Copositivity Detection of Tensors: Theory and Algorithm

TL;DR

This paper develops new theoretical criteria and a numerical method for detecting copositivity in symmetric tensors, with applications in polynomial optimization and tensor complementarity problems.

Contribution

It introduces novel copositivity criteria based on barycentric coordinates and simplicial partitions, advancing both theory and computational detection methods.

Findings

New necessary conditions for copositivity

Criteria that become complete with finer partitions

Preliminary numerical results support the theory

Abstract

A symmetric tensor is called copositive if it generates a multivariate form taking nonnegative values over the nonnegative orthant. Copositive tensors have found important applications in polynomial optimization and tensor complementarity problems. In this paper, we consider copositivity detection of tensors both from theoretical and computational points of view. After giving several necessary conditions for copositive tensors, we propose several new criteria for copositive tensors based on the representation of the multivariate form in barycentric coordinates with respect to the standard simplex and simplicial partitions. It is verified that, as the partition gets finer and finer, the concerned conditions eventually capture all strictly copositive tensors. Based on the obtained theoretical results with the help of simplicial partitions, we propose a numerical method to judge whether a tensor is copositive or not. The preliminary numerical results confirm our theoretical findings.