Copositive Tensor Detection and Its Applications in Physics and Hypergraphs

TL;DR

This paper introduces a new algorithm for testing tensor copositivity, with applications in physics and hypergraph theory, providing new conditions, convex subsets, and practical computational methods.

Contribution

It proposes a novel algorithm for copositivity testing of high-order tensors and demonstrates its applications in physics and hypergraph analysis.

Findings

The algorithm effectively tests tensor copositivity.

It provides an upper bound for the coclique number in hypergraphs.

Applications include particle physics and hypergraph optimization.

Abstract

Copositivity of tensors plays an important role in vacuum stability of a general scalar potential, polynomial optimization, tensor complementarity problem and tensor generalized eigenvalue complementarity problem. In this paper, we propose a new algorithm for testing copositivity of high order tensors, and then present applications of the algorithm in physics and hypergraphs. For this purpose, we first give several new conditions for copositivity of tensors based on the representative matrix of a simplex. Then a new algorithm is proposed with the help of a proper convex subcone of the copositive tensor cone, which is defined via the copositivity of Z-tensors. Furthermore, by considering a sum-of-squares program problem, we define two new subsets of the copositive tensor cone and discuss their convexity. As an application of the proposed algorithm, we prove that the coclique number of a uniform hypergraph is equivalent with an optimization problem over the completely positive tensor cone, which implies that the proposed algorithm can be applied to compute an upper bound of the coclique number of a uniform hypergraph. Then we study another application of the proposed algorithm on particle physics in testing copositivity of some potential fields. At last, various numerical examples are given to show the performance of the algorithm.