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Projected fixed-point method for vertical tensor complementarity problems

Author

Listed:
  • Ting Zhang

    (University of Science and Technology Beijing)

  • Yong Wang

    (Tianjin University)

  • Zheng-Hai Huang

    (Tianjin University)

Abstract

It is well known that the standard complementarity problem can be equivalently reformulated as a projected fixed-point equation, and this reformulation plays an important role in the theoretical and algorithmic research of complementarity problems. The vertical linear complementarity problem proposed by Cottle and Dantzig is an important generalization of the standard linear complementarity problem, and recently it has been further generalized to the case of tensors, called the vertical tensor complementarity problem (VTCP). In this paper, we give a projected fixed-point reformulation of the VTCP, and then, we design a fixed-point iteration method for solving the VTCP with a vertical block implicit Z-tensor. When there are non-positive diagonal entries in the representation subtensors of the tensor involved in the VTCP, we can reduce the computational cost of our method by solving a lower dimensional VTCP. Under the assumption that the problem under consideration is feasible, we prove that the designed method converges monotonically to a solution of the problem. The numerical results show the effectiveness of the proposed method.

Suggested Citation

  • Ting Zhang & Yong Wang & Zheng-Hai Huang, 2024. "Projected fixed-point method for vertical tensor complementarity problems," Computational Optimization and Applications, Springer, vol. 89(1), pages 219-245, September.
    • Handle: RePEc:spr:coopap:v:89:y:2024:i:1:d:10.1007_s10589-024-00581-9
    DOI: 10.1007/s10589-024-00581-9
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    References listed on IDEAS

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    1. Zheng-Hai Huang & Yu-Fan Li & Yong Wang, 2023. "A fixed point iterative method for tensor complementarity problems with the implicit Z-tensors," Journal of Global Optimization, Springer, vol. 86(2), pages 495-520, June.
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    3. 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.
    4. Francesco Mezzadri & Emanuele Galligani, 2022. "Projected Splitting Methods for Vertical Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 598-620, June.
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    6. Yisheng Song & Liqun Qi, 2015. "Properties of Some Classes of Structured Tensors," Journal of Optimization Theory and Applications, Springer, vol. 165(3), pages 854-873, June.
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    9. Xuezhong Wang & Maolin Che & Yimin Wei, 2022. "Randomized Kaczmarz methods for tensor complementarity problems," Computational Optimization and Applications, Springer, vol. 82(3), pages 595-615, July.
    10. M. Seetharama Gowda & Jong-Shi Pang, 1992. "On Solution Stability of the Linear Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 17(1), pages 77-83, February.
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