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Gap functions for quasi-equilibria

Author

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  • Giancarlo Bigi

    (Università di Pisa)

  • Mauro Passacantando

    (Università di Pisa)

Abstract

An approach for solving quasi-equilibrium problems (QEPs) is proposed relying on gap functions, which allow reformulating QEPs as global optimization problems. The (generalized) smoothness properties of a gap function are analysed and an upper estimate of its Clarke directional derivative is given. Monotonicity assumptions on both the equilibrium and constraining bifunctions are a key tool to guarantee that all the stationary points of a gap function actually solve QEP. A few classes of constraints satisfying such assumptions are identified covering a wide range of situations. Relying on these results, a descent method for solving QEP is devised and its convergence proved. Finally, error bounds are given in order to guarantee the boundedness of the sequence generated by the algorithm.

Suggested Citation

  • Giancarlo Bigi & Mauro Passacantando, 2016. "Gap functions for quasi-equilibria," Journal of Global Optimization, Springer, vol. 66(4), pages 791-810, December.
    • Handle: RePEc:spr:jglopt:v:66:y:2016:i:4:d:10.1007_s10898-016-0458-9
    DOI: 10.1007/s10898-016-0458-9
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    1. Francisco Facchinei & Veronica Piccialli & Marco Sciandrone, 2011. "Decomposition algorithms for generalized potential games," Computational Optimization and Applications, Springer, vol. 50(2), pages 237-262, October.
    2. 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.
    3. Anna Heusinger & Christian Kanzow, 2009. "Optimization reformulations of the generalized Nash equilibrium problem using Nikaido-Isoda-type functions," Computational Optimization and Applications, Springer, vol. 43(3), pages 353-377, July.
    4. D. Aussel & R. Correa & M. Marechal, 2011. "Gap Functions for Quasivariational Inequalities and Generalized Nash Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 151(3), pages 474-488, December.
    5. Rachana Gupta & Aparna Mehra, 2012. "Gap functions and error bounds for quasi variational inequalities," Journal of Global Optimization, Springer, vol. 53(4), pages 737-748, August.
    6. Masao Fukushima, 2011. "Restricted generalized Nash equilibria and controlled penalty algorithm," Computational Management Science, Springer, vol. 8(3), pages 201-218, August.
    7. 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.
    8. Giancarlo Bigi & Mauro Passacantando, 2012. "Gap functions and penalization for solving equilibrium problems with nonlinear constraints," Computational Optimization and Applications, Springer, vol. 53(2), pages 323-346, October.
    9. 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.
    10. Axel Dreves & Christian Kanzow, 2011. "Nonsmooth optimization reformulations characterizing all solutions of jointly convex generalized Nash equilibrium problems," Computational Optimization and Applications, Springer, vol. 50(1), pages 23-48, September.
    11. Giancarlo Bigi & Mauro Passacantando, 2015. "Descent and Penalization Techniques for Equilibrium Problems with Nonlinear Constraints," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 804-818, March.
    12. Francisco Facchinei & Christian Kanzow, 2010. "Generalized Nash Equilibrium Problems," Annals of Operations Research, Springer, vol. 175(1), pages 177-211, March.
    13. NESTEROV, Yurii & SCRIMALI, Laura, 2011. "Solving strongly monotone variational and quasi-variational inequalities," LIDAM Reprints CORE 2357, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    14. A. Heusinger & C. Kanzow, 2009. "Relaxation Methods for Generalized Nash Equilibrium Problems with Inexact Line Search," Journal of Optimization Theory and Applications, Springer, vol. 143(1), pages 159-183, October.
    15. Bigi, Giancarlo & Castellani, Marco & Pappalardo, Massimo & Passacantando, Mauro, 2013. "Existence and solution methods for equilibria," European Journal of Operational Research, Elsevier, vol. 227(1), pages 1-11.
    16. D. Chan & J. S. Pang, 1982. "The Generalized Quasi-Variational Inequality Problem," Mathematics of Operations Research, INFORMS, vol. 7(2), pages 211-222, May.
    17. Harker, Patrick T., 1991. "Generalized Nash games and quasi-variational inequalities," European Journal of Operational Research, Elsevier, vol. 54(1), pages 81-94, September.
    18. William Hogan, 1973. "Directional Derivatives for Extremal-Value Functions with Applications to the Completely Convex Case," Operations Research, INFORMS, vol. 21(1), pages 188-209, February.
    19. 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.
    20. Alexey Izmailov & Mikhail Solodov, 2014. "On error bounds and Newton-type methods for generalized Nash equilibrium problems," Computational Optimization and Applications, Springer, vol. 59(1), pages 201-218, October.
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    3. Nguyen Van Hung & Vicente Novo & Vo Minh Tam, 2022. "Error bound analysis for vector equilibrium problems with partial order provided by a polyhedral cone," Journal of Global Optimization, Springer, vol. 82(1), pages 139-159, January.
    4. Simone Sagratella, 2017. "Algorithms for generalized potential games with mixed-integer variables," Computational Optimization and Applications, Springer, vol. 68(3), pages 689-717, December.

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