IDEAS home Printed from https://ideas.repec.org/a/spr/mathme/v80y2014i2p139-159.html
   My bibliography  Save this article

Finding all solutions of affine generalized Nash equilibrium problems with one-dimensional strategy sets

Author

Listed:
  • Axel Dreves

Abstract

We consider a class of generalized Nash equilibrium problems with quadratic cost functions and common linear constraints for all players. Further we focus on the case where every player has a single strategy variable within a bounded set. For this problem class we present an algorithm that is able to compute all solutions and that terminates finitely. Our method is based on a representation of the solution set as a finite union of polyhedral sets using sign conditions for the derivatives of the cost and constraint functions. The effectiveness of the algorithm is shown in various examples from literature. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • 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.
    • Handle: RePEc:spr:mathme:v:80:y:2014:i:2:p:139-159
    DOI: 10.1007/s00186-014-0473-x
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s00186-014-0473-x
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s00186-014-0473-x?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. C. E. Lemke, 1965. "Bimatrix Equilibrium Points and Mathematical Programming," Management Science, INFORMS, vol. 11(7), pages 681-689, May.
    2. Axel Dreves & Christian Kanzow & Oliver Stein, 2012. "Nonsmooth optimization reformulations of player convex generalized Nash equilibrium problems," Journal of Global Optimization, Springer, vol. 53(4), pages 587-614, August.
    3. Masao Fukushima, 2011. "Restricted generalized Nash equilibria and controlled penalty algorithm," Computational Management Science, Springer, vol. 8(3), pages 201-218, August.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. Jacek Krawczyk, 2007. "Numerical solutions to coupled-constraint (or generalised Nash) equilibrium problems," Computational Management Science, Springer, vol. 4(2), pages 183-204, April.
    10. 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.
    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. 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.
    2. 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.
    3. Oliver Stein & Nathan Sudermann-Merx, 2016. "The Cone Condition and Nonsmoothness in Linear Generalized Nash Games," Journal of Optimization Theory and Applications, Springer, vol. 170(2), pages 687-709, August.
    4. 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.
    5. Simone Sagratella, 2017. "Algorithms for generalized potential games with mixed-integer variables," Computational Optimization and Applications, Springer, vol. 68(3), pages 689-717, December.

    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. Axel Dreves, 2016. "Improved error bound and a hybrid method for generalized Nash equilibrium problems," Computational Optimization and Applications, Springer, vol. 65(2), pages 431-448, November.
    2. Axel Dreves & Francisco Facchinei & Andreas Fischer & Markus Herrich, 2014. "A new error bound result for Generalized Nash Equilibrium Problems and its algorithmic application," Computational Optimization and Applications, Springer, vol. 59(1), pages 63-84, October.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. Simone Sagratella, 2017. "Algorithms for generalized potential games with mixed-integer variables," Computational Optimization and Applications, Springer, vol. 68(3), pages 689-717, December.
    8. 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.
    9. Dane A. Schiro & Benjamin F. Hobbs & Jong-Shi Pang, 2016. "Perfectly competitive capacity expansion games with risk-averse participants," Computational Optimization and Applications, Springer, vol. 65(2), pages 511-539, November.
    10. Simone Sagratella, 2017. "Computing equilibria of Cournot oligopoly models with mixed-integer quantities," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(3), pages 549-565, December.
    11. Giancarlo Bigi & Mauro Passacantando, 2016. "Gap functions for quasi-equilibria," Journal of Global Optimization, Springer, vol. 66(4), pages 791-810, December.
    12. Nadja Harms & Tim Hoheisel & Christian Kanzow, 2015. "On a Smooth Dual Gap Function for a Class of Player Convex Generalized Nash Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 659-685, August.
    13. J. Contreras & J. B. Krawczyk & J. Zuccollo, 2016. "Economics of collective monitoring: a study of environmentally constrained electricity generators," Computational Management Science, Springer, vol. 13(3), pages 349-369, July.
    14. 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.
    15. Anna Schwele & Christos Ordoudis & Pierre Pinson & Jalal Kazempour, 2021. "Coordination of power and natural gas markets via financial instruments," Computational Management Science, Springer, vol. 18(4), pages 505-538, October.
    16. 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.
    17. 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.
    18. 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.
    19. 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.
    20. Giorgia Oggioni and Yves Smeers, 2012. "Degrees of Coordination in Market Coupling and Counter-Trading," The Energy Journal, International Association for Energy Economics, vol. 0(Number 3).

    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:spr:mathme:v:80:y:2014:i:2:p:139-159. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.