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An Iterative Method for Finding the Least Solution to the Tensor Complementarity Problem

Author

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  • Shui-Lian Xie

    (South China Normal University)

  • Dong-Hui Li

    (South China Normal University)

  • Hong-Ru Xu

    (Jiaying University)

Abstract

In this paper, we are concerned with finding the least solution to the tensor complementarity problem. When the involved tensor is strongly monotone, we present a way to estimate the nonzero elements of the solution in a successive manner. The procedure for identifying the nonzero elements of the solution gives rise to an iterative method of solving the tensor complementarity problem. In each iteration, we obtain an iterate by solving a lower-dimensional tensor equation. After finitely many iterations, the method terminates with a solution to the problem. Moreover, the sequence generated by the method is monotonically convergent to the least solution to the problem. We then extend this idea for general case and propose a sequential mathematical programming method for finding the least solution to the problem. Since the least solution to the tensor complementarity problem is the sparsest solution to the problem, the method can be regarded as an extension of a recent result by Luo et al. (Optim Lett 11:471–482, 2017). Our limited numerical results show that the method can be used to solve the tensor complementarity problem efficiently.

Suggested Citation

  • Shui-Lian Xie & Dong-Hui Li & Hong-Ru Xu, 2017. "An Iterative Method for Finding the Least Solution to the Tensor Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 175(1), pages 119-136, October.
    • Handle: RePEc:spr:joptap:v:175:y:2017:i:1:d:10.1007_s10957-017-1157-5
    DOI: 10.1007/s10957-017-1157-5
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    References listed on IDEAS

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    1. Yisheng Song & Gaohang Yu, 2016. "Properties of Solution Set of Tensor Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 170(1), pages 85-96, July.
    2. Fabien Lauer & Henrik Ohlsson, 2015. "Finding sparse solutions of systems of polynomial equations via group-sparsity optimization," Journal of Global Optimization, Springer, vol. 62(2), pages 319-349, June.
    3. 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.
    4. 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.
    5. 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.
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    Citations

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    Cited by:

    1. Yan, Weijie & Ling, Chen & Ling, Liyun & He, Hongjin, 2019. "Generalized tensor equations with leading structured tensors," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 311-324.
    2. Vu Trung Hieu, 2019. "On the R0-Tensors and the Solution Map of Tensor Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 163-183, April.
    3. Qilong Liu & Qingshui Liao, 2023. "Computing Nash Equilibria for Multiplayer Symmetric Games Based on Tensor Form," Mathematics, MDPI, vol. 11(10), pages 1-17, May.
    4. Ping-Fan Dai & Shi-Liang Wu, 2022. "The GUS-Property and Modulus-Based Methods for Tensor Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 976-1006, December.
    5. Hong-Bo Guan & Dong-Hui Li, 2020. "Linearized Methods for Tensor Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 972-987, March.
    6. 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.
    7. 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.
    8. 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.
    9. Shouqiang Du & Maolin Che & Yimin Wei, 2020. "Stochastic structured tensors to stochastic complementarity problems," Computational Optimization and Applications, Springer, vol. 75(3), pages 649-668, April.
    10. 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.
    11. Shouqiang Du & Liyuan Cui & Yuanyuan Chen & Yimin Wei, 2022. "Stochastic Tensor Complementarity Problem with Discrete Distribution," Journal of Optimization Theory and Applications, Springer, vol. 192(3), pages 912-929, March.
    12. Shouqiang Du & Liping Zhang, 2019. "A mixed integer programming approach to the tensor complementarity problem," Journal of Global Optimization, Springer, vol. 73(4), pages 789-800, April.

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