Hypergraph characterizations of copositive tensors

Abstract

A real symmetric tensor ${\mathscr{A}} = ({a_{{i_1} \ldots {i_m}}}) \in {\mathbb{R}^{[m,n]}}$ is copositive (resp., strictly copositive) if ${\mathscr{A}}\;{x^m} \geqslant 0$ (resp.,${\mathscr{A}}\;{x^m} > 0$) for any nonzero nonnegative vector x ∈ ℝⁿ. By using the associated hypergraph of ${\mathscr{A}}$, we give necessary and sufficient conditions for the copositivity of ${\mathscr{A}}$. For a real symmetric tensor ${\mathscr{A}}$ satisfying the associated negative hypergraph ${H_ - }({\mathscr{A}})$ and associated positive hypergraph ${H_ + }({\mathscr{A}})$ are edge disjoint subhypergraphs of a supertree or cored hypergraph, we derive criteria for the copositivity of ${\mathscr{A}}$. We also use copositive tensors to study the positivity of tensor systems.