Hypergraph characterizations of copositive tensors

Abstract

A real symmetric tensor \mathcal{A}=(a_{i_{\mathrm{1}}}\cdots i_m)\in {ℝ}^{[mn]} is copositive (resp., strictly copositive) if \mathcal{A}{x}^m\ge \mathrm{0} (resp., \mathcal{A}{x}^m>\mathrm{0}) for any nonzero nonnegative vectorx\in {ℝ}^n: By using the associated hypergraph of \mathcal{A}, we give necessary and sufficient conditions for the copositivity of \mathcal{A}: For a real symmetric tensor \mathcal{A}satisfying the associated negative hypergraph H\_(\mathcal{A}) and associated positive hypergraph H_{+}(\mathcal{A}) are edge disjoint subhypergraphs of a supertree or cored hypergraph, we derive criteria for the copositivity of \mathcal{A}: We also use copositive tensors to study the positivity of tensor systems.

Key words： Copositive tensor; hypergraph; positive system

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