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Copositivity Detection of Tensors: Theory and Algorithm

Author

Listed:
  • Haibin Chen

    (Qufu Normal University)

  • Zheng-Hai Huang

    (Tianjin University)

  • Liqun Qi

    (The Hong Kong Polytechnic University)

Abstract

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.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:174:y:2017:i:3:d:10.1007_s10957-017-1131-2
    DOI: 10.1007/s10957-017-1131-2
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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. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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. Ya-nan Zheng & Wei Wu, 2018. "On a Class of Semi-Positive Tensors in Tensor Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 177(1), pages 127-136, April.
    4. 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.
    5. 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.
    6. 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.
    7. 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.

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