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Nonemptiness and Compactness of Solution Sets to Generalized Polynomial Complementarity Problems

Author

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  • Meng-Meng Zheng

    (Tianjin University)

  • Zheng-Hai Huang

    (Tianjin University)

  • Xiao-Xiao Ma

    (Tianjin University)

Abstract

In this paper, we investigate the generalized polynomial complementarity problem, which is a subclass of generalized complementarity problems with the involved map pairs being two polynomials. Based on the analysis on two structured tensor pairs located in the heading items of polynomials involved, and by using the degree theory, we achieve several results on the nonemptiness and compactness of solution sets. When generalized polynomial complementarity problems reduce to polynomial complementarity problems (or tensor complementarity problems), our results reduce to the existing ones. In particular, one of our results broadens the one proposed in a very recent paper to guarantee the nonemptiness and compactness of solution sets to generalized polynomial complementarity problems. Furthermore, we establish several existence and uniqueness results, which enrich the theory of generalized complementarity problems with the observation that some known conditions to guarantee the existence and uniqueness of solutions may not hold for a lot of generalized polynomial complementarity problems.

Suggested Citation

  • Meng-Meng Zheng & Zheng-Hai Huang & Xiao-Xiao Ma, 2020. "Nonemptiness and Compactness of Solution Sets to Generalized Polynomial Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 185(1), pages 80-98, April.
    • Handle: RePEc:spr:joptap:v:185:y:2020:i:1:d:10.1007_s10957-020-01645-6
    DOI: 10.1007/s10957-020-01645-6
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    References listed on IDEAS

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    1. Vyacheslav Kalashnikov & George Isac, 2002. "Solvability of Implicit Complementarity Problems," Annals of Operations Research, Springer, vol. 116(1), pages 199-221, October.
    2. Yong Wang & Zheng-Hai Huang & Liqun Qi, 2018. "Global Uniqueness and Solvability of Tensor Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 177(1), pages 137-152, April.
    3. Jie Wang & Shenglong Hu & Zheng-Hai Huang, 2018. "Solution Sets of Quadratic Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 176(1), pages 120-136, January.
    4. 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.
    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. 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.
    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. 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.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. Shenglong Hu & Jie Wang & Zheng-Hai Huang, 2018. "Error Bounds for the Solution Sets of Quadratic Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 983-1000, December.
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    Cited by:

    1. Meng-Meng Zheng & Zheng-Hai Huang & Xue-Li Bai, 2021. "Nonemptiness and Compactness of Solution Sets to Weakly Homogeneous Generalized Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 189(3), pages 919-937, June.
    2. Tong-tong Shang & Jing Yang & Guo-ji Tang, 2022. "Generalized Polynomial Complementarity Problems over a Polyhedral Cone," Journal of Optimization Theory and Applications, Springer, vol. 192(2), pages 443-483, February.
    3. Xueli Bai & Mengmeng Zheng & Zheng-Hai Huang, 2021. "Unique solvability of weakly homogeneous generalized variational inequalities," Journal of Global Optimization, Springer, vol. 80(4), pages 921-943, August.
    4. Tong-tong Shang & Guo-ji Tang, 2023. "Mixed polynomial variational inequalities," Journal of Global Optimization, Springer, vol. 86(4), pages 953-988, August.

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