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T-positive semidefiniteness of third-order symmetric tensors and T-semidefinite programming

Author

Listed:
  • Meng-Meng Zheng

    (Tianjin University)

  • Zheng-Hai Huang

    (Tianjin University)

  • Yong Wang

    (Tianjin University)

Abstract

The T-product for third-order tensors has been used extensively in the literature. In this paper, we first introduce first-order and second-order T-derivatives for the multi-variable real-valued function with the tensor T-product. Inspired by an equivalent characterization of a twice continuously T-differentiable multi-variable real-valued function being convex, we present a definition of the T-positive semidefiniteness of third-order symmetric tensors. After that, we extend many properties of positive semidefinite matrices to the case of third-order symmetric tensors. In particular, analogue to the widely used semidefinite programming (SDP for short), we introduce the semidefinite programming over the space of third-order symmetric tensors (T-semidefinite programming or TSDP for short), and provide a way to solve the TSDP problem by converting it into an SDP problem in the complex domain. Furthermore, we give several TSDP examples and especially some preliminary numerical results for two unconstrained polynomial optimization problems. Experiments show that finding the global minimums of polynomials via the TSDP relaxation outperforms the traditional SDP relaxation for the test examples.

Suggested Citation

  • Meng-Meng Zheng & Zheng-Hai Huang & Yong Wang, 2021. "T-positive semidefiniteness of third-order symmetric tensors and T-semidefinite programming," Computational Optimization and Applications, Springer, vol. 78(1), pages 239-272, January.
    • Handle: RePEc:spr:coopap:v:78:y:2021:i:1:d:10.1007_s10589-020-00231-w
    DOI: 10.1007/s10589-020-00231-w
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    References listed on IDEAS

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    1. Lei Yang & Zheng-Hai Huang & Shenglong Hu & Jiye Han, 2016. "An iterative algorithm for third-order tensor multi-rank minimization," Computational Optimization and Applications, Springer, vol. 63(1), pages 169-202, January.
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    Cited by:

    1. Changxin Mo & Weiyang Ding & Yimin Wei, 2024. "Perturbation Analysis on T-Eigenvalues of Third-Order Tensors," Journal of Optimization Theory and Applications, Springer, vol. 202(2), pages 668-702, August.
    2. Meng-Meng Zheng & Zheng-Hai Huang & Sheng-Long Hu, 2022. "Unconstrained minimization of block-circulant polynomials via semidefinite program in third-order tensor space," Journal of Global Optimization, Springer, vol. 84(2), pages 415-440, October.
    3. Hiroki Marumo & Sunyoung Kim & Makoto Yamashita, 2024. "T-semidefinite programming relaxation with third-order tensors for constrained polynomial optimization," Computational Optimization and Applications, Springer, vol. 89(1), pages 183-218, September.
    4. Huang, Baohua, 2024. "Conjugate gradient-type method for the tensor linear system via the T-product and its application in the calculation of Moore-Penrose inverse," Applied Mathematics and Computation, Elsevier, vol. 472(C).
    5. Xuezhong Wang & Ping Wei & Yimin Wei, 2023. "A Fixed Point Iterative Method for Third-order Tensor Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 334-357, April.

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