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Proximal-Type Algorithms for Solving Nonconvex Mixed Multivalued Quasi-Variational Inequality Problems

Author

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  • S.-M. Grad

    (Unité de Mathématiques Appliquées)

  • L. D. Muu

    (TIMAS, Thang Long University and Institute of Mathematics, VAST)

  • T. V. Thang

    (Electric Power University)

Abstract

We propose iterative algorithms of proximal point type for solving two classes of mixed multivalued quasi-variational inequality problems in real Euclidean spaces involving nonconvex functions, for which no similar algorithms are currently known. Our proposed algorithms combine the proximal type algorithm for solving mixed variational inequalities in a nonconvex framework, the Mann iteration scheme for approximating a fixed point of certain generalized nonexpansive multivalued mappings with the infeasible projection and cutting plane techniques for variational inequalities to generate iterative sequences that converge to a solution of a mixed multivalued quasi-variational inequality problem under mild assumptions. An application to generalized Nash (quasi)equilibrium problems is discussed. Numerical experiments confirm the usability of the introduced algorithms.

Suggested Citation

  • S.-M. Grad & L. D. Muu & T. V. Thang, 2025. "Proximal-Type Algorithms for Solving Nonconvex Mixed Multivalued Quasi-Variational Inequality 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_s10957-025-02733-1
    DOI: 10.1007/s10957-025-02733-1
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    References listed on IDEAS

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    1. Sorin-Mihai Grad & Felipe Lara, 2022. "An extension of the proximal point algorithm beyond convexity," Journal of Global Optimization, Springer, vol. 82(2), pages 313-329, February.
    2. Jianzhong Zhang & Biao Qu & Naihua Xiu, 2010. "Some projection-like methods for the generalized Nash equilibria," Computational Optimization and Applications, Springer, vol. 45(1), pages 89-109, January.
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    4. E. L. Dias Júnior & P. J. S. Santos & A. Soubeyran & J. C. O. Souza, 2024. "On inexact versions of a quasi-equilibrium problem: a Cournot duopoly perspective," Journal of Global Optimization, Springer, vol. 89(1), pages 171-196, May.
    5. Giancarlo Bigi & Mauro Passacantando, 2016. "Gap functions for quasi-equilibria," Journal of Global Optimization, Springer, vol. 66(4), pages 791-810, December.
    6. F. Lara, 2022. "On Strongly Quasiconvex Functions: Existence Results and Proximal Point Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 192(3), pages 891-911, March.
    7. 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.
    8. Marco Castellani & Massimiliano Giuli & Massimo Pappalardo, 2018. "A Ky Fan Minimax Inequality for Quasiequilibria on Finite-Dimensional Spaces," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 53-64, October.
    9. Jong-Shi Pang & Masao Fukushima, 2005. "Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games," Computational Management Science, Springer, vol. 2(1), pages 21-56, January.
    10. Sorin-Mihai Grad & Felipe Lara, 2021. "Solving Mixed Variational Inequalities Beyond Convexity," Journal of Optimization Theory and Applications, Springer, vol. 190(2), pages 565-580, August.
    11. Han, Deren & Zhang, Hongchao & Qian, Gang & Xu, Lingling, 2012. "An improved two-step method for solving generalized Nash equilibrium problems," European Journal of Operational Research, Elsevier, vol. 216(3), pages 613-623.
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