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A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems

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  • Jean Strodiot
  • Thi Nguyen
  • Van Nguyen

Abstract

Generalized Nash equilibrium problems are important examples of quasi-equilibrium problems. The aim of this paper is to study a general class of algorithms for solving such problems. The method is a hybrid extragradient method whose second step consists in finding a descent direction for the distance function to the solution set. This is done thanks to a linesearch. Two descent directions are studied and for each one several steplengths are proposed to obtain the next iterate. A general convergence theorem applicable to each algorithm of the class is presented. It is obtained under weak assumptions: the pseudomonotonicity of the equilibrium function and the continuity of the multivalued mapping defining the constraint set of the quasi-equilibrium problem. Finally some preliminary numerical results are displayed to show the behavior of each algorithm of the class on generalized Nash equilibrium problems. Copyright Springer Science+Business Media, LLC. 2013

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  • Jean Strodiot & Thi Nguyen & Van Nguyen, 2013. "A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems," Journal of Global Optimization, Springer, vol. 56(2), pages 373-397, June.
    • Handle: RePEc:spr:jglopt:v:56:y:2013:i:2:p:373-397
    DOI: 10.1007/s10898-011-9814-y
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    References listed on IDEAS

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    1. SMEERS, Yves & OGGIONI, Giorgia & ALLEVI, Elisabetta & SCHAIBLE, Siegfried, 2010. "Generalized Nash Equilibrium and market coupling in the European power system," LIDAM Discussion Papers CORE 2010052, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Smeers, Y. & Oggioni, G. & Allevi, E. & Schaible, S., 2010. "Generalized Nash Equilibrium and Market Coupling in the European Power System," Cambridge Working Papers in Economics 1034, Faculty of Economics, University of Cambridge.
    3. K. Kubota & M. Fukushima, 2010. "Gap Function Approach to the Generalized Nash Equilibrium Problem," Journal of Optimization Theory and Applications, Springer, vol. 144(3), pages 511-531, March.
    4. Harker, Patrick T., 1991. "Generalized Nash games and quasi-variational inequalities," European Journal of Operational Research, Elsevier, vol. 54(1), pages 81-94, September.
    5. 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.
    6. Jong-Shi Pang & Masao Fukushima, 2009. "Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games," Computational Management Science, Springer, vol. 6(3), pages 373-375, August.
    7. Y. J. Wang & N. H. Xiu & C. Y. Wang, 2001. "Unified Framework of Extragradient-Type Methods for Pseudomonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 111(3), pages 641-656, December.
    8. Yves Smeers & Giorgia Oggioni & Elisabetta Allevi & Siegfried Schaible, 2010. "Generalized Nash Equilibrium and Market Coupling in the European Power System," Working Papers EPRG 1016, Energy Policy Research Group, Cambridge Judge Business School, University of Cambridge.
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    Cited by:

    1. L. F. Bueno & G. Haeser & F. Lara & F. N. Rojas, 2020. "An Augmented Lagrangian method for quasi-equilibrium problems," Computational Optimization and Applications, Springer, vol. 76(3), pages 737-766, July.
    2. Giancarlo Bigi & Mauro Passacantando, 2016. "Gap functions for quasi-equilibria," Journal of Global Optimization, Springer, vol. 66(4), pages 791-810, December.
    3. Yekini Shehu & Lulu Liu & Xiaolong Qin & Qiao-Li Dong, 2022. "Reflected Iterative Method for Non-Monotone Equilibrium Problems with Applications to Nash-Cournot Equilibrium Models," Networks and Spatial Economics, Springer, vol. 22(1), pages 153-180, March.
    4. Anulekha Dhara & Dinh Luc, 2014. "A solution method for linear variational relation problems," Journal of Global Optimization, Springer, vol. 59(4), pages 729-756, August.
    5. Ming Lei & Yiran He, 2021. "An Extragradient Method for Solving Variational Inequalities without Monotonicity," Journal of Optimization Theory and Applications, Springer, vol. 188(2), pages 432-446, February.
    6. Giancarlo Bigi & Massimo Pappalardo & Mauro Passacantando, 2016. "Optimization Tools for Solving Equilibrium Problems with Nonsmooth Data," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 887-905, December.
    7. Jean Strodiot & Phan Vuong & Thi Nguyen, 2016. "A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces," Journal of Global Optimization, Springer, vol. 64(1), pages 159-178, January.
    8. Phan Vuong & Jean Strodiot & Van Nguyen, 2014. "Projected viscosity subgradient methods for variational inequalities with equilibrium problem constraints in Hilbert spaces," Journal of Global Optimization, Springer, vol. 59(1), pages 173-190, May.

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