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Methods for solving generalized Nash equilibrium. (English) Zbl 1266.91047

Summary: The generalized Nash equilibrium problem (GNEP) is an extension of the standard Nash equilibrium problem (NEP), in which each player’s strategy set may depend on the rival player’s strategies. In this paper, we present two descent type methods. The algorithms are based on a reformulation of the generalized Nash equilibrium using Nikaido-Isoda function as unconstrained optimization. We prove that our algorithms are globally convergent and the convergence analysis is not based on conditions guaranteeing that every stationary point of the optimization problem is a solution of the GNEP.

MSC:

91A11 Equilibrium refinements
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[1] A. Dreves and C. Kanzow, “Nonsmooth optimization reformulations characterizing all solutions of jointly convex generalized Nash equilibrium problems,” Computational Optimization and Applications, vol. 50, no. 1, pp. 23-48, 2011. · Zbl 1227.90040
[2] A. Dreves, C. Kanzow, and O. Stein, “Nonsmooth optimization reformulations of player convex generalized Nash equilibrium problems,” Journal of Global Optimization, vol. 53, no. 4, pp. 587-614, 2012. · Zbl 1281.90055
[3] F. Facchinei and C. Kanzow, “Generalized Nash equilibrium problems,” Annals of Operations Research, vol. 175, pp. 177-211, 2010. · Zbl 1211.90228
[4] F. Facchinei, A. Fischer, and V. Piccialli, “On generalized Nash games and variational inequalities,” Operations Research Letters, vol. 35, no. 2, pp. 159-164, 2007. · Zbl 1303.91020
[5] F. Facchinei and C. Kanzow, “Generalized Nash equilibrium problems,” A Quarterly Journal of Operations Research, vol. 5, no. 3, pp. 173-210, 2007. · Zbl 1211.91162
[6] D. Han, H. Zhang, G. Qian, and L. Xu, “An improved two-step method for solving generalized Nash equilibrium problems,” European Journal of Operational Research, vol. 216, no. 3, pp. 613-623, 2012. · Zbl 1252.90081
[7] P. T. Harker, “Generalized Nash games and quasi-variational inequalities,” European Journal of Operational Research, vol. 54, no. 1, pp. 81-94, 1991. · Zbl 0754.90070
[8] A. von Heusinger and C. Kanzow, “Optimization reformulations of the generalized Nash equilibrium problem using Nikaido-Isoda-type functions,” Computational Optimization and Applications, vol. 43, no. 3, pp. 353-377, 2009. · Zbl 1170.90495
[9] B. Panicucci, M. Pappalardo, and M. Passacantando, “On solving generalized Nash equilibrium problems via optimization,” Optimization Letters, vol. 3, no. 3, pp. 419-435, 2009. · Zbl 1187.91011
[10] B. Qu and J. G. Jiang, “On the computation of normalized nash equilibrium for generalized nash equilibrium problem,” Journal of Convergence Information Technology, vol. 7, no. 22, pp. 16-21, 2012.
[11] J. Zhang, B. Qu, and N. Xiu, “Some projection-like methods for the generalized Nash equilibria,” Computational Optimization and Applications, vol. 45, no. 1, pp. 89-109, 2010. · Zbl 1198.91026
[12] S. D. Flåm and A. S. Antipin, “Equilibrium programming using proximal-like algorithms,” Mathematical Programming, vol. 78, no. 1, pp. 29-41, 1997. · Zbl 0890.90150
[13] S. D. Flåm and A. Ruszczyński, “Noncooperative convex games: computing equilibrium by partial regularization,” Working Paper 94-42, Laxenburg, Austria, 1994.
[14] M. V. Solodov and P. Tseng, “Some methods based on the D-gap function for solving monotone variational inequalities,” Computational Optimization and Applications, vol. 17, no. 2-3, pp. 255-277, 2000. · Zbl 1168.49303
[15] B. Qu, C. Y. Wang, and J. Z. Zhang, “Convergence and error bound of a method for solving variational inequality problems via the generalized D-gap function,” Journal of Optimization Theory and Applications, vol. 119, no. 3, pp. 535-552, 2003. · Zbl 1045.49015
[16] B. T. Polyak, Introduction to Optimization, Translations Series in Mathematics and Engineering, Optimization Software, New York, NY, USA, 1987.
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