Solving Nonconvex Optimization Problems using Outer Approximations of the Set-Copositive Cone

We consider the solution of nonconvex quadratic optimization problems using an outer approximation of the set-copositive cone that is iteratively strengthened with conic constraints and cutting planes. Our methodology utilizes an MILP-based oracle for a generalization of the copositive cone that considers additional linear equality constraints. In numerical testing we evaluate our algorithm on a … Read more

Approximation hierarchies for copositive cone over symmetric cone and their comparison

We first provide an inner-approximation hierarchy described by a sum-of-squares (SOS) constraint for the copositive (COP) cone over a general symmetric cone. The hierarchy is a generalization of that proposed by Parrilo (2000) for the usual COP cone (over a nonnegative orthant). We also discuss its dual. Second, we characterize the COP cone over a … Read more

Finite convergence of sum-of-squares hierarchies for the stability number of a graph

We investigate a hierarchy of semidefinite bounds $\vartheta^{(r)}(G)$ for the stability number $\alpha(G)$ of a graph $G$, based on its copositive programming formulation and introduced by de Klerk and Pasechnik [SIAM J. Optim. 12 (2002), pp.875–892], who conjectured convergence to $\alpha(G)$ in $r=\alpha(G) -1$ steps. Even the weaker conjecture claiming finite convergence is still open. … Read more

Copositive Duality for Discrete Energy Markets

Optimization problems with discrete decisions are nonconvex and thus lack strong duality, which limits the usefulness of tools such as shadow prices. It was shown in Burer (2009) that mixed-binary quadratic programs can be written as completely positive programs, which are convex. Completely positive reformulations of discrete optimization problems therefore have strong duality if a … Read more

A Modified Simplex Partition Algorithm to Test Copositivity

A real symmetric matrix $A$ is copositive if $x^\top Ax\geq 0$ for all $x\geq 0$. As $A$ is copositive if and only if it is copositive on the standard simplex, algorithms to determine copositivity, such as those in Sponsel et al. (J Glob Optim 52:537–551, 2012) and Tanaka and Yoshise (Pac J Optim 11:101–120, 2015) … Read more

Testing Copositivity via Mixed-Integer Linear Programming

We describe a simple method to test if a given matrix is copositive by solving a single mixed-integer linear programming (MILP) problem. This methodology requires no special coding to implement and takes advantage of the computational power of modern MILP solvers. Numerical experiments demonstrate that the method is robust and efficient. Citation Dept. of Business … Read more

On Standard Quadratic Programs with Exact and Inexact Doubly Nonnegative Relaxations

The problem of minimizing a (nonconvex) quadratic form over the unit simplex, referred to as a standard quadratic program, admits an exact convex conic formulation over the computationally intractable cone of completely positive matrices. Replacing the intractable cone in this formulation by the larger but tractable cone of doubly nonnegative matrices, i.e., the cone of … Read more

Gaddum’s test for symmetric cones

A real symmetric matrix “A” is copositive if the inner product if Ax and x is nonnegative for all x in the nonnegative orthant. Copositive programming has attracted a lot of attention since Burer showed that hard nonconvex problems can be formulated as completely-positive programs. Alas, the power of copositive programming is offset by its … Read more

Generating irreducible copositive matrices using the stable set problem

In this paper it is considered how graphs can be used to generate copositive matrices, and necessary and sufficient conditions are given for these generated matrices to then be irreducible with respect to the set of positive semidefinite plus nonnegative matrices. This is done through combining the well known copositive formulation of the stable set … Read more

Non-convex min-max fractional quadratic problems under quadratic constraints: copositive relaxations

In this paper we address a min-max problem of fractional quadratic (not necessarily convex) over linear functions on a feasible set described by linear and (not necessarily convex) quadratic functions. We propose a conic reformulation on the cone of completely positive matrices. By relaxation, a doubly non negative conic formulation is used to provide lower … Read more