Copositive tensor detection and its applications in physics and hypergraphs

Haibin Chen (), Zheng-Hai Huang () and Liqun Qi ()
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Haibin Chen: Qufu Normal University
Zheng-Hai Huang: Tianjin University
Liqun Qi: The Hong Kong Polytechnic University

Computational Optimization and Applications, 2018, vol. 69, issue 1, No 6, 133-158

Abstract: 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 to 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.

Keywords: Symmetric tensor; Strictly copositive tensor; Positive semi-definiteness; Simplicial partition; Particle physics; Hypergraphs; 65H17; 15A18; 90C30 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (5)

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DOI: 10.1007/s10589-017-9938-1

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