QCQP

QCQP is a package for modeling and nonconvex solving quadratically constrained quadratic programs (QCQPs) using relaxations and local search heuristics. Our heuristics are based on the Suggest-and-Improve framework:

• Suggest method finds a candidate point for a local method.
• Improve method takes a point from the Suggest method and performs a local search to find a better point.

The notion of better points is defined by the maximum violation of a point and the objective value. See our associated paper for more information on the Suggest-and-Improve framework.

QCQP is built on top of CVXPY, a domain-specific language for convex optimization embedded in Python.

Installation

NOTE: QCQP was developed before the release of CVXPY 1.0, which is not backward compatible with the previous version CVXPY 0.4. As of August 2018, QCQP is only compatible with CVXPY 0.4.

You should first install CVXPY 0.4, following the instructions here. If you already have CVXPY, make sure you have the version compatible with QCQP by running conda list cvxpy or pip show cvxpy. You can install the compatible version of CVXPY by running the following.

conda install -c cvxgrp cvxpy=0.4.9

The simplest and recommended way of installing QCQP is to run pip install qcqp. To install the package from source, run python setup.py install in the source directory. You may need to run the commands with the sudo privilege.

Example

The following code uses semidefinite relaxation (SDR) to get a lower bound on a random instance of the Boolean least squares problem. Then, using a candidate point generated from the SDR, it runs a coordinate descent method to attempt to find a feasible point with better objective value.

from numpy.random import randn
import cvxpy as cvx
from qcqp import *

n, m = 10, 15
A = randn(m, n)
b = randn(m, 1)

# Form a nonconvex problem.
x = cvx.Variable(n)
obj = cvx.sum_squares(A*x - b)
cons = [cvx.square(x) == 1]
prob = cvx.Problem(cvx.Minimize(obj), cons)

# Create a QCQP handler.
qcqp = QCQP(prob)

# Solve the SDP relaxation and get a starting point to a local method
qcqp.suggest(SDR)
print("SDR lower bound: %.3f" % qcqp.sdr_bound)

# Attempt to improve the starting point given by the suggest method
f_cd, v_cd = qcqp.improve(COORD_DESCENT)
print("Coordinate descent: objective %.3f, violation %.3f" % (f_cd, v_cd))
print(x.value)

Quadratic expressions

The quadraticity of an expression e can be tested using e.is_quadratic(). Below is a list of expressions that CVXPY recognizes as a quadratic expression. Refer to the CVXPY documentation for the specifications of the functions.

• Any constant or affine expression
• Any affine transformation applied to a quadratic expression, e.g., (quadratic) + (quadratic) or (constant) * (quadratic)
• (affine) * (affine)
• power(affine, 2)
• square(affine)
• sum_squares(affine)
• quad_over_lin(affine, constant)
• matrix_frac(affine, constant)
• quad_form(affine, constant)

Constructing and solving problems

QCQPs must be represented using the standard CVXPY syntax. In order for the problem to be accepted by QCQP, the problem must have a quadratic objective function and quadratic constraints. To apply the Suggest and Improve methods on a QCQP, the corresponding CVXPY problem object must be passed to the QCQP constructor first. For example, if problem is a CVXPY problem object describing a QCQP, then the following code checks the validity and prepares the Suggest and Improve methods:

qcqp = QCQP(problem)

Currently two Suggest methods are available for QCQPs:

• qcqp.suggest() or qcqp.suggest(RANDOM) fills the values of the variables using independent and identically distributed Gaussian random variables.
• qcqp.suggest(SPECTRAL) adds all the constraints to a single (possibly nonconvex) constraint, and solves the resulting QCQP with that one constraint. The solution of the relaxation is then stored to the values of the variables. Then, a lower bound (or an upper bound, in the case of a maximization problem) on the optimal value is saved to qcqp.spectral_bound. The performance of this method is yet to be optimized.
• qcqp.suggest(SDR) fills the values of the variables drawn from an optimal probability distribution given by the semidefinite relaxation. Then, a lower bound (or an upper bound, in the case of a maximization problem) on the optimal value is saved to qcqp.sdr_bound. Note: For larger problem instances, QCQP may fail while solving the semidefinite relaxation. In this case, specifying the MOSEK solver may help: qcqp.suggest(SDR, solver=cvx.MOSEK). For more information on how to choose solvers, please see the CVXPY guide.

Below is a list of available solve methods for QCQPs:

• qcqp.improve(ADMM) attempts to improve the given point via consensus alternating directions method of multipliers (ADMM). An optional parameter rho can be specified.
• qcqp.improve(DCCP) automatically splits indefinite quadratic functions to convex and concave parts, then invokes the DCCP package, using the given point as a starting point. An optional parameter tau can be specified. In order to use this method, DCCP must be installed first.
• qcqp.improve(COORD_DESCENT) performs a two-stage coordinate descent algorithm. The first stage tries to find a feasible point. If a feasible point is found, then the second stage tries to optimize the objective function over the set of feasible points.
• qcqp.improve(IPOPT) invokes the global optimizer IPOPT package, via PyIpopt as the interface. In order to use this method, both IPOPT and PyIpopt must be installed first.

Both improve() and suggest() methods return a pair (f, v), where f represents the current objective value, and v represents the maximum constraint violation of the current point.