Copositive Matrices with Ordered Off-Diagonal Entries

TL;DR

This paper characterizes copositive matrices with ordered off-diagonal entries, showing they admit a specific decomposition and applying this to quadratic form optimization, resolving an open problem.

Contribution

It proves that copositive matrices with nondecreasing off-diagonal entries admit a positive semidefinite plus nonnegative decomposition, impacting quadratic optimization.

Findings

Ordered off-diagonal copositive matrices admit a specific decomposition.

The relaxation of quadratic form optimization over the simplex is tight for separable objectives.

Resolved an open question by Dey and Kocuk regarding quadratic optimization.

Abstract

We study copositive matrices which admit a decomposition into a sum of a positive semidefinite matrix and a matrix with nonnegative entries. Our main result shows that if the off-diagonal entries of a copositive matrix are nondecreasing in rows and in columns, then it admits such a decomposition. We apply this result to study optimization of quadratic forms over the standard simplex. As a corollary, we obtain that a natural relaxation of this problem is tight when the objective function is separable, resolving an open question of Dey and Kocuk.