Max_iter_reached Solver reached iteration limit (set with Settings.max_iter)\n:Time_limit_reached Solver reached time limit (set with Settings.time_limit)\n:Primal_infeasible Problem is primal infeasible\n:Dual_infeasible Problem is dual infeasible","category":"page"},{"location":"getting_started/#Timings","page":"Getting Started","title":"Timings","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"If settings.verbose_timing is set to true, COSMO will report the following times in result.times:","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"COSMO.ResultTimes","category":"page"},{"location":"getting_started/#COSMO.ResultTimes","page":"Getting Started","title":"COSMO.ResultTimes","text":"ResultTimes\n\nPart of the Result object returned by the solver. ResultTimes contains timing results for certain parts of the algorithm:\n\nTime Name Description\nsolver_time Total time used to solve the problem\nsetup_time Setup time = graph_time + init_factor_time + scaling_time\nscaling_time Time to scale the problem data\ngraph_time Time used to perform chordal decomposition\ninit_factor_time Time used for initial factorisation of the system of linear equations\nfactor_update_time Sum of times used to refactor the system of linear equations due to rho\niter_time Time spent in iteration loop\nproj_time Time spent in projection functions\npost_time Time used for post processing\nupdate_time Time spent in the update! function of the accelerator\naccelerate_time Time spent in the accelerate! function of the accelerator\n\nBy default COSMO only measures solver_time, setup_time and proj_time. To measure the other times set verbose_timing = true.\n\n\n\n\n\n","category":"type"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"It holds: solver_time = setup_time+ iter_time + factor_update_time + post_time,","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"setup_time = graph_time+ init_factor_time + scaling_time,","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"proj_time is a subset of iter_time.","category":"page"},{"location":"getting_started/#Updating-the-model","page":"Getting Started","title":"Updating the model","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"In some cases we want to solve a large number of similar models. COSMO allows you to update the model problem data vectors q and b after the first call of optimize!(). After changing the problem data, COSMO can reuse the factorisation step of the KKT matrix from the previous problem which can save a lot of time in the case of LPs and QPs.","category":"page"},{"location":"getting_started/","page":"Getting Started","title":"Getting Started","text":"COSMO.update!","category":"page"},{"location":"getting_started/#COSMO.update!","page":"Getting Started","title":"COSMO.update!","text":"update!(model, q = nothing, b = nothing)\n\nUpdates the model data for b or q. This can be done without refactoring the KKT matrix. The vectors will be appropriatly scaled.\n\n\n\n\n\n","category":"function"},{"location":"method/#Method","page":"Method","title":"Method","text":"","category":"section"},{"location":"method/","page":"Method","title":"Method","text":"This section describes COSMO's underlying ADMM algorithm and how the user can use the settings to adjust this algorithm. For a more detailed explanation take a look at the associated publication in Citing COSMO.","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"COSMO solves problems with quadratic objective function and a number of conic constraints in the following form:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"beginarrayll textminimize textstylefrac12x^top Px + q^top x textsubject to Ax + s = b s in mathcalK endarray","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"with primal decision variable x in mathbbR^n, primal slack variable s in mathbbR^m. The objective function is defined by positive semidefinite matrix P=P^top succeq 0 and vector q in mathbbR^n. The constraints are defined by matrix A in mathbbR^m times n, vector b in mathbbR^m and a non-empty, closed convex set mathcalK. The convex set itself can be a Cartesian product of convex sets in the form:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":" mathcalK = mathcalK_1^m_1 times mathcalK_2^m_2 times cdots times mathcalK_N^m_N","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"Accordingly, by an appropriate choice of convex sets one can represent any LP, QP, SOCP or SDP.","category":"page"},{"location":"method/#Dual-problem","page":"Method","title":"Dual problem","text":"","category":"section"},{"location":"method/","page":"Method","title":"Method","text":"The dual problem of the optimisation problem above is given by:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"beginarrayll\ntextmaximize -textstylefrac12x^top Px - b^top y - textsup_s in mathcalK(-y^top s ) \ntextsubject to Px + A^top y = -q\n y in (mathcalK^infty)^* \nendarray","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"with dual variable y in mathbbR^m.","category":"page"},{"location":"method/#Algorithm","page":"Method","title":"Algorithm","text":"","category":"section"},{"location":"method/","page":"Method","title":"Method","text":"The algorithm considers a slightly transformed problem. By introducing two dummy variables tildex = x and tildes = s we can rewrite the original problem:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"beginarrayll\ntextminimize textstylefrac12tildex^top P tildex + q^top tildex + mathcalI_Ax+s=b(tildextildes) + mathcalI_mathcalK(s)\ntextsubject to (tildextildes) = (xs)\nendarray","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"where indicator functions mathcalI were used to move the constraints into the objective function. The resulting problem is now in the right format to apply the alternating direction method of multipliers (ADMM). To apply ADMM we first find the augmented Lagrangian L:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":" L(xstildextildeslambday) = textstylefrac12tildex^top Ptildex + q^top tildex + mathcalI_Ax+s=b(tildextildes) + mathcalI_mathcalK(s) + fracsigma2 tildex - x + textstylefrac1sigma λ _2^2 + fracrho2 tildes - s + textstylefrac1rho y _2^2","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"Minimizing the Lagrangian in an alternating fashion with respect to the two variable pairs (tildextildes) and (xs) yields the following algorithm steps:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"beginaligned\n ( tildex^k+1tildes^k+1) rarr undersettildextildestextargmin Lleft( tildextildesx^ks^ky^k right)\n x^k+1 larr tildex^k+1\n s^k+1 larr undersetstextargmin fracrho2 tildes^k+1 - s+textstylefrac1rhoy^k _2^2 + I_mathcalK(s) \n y^k+1 larr y^k + rho left( tildes^k+1 -s^k+1 right)\nendaligned","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"By the construction of the ADMM method those iterates are converging to the global solution. These steps are executed in a loop until convergence. Two important parameters are the ADMM steps sizes rho (Settings.rho) and sigma (Settings.sigma) which can be adjusted via the solver settings.","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"The two most important steps of the algorithm happen in the first and third line. The evaluation of the first line turns out to be an equality constrained quadratic program. We get a solution for ( tildex^k+1tildes^k+1) at every iteration by solving the following linear system:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"beginaligned\nbeginbmatrix\nP + sigma I A^top A - frac1rhoI\n endbmatrixbeginbmatrixtildex^k+1 nu^k+1endbmatrix= beginbmatrix-q+sigma x^k b-s^k+frac1rhoy^kendbmatrix\ntildes^k+1 = s^k - frac1rholeft(nu^k+1 + y^kright)\nendaligned","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"Fortunately, the left hand matrix doesn't change, which is why COSMO only has to factor the matrix once at the start of the algorithm.","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"The second important step in the algorithm is the update equation for s^k+1 which can be interpreted as a projection onto the constraint set mathcalK:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"s^k+1 = Pi_mathcalKleft( tildes^k+1 + frac1rhoy^kright)","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"The computational cost of this projection is highly dependent on the constraints of the problem. While projections onto the zero set or the nonnegative orthant are inexpensive, projections onto the positive semidefinite cone of order N involve an eigen-decomposition. Since methods for eigen-decompositions have a complexity of mathcalO(N^3) the projection can become the computationally most expensive operation of the algorithm.","category":"page"},{"location":"method/#Scaling","page":"Method","title":"Scaling","text":"","category":"section"},{"location":"method/","page":"Method","title":"Method","text":"The convergence of ADMM-based algorithms depends on the relative scaling of the problem data. Especially to improve the convergence of badly scaled problems, COSMO tries to rescale the data in a preprocessing step.","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"We rescale the equality constraints with diagonal positive semidefinite matrices D and E. The scaled problem is given by:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"beginarrayll\ntextminimize textstylefrac12 hatx^top hatP hatx + hatq^top hatx\ntextsubject to hatA hatx + hats = hatb \n hats in EmathcalK\nendarray","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"with scaled problem data","category":"page"},{"location":"method/","page":"Method","title":"Method","text":" hatP=DPD quad hatq=Dq quadhatA=EAD quad hatb=Eb","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"and the scaled convex cone EmathcalK = Ev in mathbbR^m mid v in mathcalK . After solving the scaled problem the original solution is obtained by reversing the scaling:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":" x = Dhatx quad s = E^-1hats quad y = Ehaty","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"To obtain the scaling matrices D and E we use a modified Ruiz equilibration algorithm which involves a certain number of scaling iterations to equilibrate the column norms of the data matrices P and A. The number of these iterations can be adjusted by the user with scaling. To disable the scaling step set scaling = 0.","category":"page"},{"location":"method/#Termination-criteria","page":"Method","title":"Termination criteria","text":"","category":"section"},{"location":"method/","page":"Method","title":"Method","text":"The COSMO algorithm can terminate for four reasons:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"The maximum number of allowed iterations has been reached. The user can specify this value in the solver settings with max_iter.\nThe solver runtime reaches the time limit specified by the user (time_limit).\nCOSMO detects an infeasible problem.\nThe iterates fulfil the termination criteria for convergence.","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"COSMO uses the primal and dual residuals of the problem to determine if the algorithm has converged. The primal and dual residuals are given by:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"beginaligned\nr_p = Ax + s -b\nr_d = Px + q + A^T y\nendaligned","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"The solver terminates when the infty-norms of the residuals lie below a specified tolerance. COSMO uses the sum of an absolute and relative tolerance term:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"beginaligned\n r_p^k _infty leq epsilon_mathrmabs + epsilon_mathrmrel textmax left Ax^k _inftys^k_infty b_infty right\n r_d^k_infty leq epsilon_mathrmabs + epsilon_mathrmrel textmax leftPx^k_inftyq_infty A^top y^k_infty right\nendaligned","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"The absolute and relative tolerances epsilon_mathrmabsand epsilon_mathrmrel can be set by the user by specifying eps_abs and eps_rel. Furthermore, the user can adjust the number of iterations after which the convergence criteria are checked (check_termination).","category":"page"},{"location":"method/#Infeasibility-detection","page":"Method","title":"Infeasibility detection","text":"","category":"section"},{"location":"method/","page":"Method","title":"Method","text":"COSMO uses conditions based on separating hyperplanes to detect infeasible problems. The conditions for COSMO's problem format have been developed in [1]. Define the convex set mathcalC = mathcal-K + b then we can use the following infeasibility conditions:","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"beginaligned\nmathcalP = leftx in mathbbR^n mid Px = 0 Ax in mathcalC^infty langle qx rangle 0 right \nmathcalD = lefty in mathbbR^m mid A^top y = 0 sigma_mathcalC(y) 0 right\nendaligned","category":"page"},{"location":"method/","page":"Method","title":"Method","text":"The existence of some y in mathcalD is a certificate that the problem is primal infeasible, while the existence of some x in mathcalP is a certificate for dual infeasibility. COSMO regularly checks above conditions to detect infeasible problems. If the detection is successful, the solver terminates and returns the status codes :Primal_infeasible or :Dual_infeasible. COSMO checks the conditions every check_infeasibility iterations, which can be adjusted by the user.","category":"page"},{"location":"method/#References","page":"Method","title":"References","text":"","category":"section"},{"location":"method/","page":"Method","title":"Method","text":"[1] Banjac, G. et al. Infeasibility detection in the alternating direction method of multipliers for convex optimization. Preprint, 2017.","category":"page"},{"location":"examples/#Examples","page":"Examples","title":"Examples","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"Some example problems that are solved with COSMO can be found in the /examples folder.","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"In the first example the native solver interface is used to define and solve a Linear Program. In the second example JuMP is used to describe the problem and COSMO is set as the solver backend.","category":"page"},{"location":"examples/#Linear-Program","page":"Examples","title":"Linear Program","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"We want to solve the following linear program with decision variable x:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"beginarrayll textminimize c^top x\ntextsubject to A x leq b \n x geq 1 \n x_2 geq 5 \n x_1 + x_3 geq 4\nendarray","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"The problem can be solved with COSMO in the following way:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"using COSMO, LinearAlgebra, SparseArrays, Test\n\nc = [1; 2; 3; 4.]\nA = Matrix(1.0I, 4, 4)\nb = [10; 10; 10; 10]\nn = 4\n# -------------------\n# create constraints A * x + b in set\n# -------------------\n# Ax <= b\nc1 = COSMO.Constraint(-A, b, COSMO.Nonnegatives)\n# x >= 1\nc2 = COSMO.Constraint(Matrix(1.0I, n, n), -ones(n), COSMO.Nonnegatives)\n# x2 >= 5\nc3 = COSMO.Constraint(1, -5, COSMO.Nonnegatives, n, 2:2)\n# x1 + x3 >= 4\nc4 = COSMO.Constraint([1 0 1 0], -4, COSMO.Nonnegatives)\n\n# -------------------\n# define cost function\n# -------------------\nP = spzeros(4, 4)\nq = c\n\n# -------------------\n# assemble solver model\n# -------------------\nsettings = COSMO.Settings(max_iter=2500, verbose=true, eps_abs = 1e-4, eps_rel = 1e-5)\nmodel = COSMO.Model()\nassemble!(model, P, q, [c1; c2; c3; c4], settings = settings)\nres = COSMO.optimize!(model);\n\n@testset \"Linear Problem\" begin\n @test isapprox(res.x[1:4], [3; 5; 1; 1], atol=1e-2, norm = (x -> norm(x, Inf)))\n @test isapprox(res.obj_val, 20.0, atol=1e-2)\nend","category":"page"},{"location":"examples/#Closest-Correlation-Matrix","page":"Examples","title":"Closest Correlation Matrix","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"We consider the problem of finding the closest correlation matrix X to a given random matrix C. With closest correlation matrix we mean a positive semidefinite matrix with ones on the diagonal. The problem is given by:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"beginarrayll textminimize frac12X - C_F^2\ntextsubject to X_ii = 1 quad i=1dotsn \n X succeq 0\nendarray","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Notice how JuMP is used to describe the problem. COSMO is chosen as the backend solver using JuMP's optimizer_with_attributes() function.","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"using COSMO, JuMP, LinearAlgebra, SparseArrays, Test, Random\nrng = Random.MersenneTwister(12345);\n\n# create a random test matrix C\nn = 8\nC = -1 .+ rand(rng, n, n) .* 2;\nc = vec(C);\n\n# define problem in JuMP\nq = -vec(C);\nr = 0.5 * vec(C)' * vec(C);\nm = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"verbose\" => true, \"eps_abs\" => 1e-4));\n@variable(m, X[1:n, 1:n], PSD);\nx = vec(X);\n@objective(m, Min, 0.5 * x' * x + q' * x + r)\nfor i = 1:n\n @constraint(m, X[i, i] == 1.)\nend\n\n# solve and get results\nstatus = JuMP.optimize!(m)\nobj_val = JuMP.objective_value(m)\nX_sol = JuMP.value.(X)","category":"page"},{"location":"examples/#Logistic-Regression","page":"Examples","title":"Logistic Regression","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"Logistic regression problems can be solved using exponential cone constraints. An example on how to use COSMO to solve a logistic regression problem is presented in /examples/logistic_regression_regularization.ipynb.","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"EditURL = \"../../../examples/qp.jl\"","category":"page"},{"location":"examples/qp/#Quadratic-Program","page":"Quadratic Program","title":"Quadratic Program","text":"","category":"section"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"We want to solve the following quadratic program with decision variable x:","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"beginarrayll textminimize 12 x^top P x + q^top x \ntextsubject to l leq A x leq u\nendarray","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"The problem can be solved with COSMO in the following way. Start by defining the problem data","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"using COSMO, SparseArrays, LinearAlgebra, Test\n\nq = [1; 1.];\nP = sparse([4. 1; 1 2]);\nA = [1. 1; 1 0; 0 1];\nl = [1.; 0; 0];\nu = [1; 0.7; 0.7];\nnothing #hide","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"First, we decide to solve the problem with two one-sided constraints using COSMO.Nonnegatives as the convex set:","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"Aa = [-A; A]\nba = [u; -l]\nconstraint1 = COSMO.Constraint(Aa, ba, COSMO.Nonnegatives);\nnothing #hide","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"Next, we define the settings object, the model and then assemble everything:","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"settings = COSMO.Settings(verbose=true);\nmodel = COSMO.Model();\nassemble!(model, P, q, constraint1, settings = settings);\nres = COSMO.optimize!(model);\nnothing #hide","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"Alternatively, we can also use two-sided constraints with COSMO.Box:","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"constraint1 = COSMO.Constraint(A, zeros(3), COSMO.Box(l, u));\n\nmodel = COSMO.Model();\nassemble!(model, P, q, constraint1, settings = settings);\nres_box = COSMO.optimize!(model);\nnothing #hide","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"Let's check that the solution is correct:","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"@testset \"QP Problem\" begin\n @test norm(res.x[1:2] - [0.3; 0.7], Inf) < 1e-3\n @test norm(res_box.x[1:2] - [0.3; 0.7], Inf) < 1e-3\n @test abs(res.obj_val - 1.88) < 1e-3\n @test abs(res_box.obj_val - 1.88) < 1e-3\nend\nnothing","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"","category":"page"},{"location":"examples/qp/","page":"Quadratic Program","title":"Quadratic Program","text":"This page was generated using Literate.jl.","category":"page"},{"location":"lin_solver/#Linear-System-Solver","page":"Linear System Solver","title":"Linear System Solver","text":"","category":"section"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"One major step of COSMO's ADMM algorithm is solving a linear system of equations at each iteration. Fortunately, the left-hand matrix is only dependent on the problem data and therefore only needs to be factored once. Depending on the problem class this factorisation can be the computationally most expensive step of the algorithm (LPs, QPs). See the Method section for a more detailed description of the linear system.","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"COSMO allows you to specify the linear system solver that performs the factorisation and back-substitution. We also support indirect system solver which are useful for very large problems where a factorisation becomes inefficient. The table below shows the currently supported linear system solver:","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"Type Solver Description\nCholmodKKTSolver Cholmod Julia's default linear system solver (from SuiteSparse)\nQdldlKKTSolver QDLDL For more information QDLDL.jl\nPardisoDirectKKTSolver Pardiso (direct) Pardiso 6.0 direct solver\nPardisoIndirectKKTSolver Pardiso (indirect) Pardiso 6.0 indirect solver\nMKLPardisoKKTSolver Intel MKL Pardiso Pardiso optimised for Intel platforms\nCGIndirectKKTSolver IterativeSolvers.jl Conjugate Gradients on the reduced KKT linear system.\nMINRESIndirectKKTSolver IterativeSolvers.jl MINRES on the (full) KKT linear system.","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"note: Note\nTo use the Pardiso and Intel MKL Pardiso solver, you have to install the respective libraries and the corresponding Julia wrapper. For more information about installing these, visit the Pardiso.jl repository page. Likewise in order to use Indirect(Reduced)KKTSolver you have to install IterativeSolvers.jl (v0.9+) and LinearMaps.jl. We are using the Requires package for lazy loading of code related to Pardiso and IterativeSolvers. This means in order to use Pardiso / IterativeSolvers, you'll have to load these packages alongside COSMO, i.e. using Pardiso and using IterativeSolvers, LinearMaps.","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"COSMO uses the Cholmod linear system solver by default. You can specify a different solver in the settings by using the kkt_solver keyword and the respective type:","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"settings = COSMO.Settings(kkt_solver = CholmodKKTSolver)\n","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"COSMO also allows you to pass in solver-specific options with the with_options(solver_type, args...; kwargs...) syntax. For example, if you want to use Pardiso with verbose printing use the following syntax:","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"settings = COSMO.Settings(kkt_solver = with_options(PardisoDirectKKTSolver, msg_level_on = true))","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"Likewise, CGIndirectKKTSolver and MINRESIndirectKKTSolver are also parameterizable with with_options(solver_type, args...; kwargs...) and accept the following arguments:","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"Argument Description Values (default)\ntol_constant::T and tol_exponent::T Parameter that defines the solution tolerance for the iterative solvers across iterations. In particular, the solution tolerance at every iteration is defined as texttol_constant textiteration^texttol_exponent 1.0, 1.5","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"This also works if you want to use this configuration with JuMP:","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"model = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"kkt_solver\" => with_options(PardisoDirectKKTSolver, msg_level_on = true));\n","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"Or alternatively:","category":"page"},{"location":"lin_solver/","page":"Linear System Solver","title":"Linear System Solver","text":"model = JuMP.Model(COSMO.Optimizer);\nset_optimizer_attribute(model, \"kkt_solver\", with_options(PardisoDirectKKTSolver, msg_level_on = true));","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"EditURL = \"../../../examples/closest_correlation_matrix.jl\"","category":"page"},{"location":"examples/closest_correlation_matrix/#Closest-Correlation-Matrix","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"","category":"section"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"We consider the problem of finding the closest correlation matrix X to a given random matrix C. With closest correlation matrix we mean a positive semidefinite matrix with ones on the diagonal. The problem is given by:","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"beginarrayll textminimize frac12X - C_F^2\ntextsubject to X_ii = 1 quad i=1dotsn \n X succeq 0\nendarray","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"Notice that we use JuMP to model the problem. COSMO is chosen as the backend solver. And COSMO-specific settings are passed using the optimizer_with_attributes() function.","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"using COSMO, JuMP, LinearAlgebra, SparseArrays, Test, Random","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"rng = Random.MersenneTwister(12345);\n# create a random test matrix C\nn = 8;\nC = -1 .+ rand(rng, n, n) .* 2;\nc = vec(C);\nnothing #hide","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"Define problem in JuMP:","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"q = -vec(C);\nr = 0.5 * vec(C)' * vec(C);\nm = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, \"verbose\" => true));\n@variable(m, X[1:n, 1:n], PSD);\nx = vec(X);\n@objective(m, Min, 0.5 * x' * x + q' * x + r);\nfor i = 1:n\n @constraint(m, X[i, i] == 1.);\nend","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"Solve the JuMP model with COSMO and query the solution X_sol:","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"status = JuMP.optimize!(m);\nobj_val = JuMP.objective_value(m);\nX_sol = JuMP.value.(X);\nnothing #hide","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"Double check result against known solution:","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"known_opt_val = 12.5406\nknown_solution = [\n 1.0 0.732562 -0.319491 -0.359985 -0.287543 -0.15578 0.0264044 -0.271438;\n 0.732562 1.0 0.0913246 -0.0386357 0.299199 -0.122733 0.126612 -0.187489;\n -0.319491 0.0913246 1.0 -0.0863377 0.432948 0.461783 -0.248641 -0.395299;\n -0.359985 -0.0386357 -0.0863377 1.0 0.503379 0.250601 0.141151 0.286088;\n -0.287543 0.299199 0.432948 0.503379 1.0 -0.0875199 0.137518 0.0262425;\n -0.15578 -0.122733 0.461783 0.250601 -0.0875199 1.0 -0.731556 0.0841783;\n 0.0264044 0.126612 -0.248641 0.141151 0.137518 -0.731556 1.0 -0.436274;\n -0.271438 -0.187489 -0.395299 0.286088 0.0262425 0.0841783 -0.436274 1.0 ];\n@test isapprox(obj_val, known_opt_val , atol=1e-3)","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"@test norm(X_sol - known_solution, Inf) < 1e-3","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"","category":"page"},{"location":"examples/closest_correlation_matrix/","page":"Closest Correlation Matrix","title":"Closest Correlation Matrix","text":"This page was generated using Literate.jl.","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"EditURL = \"../../../examples/lovasz_petersen.jl\"","category":"page"},{"location":"examples/lovasz_petersen/#Lovasz-Theta-Function","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"","category":"section"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"The Lovász theta function is an important concept in graph theory. Consider an undirected graph G = (VE) with vertex set V = 1dotsn and edge set E with E_ij = 1 if there exists an edge between the vertices i and j. A stable set (or independent set) is a subset of V such that the induced subgraph does not contain edges. The stability number alpha(G) of the graph is equal to the cardinality of the largest stable set. It is closely related to the Shannon capacity Theta(G) that models the amount of information that a noisy communication channel can carry if certain signal values can be confused with each other. Unfortunately, the determination of alpha(G) is an NP-hard problem and the computational complexity to determine Theta(G) remains unknown. The Lovász theta function vartheta(G) was introduced in [1], can be computed in polynomial time, and represents an upper bound on both the stability number and the Shannon capacity:","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"alpha(G) leq Theta(G) leq vartheta(G)","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"The value of vartheta(G) can be determined by computing the optimal value p^* of the following SDP:","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"beginarrayll textmaximize textTr(JX)\ntextsubject to textTr(X) = 1 \n X_ij = 0 quad (i j) in E \n X succeq 0\nendarray","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"with matrix variable X, matrix J in mathbfR^n times n of all ones and textTr() denoting the matrix trace. In this simple example we will compute the value of the Lovász theta function for the Petersen Graph (which is known to be 4).","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"using LinearAlgebra, COSMO, JuMP, Plots, GraphRecipes\n\n# let's define the Petersen graph\nn = 10\nE = zeros(n, n)\nE[1, 2] = E[1, 5] = E[1, 6] = 1.\nE[2, 3] = E[2, 7] = 1.\nE[3, 4] = E[3, 8] = 1.\nE[4, 5] = E[4, 9] = 1.\nE[5, 10] = 1.\nE[6, 8] = E[6, 9] = 1.\nE[7, 9] = E[7, 10] = 1.\nE[8, 10] = 1.\n\n# plot the graph\nri = 1.\nro = 2.\ncoordinates = []\nfor θ = 90:-72:-198\n push!(coordinates, [ro * cosd(θ), ro * sind(θ)])\nend\nfor θ = 90:-72:-198\n push!(coordinates, [ri * cosd(θ), ri * sind(θ)])\nend\ngraphplot(E, names = 1:n, x = getindex.(coordinates, 1), y = getindex.(coordinates, 2), fontsize = 10, nodesize = 1, nodeshape =:circle, curvature = 0.)","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"Let's solve the SDP with COSMO and JuMP:","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"model = JuMP.Model(COSMO.Optimizer);\n\n@variable(model, X[1:n, 1:n], PSD)\nx = vec(X)\n@objective(model, Max, sum(x))\n@constraint(model, tr(X)== 1.)\nfor j = 1:n\n for i = 1:j-1\n if E[i, j] == 1.\n @constraint(model, X[i, j] == 0)\n end\n end\nend\nstatus = JuMP.optimize!(model)","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"The optimal objective is given by:","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"JuMP.objective_value(model)","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"Which is the correct known value for the Petersen Graph.","category":"page"},{"location":"examples/lovasz_petersen/#References","page":"Lovász Theta Function","title":"References","text":"","category":"section"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"[1] Lovász - On the Shannon Capacity of a Graph, IEEE Transactions on Information Theory (1979)","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"","category":"page"},{"location":"examples/lovasz_petersen/","page":"Lovász Theta Function","title":"Lovász Theta Function","text":"This page was generated using Literate.jl.","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"The source files for all examples can be found in /examples.","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"EditURL = \"../../../examples/two_way_partitioning.jl\"","category":"page"},{"location":"examples/two_way_partitioning/#Relaxed-Two-Way-Partitioning-Problem","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"","category":"section"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"We consider the (nonconvex) two-way partitioning problem from Boyd and Vandenberghe (2004), p.219 [1]:","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"beginarrayll textminimize x^top W x \ntextsubject to x_i^2 = 1 quad i=1dots n\nendarray","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"with x in mathbfR^n and W in mathbfS^n. The problem can be interpreted as finding a partition of the points i = 1dots n into two sets, where the cost of two points in the same set is W_ij and -W_ij otherwise. Brute forcing the solution x^* takes 2^n attempts and becomes quickly intractable, e.g. for n geq 30. However, a lower-bound for this problem can be computed by solving the convex dual of the problem:","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"beginarrayll textmaximize -sum_i nu \ntextsubject to W + textdiag(nu) in mathbfS_+^n\nendarray","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"Solving this problem with optimal objective d^* yields a lower bound as least as good as a lower bound based on the minimum eigenvalue d^* geq lambda_textmin(W). Let's set up the problem for n = 20 and with a specially structured (banded) W:","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"using LinearAlgebra, Random, COSMO, JuMP, IterTools\n\nrng = Random.MersenneTwister(212313);\nn = 20;\n\nW = diagm(0 => randn(rng, n));\nW += diagm(1 => randn(rng, n-1));\nW = Symmetric(W)","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"As you can see, the matrix W imposes a structure on the LMI-constraint. Accordingly, COSMO will be able to use chordal decomposition to decompose the LMI constraint into multiple smaller constraints. This can make a significant difference in the performance of the algorithm for large n.","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"model = JuMP.Model(COSMO.Optimizer);\n@variable(model, ν[1:n]);\n@objective(model, Max, -ones(n)' * ν )\n@constraint(model, Symmetric(W + diagm(ν) ) in JuMP.PSDCone());\nJuMP.optimize!(model)","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"Looking at the solver output you can see how the PSD constraint was decomposed into 19 PSD constraints. Let's look at the lower bound:","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"JuMP.objective_value(model)","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"As n is small, we can verify our result by finding the optimal solution by trying out all possible combinations:","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"function brute_force_optimisation(W, n)\n opt_obj = Inf\n opt_x = Inf * ones(n)\n\n for xt in Iterators.product([[1.; -1.] for k = 1:n]...)\n x = [i for i in xt]\n obj_val = x' * W * x\n if obj_val < opt_obj\n opt_obj = obj_val\n opt_x = x\n end\n end\n return opt_obj, opt_x\nend\nopt_obj, opt_x = brute_force_optimisation(W, n)","category":"page"},{"location":"examples/two_way_partitioning/#References","page":"Relaxed Two-Way Partitioning Problem","title":"References","text":"","category":"section"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"[1] Boyd and Vandenberghe - Convex Optimization, Cambridge University Press (2004)","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"","category":"page"},{"location":"examples/two_way_partitioning/","page":"Relaxed Two-Way Partitioning Problem","title":"Relaxed Two-Way Partitioning Problem","text":"This page was generated using Literate.jl.","category":"page"},{"location":"","page":"Home","title":"Home","text":"COSMO.jl is a Julia implementation of the Conic Operator Splitting Method. The underlying ADMM-algorithm is well-suited for large convex conic problems. COSMO solves the following problem:","category":"page"},{"location":"","page":"Home","title":"Home","text":"beginarrayll textminimize textstylefrac12x^top Px + q^top x textsubject to Ax + s = b s in mathcalK\nendarray","category":"page"},{"location":"","page":"Home","title":"Home","text":"with decision variables x in mathbbR^n, s in mathbbR^m and data matrices P=P^top succeq 0, q in mathbbR^n, A in mathbbR^m times n, and b in mathbbR^m. The convex set mathcalK is a composition of convex sets and cones.","category":"page"},{"location":"#Features","page":"Home","title":"Features","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Versatile: COSMO solves linear programs, quadratic programs, second-order cone programs, semidefinite programs and problems involving exponential and power cones\nQuad SDPs: Positive semidefinite programs with quadratic objective functions are natively supported\nSafeguarded acceleration: robust and faster convergence to higher precision using COSMOAccelerators\nInfeasibility detection: Infeasible problems are detected without a homogeneous self-dual embedding of the problem\nJuMP / Convex.jl support: We provide an interface to MathOptInterface (MOI), which allows you to describe your problem in JuMP and Convex.jl.\nChordal decomposition: COSMO tries to decompose large structured PSD constraints into multiple smaller PSD constraints using chordal decomposition techniques. This often results in a significant speedup compared to solving the original problem.\nSmart clique merging: After an initial decomposition of a structured SDP, COSMO recombines overlapping cliques/blocks to speed up the algorithm.\nWarm starting: COSMO supports warm starting of the decision variables\nArbitrary precision types: You can solve problems with any floating point precision.\nOpen Source: Our code is available on GitHub and distributed under the Apache 2.0 Licence","category":"page"},{"location":"","page":"Home","title":"Home","text":"tip: Tip\nFor users: COSMO has both a native interface and can also be interfaced through JuMP. If you are interested in using COSMO to solve your optimization problem, we recommend using JuMP to define your problem. Install this package as described below, take a look at our Examples and the JuMP documentation for inspiration.","category":"page"},{"location":"","page":"Home","title":"Home","text":"tip: Tip\nFor researchers and developers: If you are interested in COSMO's algorithm, chordal decomposition or acceleration methods take a look at Method, Chordal Decomposition, and Acceleration. If you want low-level control over the algorithm, it can make sense to start with COSMO's native interface. Moreover, take a look at Contributing if you want to propose changes.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"COSMO can be installed using the Julia package manager for Julia v1.6 and higher. Inside the Julia REPL, type ] to enter the Pkg REPL mode then run","category":"page"},{"location":"","page":"Home","title":"Home","text":"pkg> add COSMO","category":"page"},{"location":"","page":"Home","title":"Home","text":"If you want to install the latest version from master run","category":"page"},{"location":"","page":"Home","title":"Home","text":"pkg> add COSMO#master","category":"page"},{"location":"#Quick-Example","page":"Home","title":"Quick Example","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Consider the following 2x2 semidefinite program with decision variable X:","category":"page"},{"location":"","page":"Home","title":"Home","text":"beginarrayll textminimize texttr(CX)\ntextsubject to texttr(A X) = b \n X succeq 0\nendarray","category":"page"},{"location":"","page":"Home","title":"Home","text":"with problem data A, b and C:","category":"page"},{"location":"","page":"Home","title":"Home","text":"A = beginbmatrix 1 5 5 2endbmatrix\nC = beginbmatrix 1 2 2 2endbmatrix\nb = 4","category":"page"},{"location":"","page":"Home","title":"Home","text":"where tr denotes the trace of a matrix. We can solve this problem either using COSMO's interface:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using COSMO, LinearAlgebra\n\nC = [1. 2; 2 2]\nA = [1. 5; 5 2]\nb = 4.0;\n\nmodel = COSMO.Model();\n\n# define the cost function\nP = zeros(4, 4)\nq = vec(C)\n\n# define the constraints\n# A x = b\ncs1 = COSMO.Constraint(vec(A)', -b, COSMO.ZeroSet)\n# X in PSD cone\ncs2 = COSMO.Constraint(Matrix(1.0I, 4, 4), zeros(4), COSMO.PsdCone)\nconstraints = [cs1; cs2]\n\n# assemble and solve the model\nassemble!(model, P, q, constraints)\nresult = COSMO.optimize!(model);\n\nX_sol = reshape(result.x, 2, 2)\nobj_value = result.obj_val","category":"page"},{"location":"","page":"Home","title":"Home","text":"or we can describe the problem using JuMP and use COSMO as the backend solver:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using COSMO, JuMP, LinearAlgebra\n\nC = [1. 2; 2 2]\nA = [1. 5; 5 2]\nb = 4.0;\nm = JuMP.Model(COSMO.Optimizer);\n@variable(m, X[1:2, 1:2], PSD)\n@objective(m, Min, tr(C * X));\n@constraint(m, tr(A * X) == b);\nJuMP.optimize!(m);\n\nstatus = JuMP.termination_status(m)\nX_sol = JuMP.value.(X)\nobj_value = JuMP.objective_value(m)","category":"page"},{"location":"#Related-talks","page":"Home","title":"Related talks","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"A video of my first presentation of COSMO at JuMP-dev (Santiago) is available here: COSMO.jl at JuMP-dev\nA more detailed presentation on chordal decomposition and clique merging is available here: Chordal decomposition and clique merging\nI gave an overview on recent features of COSMO such as acceleration methods at JuliaCon 2021: What's new in COSMO?\nMore details on the acceleration methods were presented in my ECC 2022 talk Safeguarded Anderson acceleration for parametric nonexpansive operators","category":"page"},{"location":"#Credits","page":"Home","title":"Credits","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"The following people are involved in the development of COSMO:","category":"page"},{"location":"","page":"Home","title":"Home","text":"Michael Garstka (main development)\nNikitas Rontsis (algorithm performance)\nPaul Goulart (code architecture, maths and algorithms)\nMark Cannon (maths and algorithms)","category":"page"},{"location":"","page":"Home","title":"Home","text":"*all contributors are affiliated with the University of Oxford.","category":"page"},{"location":"","page":"Home","title":"Home","text":"If this project is useful for your work please consider","category":"page"},{"location":"","page":"Home","title":"Home","text":"Citing the relevant papers\nLeaving a star on the GitHub repository","category":"page"},{"location":"#Licence","page":"Home","title":"Licence","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"COSMO.jl is licensed under the Apache License 2.0. For more details click here.","category":"page"},{"location":"citing/#Citing-COSMO","page":"Citing COSMO","title":"Citing COSMO","text":"","category":"section"},{"location":"citing/","page":"Citing COSMO","title":"Citing COSMO","text":"If you find COSMO useful in your project, we kindly request that you cite the following paper:","category":"page"},{"location":"citing/","page":"Citing COSMO","title":"Citing COSMO","text":"@Article{Garstka_2021,\n author = {Michael Garstka and Mark Cannon and Paul Goulart},\n journal = {Journal of Optimization Theory and Applications},\n title = {{COSMO}: A Conic Operator Splitting Method for Convex Conic Problems},\n volume = {190},\n number = {3},\n pages = {779--810},\n year = {2021},\n publisher = {Springer},\n doi = {10.1007/s10957-021-01896-x},\n url = {https://doi.org/10.1007/s10957-021-01896-x}\n}","category":"page"},{"location":"citing/","page":"Citing COSMO","title":"Citing COSMO","text":"The article is available under Open Access here.","category":"page"},{"location":"citing/","page":"Citing COSMO","title":"Citing COSMO","text":"The following paper describes the algorithm used for our clique graph based merging strategy:","category":"page"},{"location":"citing/","page":"Citing COSMO","title":"Citing COSMO","text":"@InProceedings{Garstka_2020,\n author = {Michael Garstka and Mark Cannon and Paul Goulart},\n title = {A clique graph based merging strategy for decomposable {SDPs}},\n year = {2020},\n note = {21th IFAC World Congress},\n number = {2},\n pages = {7355-7361},\n volume = {53},\n doi = {10.1016/j.ifacol.2020.12.1255},\n issn = {2405-8963},\n journal = {IFAC-PapersOnLine},\n}","category":"page"},{"location":"citing/","page":"Citing COSMO","title":"Citing COSMO","text":"A preprint can be downloaded here.","category":"page"},{"location":"citing/","page":"Citing COSMO","title":"Citing COSMO","text":"The acceleration strategy is described in this paper:","category":"page"},{"location":"citing/","page":"Citing COSMO","title":"Citing COSMO","text":"@InProceedings{Garstka_2022,\n title={Safeguarded Anderson acceleration for parametric nonexpansive operators},\n author={Garstka, Michael and Cannon, Mark and Goulart, Paul},\n booktitle={2022 European Control Conference (ECC)},\n pages={435--440},\n year={2022},\n organization={IEEE}\n}","category":"page"},{"location":"citing/","page":"Citing COSMO","title":"Citing COSMO","text":"A preprint can be downloaded here.","category":"page"}] }