IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v78y2021i1d10.1007_s10589-020-00231-w.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-020-00231-w
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10589-020-00231-w?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.
    3. 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.
    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Chen Ling & Gaohang Yu & Liqun Qi & Yanwei Xu, 2021. "T-product factorization method for internet traffic data completion with spatio-temporal regularization," Computational Optimization and Applications, Springer, vol. 80(3), pages 883-913, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:coopap:v:78:y:2021:i:1:d:10.1007_s10589-020-00231-w. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.