Hypergraph characterizations of copositive tensors

Yue WANG; Jihong SHEN; Changjiang BU

doi:10.1007/s11464-021-0931-8

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Front. Math. China ›› 2021, Vol. 16 ›› Issue (3) : 815-824. DOI: 10.1007/s11464-021-0931-8

RESEARCH ARTICLE

Hypergraph characterizations of copositive tensors

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Abstract

A real symmetric tensor $A = (a i 1 ⋯ i m) ∈ ℝ [m n]$ is copositive (resp., strictly copositive) if $A x m ≥ 0$ (resp., $A x m > 0$ ) for any nonzero nonnegative vector $x ∈ ℝ n$ : By using the associated hypergraph of $A$ , we give necessary and sufficient conditions for the copositivity of $A$ : For a real symmetric tensor $A$ satisfying the associated negative hypergraph $H_(A)$ and associated positive hypergraph $H + (A)$ are edge disjoint subhypergraphs of a supertree or cored hypergraph, we derive criteria for the copositivity of $A$ : We also use copositive tensors to study the positivity of tensor systems.

Keywords

Copositive tensor / hypergraph / positive system

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Yue WANG, Jihong SHEN, Changjiang BU. Hypergraph characterizations of copositive tensors. Front. Math. China, 2021, 16(3): 815‒824 https://doi.org/10.1007/s11464-021-0931-8

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