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.
Similar content being viewed by others
References
Bensoussan A.: Points de Nash dans le cas de fonctionnelles quadratiques et jeux différentiels linéaires à N personnes. SIAM J. Control 12, 460–499 (1974)
Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
Chinchuluun, A., Pardalos, P.M., Migdalas, A., Pitsoulis, L. (eds): Pareto Optimality, Game Theory and Equilibria. Springer, New York (2008)
Facchinei F., Kanzow C.: Generalized Nash equilibrium problems. Ann. Oper. Res. 175, 177–211 (2010)
Facchinei F., Pang J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)
Fukushima M.: A class of gap functions for quasi-variational inequality problems. J. Ind. Manag. Optim. 3, 165–171 (2007)
Gianessi, F., Maugeri, A., Pardalos, P.M. (eds): Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Kluwer, New York (2002)
Harker P.T.: Generalized Nash games and quasi-variational inequalities. Eur. J. Oper. Res. 54, 81–94 (1991)
Iusem A.N., Svaiter B.F.: A variant of Korpelevich’s method for variational inequalities with a new search strategy. Optimization 42, 309–321 (1997)
Khobotov E.N.: Modification of the extragradient method for solving variational inequalities and certain optimization problems. USSR Comput. Math. Phys. 27, 120–127 (1987)
Konnov I.V.: Equilibrium Models and Variational Inequalities. Mathematics in Science and Engineering. Elsevier, Amsterdam (2007)
Korpelevich G.M.: The extragradient method for finding saddle points and other problems. Matekon 12, 747–756 (1976)
Kubota K., Fukushima M.: Gap function approach to the generalized Nash equilibrium problem. J. Optim. Theory Appl. 144, 511–531 (2010)
Noor M.A.: A modified extragradient method for general monotone variational inequalities. Comput. Math. Appl. 38, 19–24 (1999)
Noor M.A.: On merit functions for quasivariational inequalities. J. Math. Inequal. 1, 259–268 (2007)
Palomar, D.P., Eldar, Y.C. (eds): Convex Optimization in Signal Processing and Communications. Cambridge University Press, Cambridge (2010)
Pang J.-S., Fukushima M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 2, 21–56 (2005)
Pang J.-S., Fukushima M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Erratum. Comput. Manag. Sci. 6, 373–375 (2009)
Pardalos, P.M., Rassias, T.M., Khan, A.A. (eds): Nonlinear Analysis and Variational Problems. Springer, New York (2010)
Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Solodov M.V., Svaiter B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999)
Smeers, Y., Oggioni, G., Allevi, E., Schaible, S.: Generalized Nash equilibrium and market coupling in the European power system. EPRG Working Paper 1016, Cambridge Working Paper in Economics 1034 (2010)
Taji, K.: On gap functions for quasi-variational inequalities. Abstract and Applied Analysis 2008, Article ID 531361, 7 pages (2008)
Tran D.Q., Le Dung M., Nguyen V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)
von Heusinger, A.: Numerical Methods for the Solution of the Generalized Nash Equilibrium Problem. PhD Thesis, University of Wuerzburg, Germany (2009)
Wang Y., Xiu N., Wang C.: Unified framework of extragradient-type methods for pseudomonotone variational inequalities. J. Optim. Theory Appl. 111, 641–656 (2001)
Wang Y., Xiu N., Wang C.: A new version of extragradient method for variational inequality problems. Comput. Math. Appl. 42, 969–979 (2001)
Wei J.Y., Smeers Y.: Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices. Oper. Res. 47, 102–112 (1999)
Xiu N., Wang Y., Zhang X.: Modified fixed-point equations and related iterative methods for variational inequalities. Comput. Math. Appl. 47, 913–920 (2004)
Xiu N., Zhang J.: Some recent advances in projection-type methods for variational inequalities. J. Comput. Appl. Math. 152, 559–585 (2003)
Zhang J., Qu B., Xiu N.: Some projection-like methods for the generalized Nash equilibria. Comput. Optim. Appl. 45, 89–109 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the Institute for Computational Science and Technology at Ho Chi Minh City, Vietnam (ICST HCMC).
Rights and permissions
About this article
Cite this article
Strodiot, J.J., Nguyen, T.T.V. & Nguyen, V.H. A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems. J Glob Optim 56, 373–397 (2013). https://doi.org/10.1007/s10898-011-9814-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-011-9814-y
Keywords
- Quasi-equilibrium problems
- Quasi-variational inequalities
- Hybrid extragradient methods
- Generalized Nash equilibrium problems