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Projected Jacobi Method for Vertical Tensor Complementarity Problems

Author

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  • Zheng-Hai Huang

    (Tianjin University)

  • Qiumei Zhao

    (Tianjin University)

  • Yong Wang

    (Tianjin University)

Abstract

As a generalization of the vertical linear complementarity problem, the vertical tensor complementarity problem (VTCP) has been recently studied . In this paper, we consider a class of VTCPs in which the involved tensors are vertical block implicitly strictly diagonally dominant by mode-1 with positive diagonal entries. By using degree theory, we show that there is at least one solution to this class of VTCPs. Moreover, we also provide a sufficient condition under which the VTCP under consideration has a unique solution. In particular, based on an equivalent reformulation of this class of VTCPs, we propose a projected Jacobi method for solving this class of VTCPs, and discuss the global linear convergence of the method under an additional condition. The proposed method is a generalization of the projected Jacobi method for the vertical linear complementarity problem proposed in the recent literature. The computations required at each iteration of the method are simple. Preliminary numerical experiments show the effectiveness of the method.

Suggested Citation

  • Zheng-Hai Huang & Qiumei Zhao & Yong Wang, 2025. "Projected Jacobi Method for Vertical Tensor Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 206(2), pages 1-26, August.
    • Handle: RePEc:spr:joptap:v:206:y:2025:i:2:d:10.1007_s10957-025-02699-0
    DOI: 10.1007/s10957-025-02699-0
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    References listed on IDEAS

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    1. Liqun Qi & Zheng-Hai Huang, 2019. "Tensor Complementarity Problems—Part II: Solution Methods," Journal of Optimization Theory and Applications, Springer, vol. 183(2), pages 365-385, November.
    2. Ge Li & Jicheng Li, 2023. "Improved Fixed Point Iterative Methods for Tensor Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 199(2), pages 787-804, November.
    3. 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.
    4. Nagae, Takeshi & Akamatsu, Takashi, 2008. "A generalized complementarity approach to solving real option problems," Journal of Economic Dynamics and Control, Elsevier, vol. 32(6), pages 1754-1779, June.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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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