A finite-termination algorithm for testing copositivity over the positive semidefinite cone

TL;DR

This paper introduces a finite-termination algorithm for testing copositivity of homogeneous polynomials over the positive semidefinite cone, utilizing a novel reformulation and semidefinite programming hierarchy.

Contribution

It presents a new algorithm that guarantees finite termination for copositivity testing over the positive semidefinite cone, with extensions to product cones and practical certificates.

Findings

Algorithm always terminates finitely

Provides certificates or refutations of copositivity

Preliminary experiments show effectiveness

Abstract

This paper proposes an efficient algorithm for testing copositivity of homogeneous polynomials over the positive semidefinite cone. The algorithm is based on a novel matrix optimization reformulation and requires solving a hierarchy of semidefinite programs. Notably, it always terminates in finitely many iterations. If a homogeneous polynomial is copositive over the positive semidefinite cone, the algorithm provides a certificate; otherwise, it returns a vector that refutes copositivity. Building on a similar idea, we further propose an algorithm to test copositivity over the direct product of the positive semidefinite cone and the nonnegative orthant. Preliminary numerical experiments demonstrate the effectiveness of the proposed methods.