Copositivity Detection of Tensors: Theory and Algorithm

Haibin Chen, Zheng Hai Huang, Liqun Qi

Research output: Journal article publicationJournal articleAcademic researchpeer-review

39 Citations (Scopus)


A symmetric tensor is called copositive if it generates a multivariate form taking nonnegative values over the nonnegative orthant. Copositive tensors have found important applications in polynomial optimization, tensor complementarity problems and vacuum stability of a general scalar potential. In this paper, we consider copositivity detection of tensors from both theoretical and computational points of view. After giving several necessary conditions for copositive tensors, we propose several new criteria for copositive tensors based on the representation of the multivariate form in barycentric coordinates with respect to the standard simplex and simplicial partitions. It is verified that, as the partition gets finer and finer, the concerned conditions eventually capture all strictly copositive tensors. Based on the obtained theoretical results with the help of simplicial partitions, we propose a numerical method to judge whether a tensor is copositive or not. The preliminary numerical results confirm our theoretical findings.
Original languageEnglish
Pages (from-to)746-761
Number of pages16
JournalJournal of Optimization Theory and Applications
Issue number3
Publication statusPublished - 1 Sept 2017


  • Positive semi-definiteness
  • Simplicial partition
  • Strictly copositive tensor
  • Symmetric tensor

ASJC Scopus subject areas

  • Control and Optimization
  • Management Science and Operations Research
  • Applied Mathematics


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