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A mixed integer programming approach to the tensor complementarity problem

Author

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  • Shouqiang Du

    (Qingdao University)

  • Liping Zhang

    (Tsinghua University)

Abstract

The tensor complementarity problem is a special instance of nonlinear complementarity problems, which has many applications. How to solve the tensor complementarity problem, via analyzing the structure of the related tensor, is one of very important research issues. In this paper, we propose a mixed integer programming approach for solving the tensor complementarity problem. We reformulate the tensor complementarity problem as an equivalent mixed integer feasibility problem. Based on the reformulation, some conditions for the solution existence and some solution properties of the tensor complementarity problem are given. We also prove that the tensor complementarity problem, corresponding to a positive definite diagonal tensor, has a unique solution. Finally, numerical results are reported to indicate the efficiency of the proposed algorithm.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:73:y:2019:i:4:d:10.1007_s10898-018-00731-4
    DOI: 10.1007/s10898-018-00731-4
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    References listed on IDEAS

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    1. Chen Ling & Hongjin He & Liqun Qi, 2016. "On the cone eigenvalue complementarity problem for higher-order tensors," Computational Optimization and Applications, Springer, vol. 63(1), pages 143-168, January.
    2. 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.
    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. Zheng-Hai Huang & Liqun Qi, 2017. "Formulating an n-person noncooperative game as a tensor complementarity problem," Computational Optimization and Applications, Springer, vol. 66(3), pages 557-576, April.
    5. 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.
    6. Yisheng Song & Liqun Qi, 2016. "Tensor Complementarity Problem and Semi-positive Tensors," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 1069-1078, June.
    7. 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.
    8. 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.
    9. 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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    Cited by:

    1. 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.
    2. 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.
    3. Zhenyu Ming & Liping Zhang & Liqun Qi, 2020. "Expected residual minimization method for monotone stochastic tensor complementarity problem," Computational Optimization and Applications, Springer, vol. 77(3), pages 871-896, December.
    4. 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.
    5. Shouqiang Du & Weiyang Ding & Yimin Wei, 2021. "Acceptable Solutions and Backward Errors for Tensor Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 188(1), pages 260-276, January.
    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.

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