IDEAS home Printed from https://ideas.repec.org/a/eee/ejores/v283y2020i3p1136-1147.html
   My bibliography  Save this article

A parametrized variational inequality approach to track the solution set of a generalized nash equilibrium problem

Author

Listed:
  • Migot, Tangi
  • Cojocaru, Monica-G.

Abstract

In this paper, we present a numerical method to describe the solution set of a generalized Nash equilibrium problem (GNEP). Previous approaches show how to reformulate the GNEP as a family of parametric variational inequalities in the special case where the game has shared constraints. We extend this result to generalized Nash problems by means of an umbrella shared constraint approximation of the game. We show the validity of our approach on numerical examples from the literature, and we provide new results that pinpoint the handling of the algorithm’s parameters for its implementation. Last but not least, we extend, solve, and discuss an applied example of a generalized Nash equilibrium problem of environmental accords between countries.

Suggested Citation

  • Migot, Tangi & Cojocaru, Monica-G., 2020. "A parametrized variational inequality approach to track the solution set of a generalized nash equilibrium problem," European Journal of Operational Research, Elsevier, vol. 283(3), pages 1136-1147.
    • Handle: RePEc:eee:ejores:v:283:y:2020:i:3:p:1136-1147
    DOI: 10.1016/j.ejor.2019.11.054
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S037722171930966X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.ejor.2019.11.054?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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. WEI, Jing-Yuan & SMEERS, Yves, 1999. "Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices," LIDAM Reprints CORE 1454, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Koichi Nabetani & Paul Tseng & Masao Fukushima, 2011. "Parametrized variational inequality approaches to generalized Nash equilibrium problems with shared constraints," Computational Optimization and Applications, Springer, vol. 48(3), pages 423-452, April.
    4. 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.
    5. Breton, Michele & Zaccour, Georges & Zahaf, Mehdi, 2006. "A game-theoretic formulation of joint implementation of environmental projects," European Journal of Operational Research, Elsevier, vol. 168(1), pages 221-239, January.
    6. Francisco Facchinei & Christian Kanzow, 2010. "Generalized Nash Equilibrium Problems," Annals of Operations Research, Springer, vol. 175(1), pages 177-211, March.
    7. Stein, Oliver & Sudermann-Merx, Nathan, 2018. "The noncooperative transportation problem and linear generalized Nash games," European Journal of Operational Research, Elsevier, vol. 266(2), pages 543-553.
    8. Axel Dreves, 2014. "Finding all solutions of affine generalized Nash equilibrium problems with one-dimensional strategy sets," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 80(2), pages 139-159, October.
    9. Axel Dreves, 2017. "Computing all solutions of linear generalized Nash equilibrium problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 85(2), pages 207-221, April.
    10. Anthony Downs, 1957. "An Economic Theory of Political Action in a Democracy," Journal of Political Economy, University of Chicago Press, vol. 65(2), pages 135-135.
    11. Anselmi, Jonatha & Ardagna, Danilo & Passacantando, Mauro, 2014. "Generalized Nash equilibria for SaaS/PaaS Clouds," European Journal of Operational Research, Elsevier, vol. 236(1), pages 326-339.
    12. Harker, Patrick T., 1991. "Generalized Nash games and quasi-variational inequalities," European Journal of Operational Research, Elsevier, vol. 54(1), pages 81-94, September.
    13. 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.
    14. Benjamin F. Hobbs & J. S. Pang, 2007. "Nash-Cournot Equilibria in Electric Power Markets with Piecewise Linear Demand Functions and Joint Constraints," Operations Research, INFORMS, vol. 55(1), pages 113-127, February.
    15. Wei Jing-Yuan & Yves Smeers, 1999. "Spatial Oligopolistic Electricity Models with Cournot Generators and Regulated Transmission Prices," Operations Research, INFORMS, vol. 47(1), pages 102-112, February.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Cao, Yiyin & Dang, Chuangyin & Xiao, Zhongdong, 2022. "A differentiable path-following method to compute subgame perfect equilibria in stationary strategies in robust stochastic games and its applications," European Journal of Operational Research, Elsevier, vol. 298(3), pages 1032-1050.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. 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.
    3. Axel Dreves, 2018. "How to Select a Solution in Generalized Nash Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 178(3), pages 973-997, September.
    4. Axel Dreves & Simone Sagratella, 2020. "Nonsingularity and Stationarity Results for Quasi-Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 711-743, June.
    5. Christian Kanzow & Daniel Steck, 2018. "Augmented Lagrangian and exact penalty methods for quasi-variational inequalities," Computational Optimization and Applications, Springer, vol. 69(3), pages 801-824, April.
    6. Simone Sagratella, 2017. "Algorithms for generalized potential games with mixed-integer variables," Computational Optimization and Applications, Springer, vol. 68(3), pages 689-717, December.
    7. Francisco Facchinei & Christian Kanzow, 2010. "Generalized Nash Equilibrium Problems," Annals of Operations Research, Springer, vol. 175(1), pages 177-211, March.
    8. Giancarlo Bigi & Mauro Passacantando, 2016. "Gap functions for quasi-equilibria," Journal of Global Optimization, Springer, vol. 66(4), pages 791-810, December.
    9. 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.
    10. Francisco Facchinei & Jong-Shi Pang & Gesualdo Scutari, 2014. "Non-cooperative games with minmax objectives," Computational Optimization and Applications, Springer, vol. 59(1), pages 85-112, October.
    11. Derek W. Bunn & Fernando S. Oliveira, 2008. "Modeling the Impact of Market Interventions on the Strategic Evolution of Electricity Markets," Operations Research, INFORMS, vol. 56(5), pages 1116-1130, October.
    12. Sreekumaran, Harikrishnan & Hota, Ashish R. & Liu, Andrew L. & Uhan, Nelson A. & Sundaram, Shreyas, 2021. "Equilibrium strategies for multiple interdictors on a common network," European Journal of Operational Research, Elsevier, vol. 288(2), pages 523-538.
    13. Rodica Ioana Lung & Noémi Gaskó & Mihai Alexandru Suciu, 2020. "Pareto-based evolutionary multiobjective approaches and the generalized Nash equilibrium problem," Journal of Heuristics, Springer, vol. 26(4), pages 561-584, August.
    14. Benjamin F. Hobbs & J. S. Pang, 2007. "Nash-Cournot Equilibria in Electric Power Markets with Piecewise Linear Demand Functions and Joint Constraints," Operations Research, INFORMS, vol. 55(1), pages 113-127, February.
    15. Stein, Oliver & Sudermann-Merx, Nathan, 2018. "The noncooperative transportation problem and linear generalized Nash games," European Journal of Operational Research, Elsevier, vol. 266(2), pages 543-553.
    16. Masao Fukushima, 2011. "Restricted generalized Nash equilibria and controlled penalty algorithm," Computational Management Science, Springer, vol. 8(3), pages 201-218, August.
    17. David Pozo & Enzo Sauma & Javier Contreras, 2017. "Basic theoretical foundations and insights on bilevel models and their applications to power systems," Annals of Operations Research, Springer, vol. 254(1), pages 303-334, July.
    18. Jian Yao & Ilan Adler & Shmuel S. Oren, 2008. "Modeling and Computing Two-Settlement Oligopolistic Equilibrium in a Congested Electricity Network," Operations Research, INFORMS, vol. 56(1), pages 34-47, February.
    19. Sagratella, Simone & Schmidt, Marcel & Sudermann-Merx, Nathan, 2020. "The noncooperative fixed charge transportation problem," European Journal of Operational Research, Elsevier, vol. 284(1), pages 373-382.
    20. Giorgia Oggioni & Yves Smeers & Elisabetta Allevi & Siegfried Schaible, 2012. "A Generalized Nash Equilibrium Model of Market Coupling in the European Power System," Networks and Spatial Economics, Springer, vol. 12(4), pages 503-560, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:ejores:v:283:y:2020:i:3:p:1136-1147. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/eor .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.