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Relaxation Methods for Generalized Nash Equilibrium Problems with Inexact Line Search

Author

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  • A. Heusinger

    (University of Würzburg)

  • C. Kanzow

    (University of Würzburg)

Abstract

The generalized Nash equilibrium problem (GNEP) is an extension of the standard Nash game where, in addition to the cost functions, also the strategy spaces of each player depend on the strategies chosen by all other players. This problem is rather difficult to solve and there are only a few methods available in the literature. One of the most popular ones is the so-called relaxation method, which is known to be globally convergent under a set of assumptions. Some of these assumptions, however, are rather strong or somewhat difficult to understand. Here, we present a modified relaxation method for the solution of a certain class of GNEPs. The convergence analysis uses completely different arguments based on a certain descent property and avoids some of the technical conditions for the original relaxation method. Moreover, numerical experiments indicate that the modified relaxation method performs quite well on a number of different examples taken from the literature.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:143:y:2009:i:1:d:10.1007_s10957-009-9553-0
    DOI: 10.1007/s10957-009-9553-0
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    References listed on IDEAS

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    1. Steffan Berridge & Jacek Krawczyk, "undated". "Relaxation Algorithms in Finding Nash Equilibrium," Computing in Economics and Finance 1997 159, Society for Computational Economics.
    2. 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.
    3. Krawczyk, Jacek B., 2005. "Coupled constraint Nash equilibria in environmental games," Resource and Energy Economics, Elsevier, vol. 27(2), pages 157-181, June.
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    Cited by:

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    2. J. Y. Bello Cruz & P. S. M. Santos & S. Scheimberg, 2013. "A Two-Phase Algorithm for a Variational Inequality Formulation of Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 159(3), pages 562-575, December.
    3. Giancarlo Bigi & Mauro Passacantando, 2016. "Gap functions for quasi-equilibria," Journal of Global Optimization, Springer, vol. 66(4), pages 791-810, December.
    4. Xiao, Jiang-Wen & Yang, Yan-Bing & Cui, Shichang & Liu, Xiao-Kang, 2022. "A new energy storage sharing framework with regard to both storage capacity and power capacity," Applied Energy, Elsevier, vol. 307(C).
    5. Francisco Facchinei & Christian Kanzow, 2010. "Generalized Nash Equilibrium Problems," Annals of Operations Research, Springer, vol. 175(1), pages 177-211, March.
    6. Axel Dreves & Anna Heusinger & Christian Kanzow & Masao Fukushima, 2013. "A globalized Newton method for the computation of normalized Nash equilibria," Journal of Global Optimization, Springer, vol. 56(2), pages 327-340, June.
    7. Napat Harnpornchai & Wiriyaporn Wonggattaleekam, 2021. "A Nikaido Isoda-Based Hybrid Genetic Algorithm and Relaxation Method for Finding Nash Equilibrium," Mathematics, MDPI, vol. 10(1), pages 1-17, December.
    8. Jiawang Nie & Xindong Tang & Lingling Xu, 2021. "The Gauss–Seidel method for generalized Nash equilibrium problems of polynomials," Computational Optimization and Applications, Springer, vol. 78(2), pages 529-557, March.
    9. Axel Dreves & Joachim Gwinner, 2016. "Jointly Convex Generalized Nash Equilibria and Elliptic Multiobjective Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 1065-1086, March.
    10. Riccardi, R. & Bonenti, F. & Allevi, E. & Avanzi, C. & Gnudi, A., 2015. "The steel industry: A mathematical model under environmental regulations," European Journal of Operational Research, Elsevier, vol. 242(3), pages 1017-1027.
    11. Mauro Passacantando & Danilo Ardagna & Anna Savi, 2016. "Service Provisioning Problem in Cloud and Multi-Cloud Systems," INFORMS Journal on Computing, INFORMS, vol. 28(2), pages 265-277, May.

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