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Copositive tensor detection and its applications in physics and hypergraphs

Author

Listed:
  • Haibin Chen

    (Qufu Normal University)

  • Zheng-Hai Huang

    (Tianjin University)

  • Liqun Qi

    (The Hong Kong Polytechnic University)

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.

Suggested Citation

  • Haibin Chen & Zheng-Hai Huang & Liqun Qi, 2018. "Copositive tensor detection and its applications in physics and hypergraphs," Computational Optimization and Applications, Springer, vol. 69(1), pages 133-158, January.
    • Handle: RePEc:spr:coopap:v:69:y:2018:i:1:d:10.1007_s10589-017-9938-1
    DOI: 10.1007/s10589-017-9938-1
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    References listed on IDEAS

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    1. Zheng-Hai Huang & Liqun Qi, 2017. "Formulating an n-person noncooperative game as a tensor complementarity problem," Computational Optimization and Applications, Springer, vol. 66(3), pages 557-576, April.
    2. Chen Ling & Hongjin He & Liqun Qi, 2016. "On the cone eigenvalue complementarity problem for higher-order tensors," Computational Optimization and Applications, Springer, vol. 63(1), pages 143-168, January.
    3. Yisheng Song & Liqun Qi, 2016. "Eigenvalue analysis of constrained minimization problem for homogeneous polynomial," Journal of Global Optimization, Springer, vol. 64(3), pages 563-575, March.
    4. Yisheng Song & Liqun Qi, 2016. "Tensor Complementarity Problem and Semi-positive Tensors," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 1069-1078, June.
    5. Haibin Chen & Zheng-Hai Huang & Liqun Qi, 2017. "Copositivity Detection of Tensors: Theory and Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 174(3), pages 746-761, September.
    6. Xue-Li Bai & Zheng-Hai Huang & Yong Wang, 2016. "Global Uniqueness and Solvability for Tensor Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 170(1), pages 72-84, July.
    7. Julia Sponsel & Stefan Bundfuss & Mirjam Dür, 2012. "An improved algorithm to test copositivity," Journal of Global Optimization, Springer, vol. 52(3), pages 537-551, March.
    8. Maolin Che & Liqun Qi & Yimin Wei, 2016. "Positive-Definite Tensors to Nonlinear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 168(2), pages 475-487, February.
    9. Peter Dickinson & Luuk Gijben, 2014. "On the computational complexity of membership problems for the completely positive cone and its dual," Computational Optimization and Applications, Springer, vol. 57(2), pages 403-415, March.
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    Cited by:

    1. Zheng-Hai Huang & Liqun Qi, 2019. "Tensor Complementarity Problems—Part I: Basic Theory," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 1-23, October.
    2. Zheng-Hai Huang & Liqun Qi, 2019. "Tensor Complementarity Problems—Part III: Applications," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 771-791, December.
    3. Haibin Chen & Hongjin He & Yiju Wang & Guanglu Zhou, 2022. "An efficient alternating minimization method for fourth degree polynomial optimization," Journal of Global Optimization, Springer, vol. 82(1), pages 83-103, January.
    4. Xue-Li Bai & Zheng-Hai Huang & Xia Li, 2019. "Stability of Solutions and Continuity of Solution Maps of Tensor Complementarity Problems," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 36(02), pages 1-19, April.
    5. Yisheng Song & Xudong Li, 2022. "Copositivity for a Class of Fourth-Order Symmetric Tensors Given by Scalar Dark Matter," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 334-346, October.

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