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[docs] add more tips and tricks for linear programs
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odow committed Dec 7, 2022
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1 change: 1 addition & 0 deletions docs/make.jl
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Expand Up @@ -88,6 +88,7 @@ if !_FAST
for file in [
joinpath("getting_started", "getting_started_with_julia.md"),
joinpath("getting_started", "getting_started_with_JuMP.md"),
joinpath("linear", "tips_and_tricks.md"),
]
filename = joinpath(@__DIR__, "src", "tutorials", file)
content = read(filename, String)
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2 changes: 2 additions & 0 deletions docs/src/changelog.md
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Expand Up @@ -12,6 +12,8 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
### Other

- Added Benders tutorial with in-place resolves (#3145)
- Added more [Tips and tricks](@id linear_tips_and_tricks) for linear programs
(#3144)

## Version 1.5.0 (December 8, 2022)

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112 changes: 98 additions & 14 deletions docs/src/tutorials/linear/tips_and_tricks.jl
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Expand Up @@ -30,6 +30,90 @@

using JuMP

# ## Absolute value

# To model the absolute value function ``t \ge |x|``, there are a few options.
# In all cases, these reformulations only work if you are minimizing ``t``
# "down" into ``|x|``. They do not work if you are trying to maximize ``|x|``.

# ### Option 1

# This option adds two linear inequality constraints:

model = Model();
@variable(model, x)
@variable(model, t)
@constraint(model, t >= x)
@constraint(model, t >= -x)

# ### Option 2

# This option uses two non-negative variables and forms expressions for ``x``
# and ``t``:

model = Model();
@variable(model, z[1:2] >= 0)
@expression(model, t, z[1] + z[2])
@expression(model, x, z[1] - z[2])

# ### Option 3

# This option uses [`MOI.NormOneCone`](@ref) and lets JuMP choose the
# reformulation:

model = Model();
@variable(model, x)
@variable(model, t)
@constraint(model, [t; x] in MOI.NormOneCone(2))

# ## L1-norm

# To model ``\min ||x||_1``, that is, ``\min \sum\limits_i |x_i|``, use the
# [`MOI.NormOneCone`](@ref):

model = Model();
@variable(model, x[1:3])
@variable(model, t)
@constraint(model, [t; x] in MOI.NormOneCone(1 + length(x)))
@objective(model, Min, t)

# ## Infinity-norm

# To model ``\min ||x||_\infty``, that is, ``\min \max\limits_i |x_i|``, use the
# [`MOI.NormInfinityCone`](@ref):

model = Model();
@variable(model, x[1:3])
@variable(model, t)
@constraint(model, [t; x] in MOI.NormInfinityCone(1 + length(x)))
@objective(model, Min, t)

# ## Max

# To model ``t \ge \max\{x, y\}``, do:

model = Model();
@variable(model, t)
@variable(model, x)
@variable(model, y)
@constraint(model, t >= x)
@constraint(model, t >= y)

# This reformulation does not work for ``t \ge \min\{x, y\}``.

# ## Max

# To model ``t \le \min\{x, y\}``, do:

model = Model();
@variable(model, t)
@variable(model, x)
@variable(model, y)
@constraint(model, t <= x)
@constraint(model, t <= y)

# This reformulation does not work for ``t \le \max\{x, y\}``.

# ## Boolean operators

# Binary variables can be used to construct logical operators. Here are some
Expand All @@ -39,7 +123,7 @@ using JuMP

# $$x_3 = x_1 \lor x_2$$

model = Model()
model = Model();
@variable(model, x[1:3], Bin)
@constraints(model, begin
x[1] <= x[3]
Expand All @@ -51,7 +135,7 @@ end)

# $$x_3 = x_1 \land x_2$$

model = Model()
model = Model();
@variable(model, x[1:3], Bin)
@constraints(model, begin
x[3] <= x[1]
Expand All @@ -63,15 +147,15 @@ end)

# $$x_1 \neg x_2$$

model = Model()
model = Model();
@variable(model, x[1:2], Bin)
@constraint(model, x[1] == 1 - x[2])

# ### Implies

# $$x_1 \implies x_2$$

model = Model()
model = Model();
@variable(model, x[1:2], Bin)
@constraint(model, x[1] <= x[2])

Expand All @@ -89,7 +173,7 @@ model = Model()

# **Example** Either $x_1 \leq 1$ or $x_2 \leq 2$.

model = Model()
model = Model();
@variable(model, x[1:2])
@variable(model, y, Bin)
M = 100
Expand Down Expand Up @@ -118,14 +202,14 @@ M = 100

# **Example** $x_1 + x_2 \leq 1$ if $z = 1$.

model = Model()
model = Model();
@variable(model, x[1:2])
@variable(model, z, Bin)
@constraint(model, z => {sum(x) <= 1})

# **Example** $x_1 + x_2 \leq 1$ if $z = 0$.

model = Model()
model = Model();
@variable(model, x[1:2])
@variable(model, z, Bin)
@constraint(model, !z => {sum(x) <= 1})
Expand All @@ -137,15 +221,15 @@ model = Model()

# **Example** $x_1 + x_2 \leq 1$ if $z = 1$.

model = Model()
model = Model();
@variable(model, x[1:2])
@variable(model, z, Bin)
M = 100
@constraint(model, sum(x) <= 1 + M * (1 - z))

# **Example** $x_1 + x_2 \leq 1$ if $z = 0$.

model = Model()
model = Model();
@variable(model, x[1:2])
@variable(model, z, Bin)
M = 100
Expand All @@ -169,7 +253,7 @@ M = 100

# **Example** $$x \in \{0\}\cup [1, 2]$$

model = Model()
model = Model();
@variable(model, x in MOI.Semicontinuous(1.0, 2.0))

# ## Semi-integer variables
Expand All @@ -178,7 +262,7 @@ model = Model()
# bounds $[l,u]$ and can also assume the value zero:
# $$x \in \{0\} \cup [l, u] \cap \mathbb{Z}.$$

model = Model()
model = Model();
@variable(model, x in MOI.Semiinteger(5.0, 10.0))

# ## Special Ordered Sets of Type I
Expand All @@ -190,7 +274,7 @@ model = Model()
# variables. In other words, we have to choose at most one from a set of
# possibilities.

model = Model()
model = Model();
@variable(model, x[1:3], Bin)
@constraint(model, x in SOS1())

Expand All @@ -208,7 +292,7 @@ model = Model()
# at most two can be non-zero, and if two are non-zero these must be consecutive
# in their ordering.

model = Model()
model = Model();
@variable(model, x[1:3])
@constraint(model, x in SOS2([3.0, 1.0, 2.0]))

Expand All @@ -234,7 +318,7 @@ ŷ = x̂ .^ 2
# as convex combinations of `x̂` and `ŷ`.

N = length(x̂)
model = Model()
model = Model();
@variable(model, -1 <= x <= 2)
@variable(model, y)
@variable(model, 0 <= λ[1:N] <= 1)
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