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Mixed polynomial variational inequalities

Author

Listed:
  • Tong-tong Shang

    (Guizhou University
    Guangxi Minzu University)

  • Guo-ji Tang

    (Guangxi Minzu University)

Abstract

The goal of this paper is to introduce a class of mixed polynomial variational inequalities, which is a natural generalization of the affine variational inequality and the tensor variational inequality, and a special case of the mixed variational inequality. It is shown that a class of polynomial optimization problem and a class of m-person noncooperative game can be reformulated as a mixed polynomial variational inequality. Firstly, some classes of structured tensor tuples are introduced and the relationship between them is discussed. Then, a new asymptotic function (denoted by m-asymptotic function) is introduced and some basic properties are investigated. An equivalent characterization for the nonexistence of solutions is given by using the exceptional family of elements. Finally, the nonemptiness and compactness of the solution sets of the mixed polynomial variational inequalities with some special structured tensors and m-asymptotic function are proved and then the uniqueness of the solution is further investigated.

Suggested Citation

  • Tong-tong Shang & Guo-ji Tang, 2023. "Mixed polynomial variational inequalities," Journal of Global Optimization, Springer, vol. 86(4), pages 953-988, August.
    • Handle: RePEc:spr:jglopt:v:86:y:2023:i:4:d:10.1007_s10898-023-01298-5
    DOI: 10.1007/s10898-023-01298-5
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    References listed on IDEAS

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    1. 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.
    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. Nina Ovcharova & Joachim Gwinner, 2016. "Semicoercive Variational Inequalities: From Existence to Numerical Solution of Nonmonotone Contact Problems," Journal of Optimization Theory and Applications, Springer, vol. 171(2), pages 422-439, November.
    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, 2017. "Formulating an n-person noncooperative game as a tensor complementarity problem," Computational Optimization and Applications, Springer, vol. 66(3), pages 557-576, April.
    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.
    7. D. Goeleven, 2008. "Existence and Uniqueness for a Linear Mixed Variational Inequality Arising in Electrical Circuits with Transistors," Journal of Optimization Theory and Applications, Springer, vol. 138(3), pages 397-406, September.
    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. Alfredo Iusem & Felipe Lara, 2019. "Existence Results for Noncoercive Mixed Variational Inequalities in Finite Dimensional Spaces," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 122-138, October.
    11. 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.
    12. Vu Trung Hieu, 2020. "Solution maps of polynomial variational inequalities," Journal of Global Optimization, Springer, vol. 77(4), pages 807-824, August.
    13. Khalid Addi & Daniel Goeleven, 2017. "Complementarity and Variational Inequalities in Electronics," Springer Optimization and Its Applications, in: Nicholas J. Daras & Themistocles M. Rassias (ed.), Operations Research, Engineering, and Cyber Security, pages 1-43, Springer.
    14. Wenjie Mu & Jianghua Fan, 2022. "Existence results for solutions of mixed tensor variational inequalities," Journal of Global Optimization, Springer, vol. 82(2), pages 389-412, February.
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    1. Waqar Afzal & Mujahid Abbas & Omar Mutab Alsalami, 2024. "Bounds of Different Integral Operators in Tensorial Hilbert and Variable Exponent Function Spaces," Mathematics, MDPI, vol. 12(16), pages 1-33, August.

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