A Complete Semidefinite Algorithm for Detecting Copositive Matrices and Tensors

TL;DR

This paper introduces a comprehensive semidefinite relaxation algorithm capable of definitively detecting copositivity in matrices and tensors, providing certificates or refutations, despite the NP-hard nature of the problem.

Contribution

It presents a complete semidefinite relaxation method that can conclusively determine copositivity for matrices and tensors, including certificates and refutations.

Findings

Algorithm can detect copositivity or refute it.

Detection is achieved through solving a finite sequence of semidefinite relaxations.

Method applies to all symmetric matrices and tensors, regardless of size.

Abstract

A real symmetric matrix (resp., tensor) is said to be copositive if the associated quadratic (resp., homogeneous) form is greater than or equal to zero over the nonnegative orthant. The problem of detecting their copositivity is NP-hard. This paper proposes a complete semidefinite relaxation algorithm for detecting the copositivity of a matrix or tensor. If it is copositive, the algorithm can get a certificate for the copositivity. If it is not, the algorithm can get a point that refutes the copositivity. We show that the detection can be done by solving a finite number of semidefinite relaxations, for all matrices and tensors.