# American Institute of Mathematical Sciences

January  2012, 8(1): 51-65. doi: 10.3934/jimo.2012.8.51

## A penalty method for generalized Nash equilibrium problems

 1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China, China, China

Received  December 2010 Revised  July 2011 Published  November 2011

This paper considers a penalty algorithm for solving the generalized Nash equilibrium problem (GNEP). Under the GNEP-MFCQ at a limiting point of the sequence generated by the algorithm, we demonstrate that the limit point is a solution to the GNEP and the parameter becomes a constant after a finite iterations. We formulate the Karush-Kuhn-Tucker conditions for a penalized problem into a system of nonsmooth equations and demonstrate the nonsingularity of its Clarke’s generalized Jacobian at the KKT point under the so-called GNEP-Second Order Sufficient Condition. The nonsingularity facilitates the application of the smoothing Newton method for the global convergence and local quadratic rate. Finally, the numerical results are reported to show the effectiveness of the penalty method in solving generalized Nash equilibrium problem.
Citation: Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A penalty method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2012, 8 (1) : 51-65. doi: 10.3934/jimo.2012.8.51
##### References:
 [1] K. J. Arrow and G. Debreu, Existence of an equilibrium for a competitive economy,, Econometrica, 22 (1954), 265.  doi: 10.2307/1907353.  Google Scholar [2] D. P. Bertsekas, "Constrained Optimization and Lagrange Multiplier Methods,", Athena Scientific, (1996).   Google Scholar [3] B. Chen, X. Chen and C. Kanzow, A penalized Fischer-Burmeister NCP-function,, Mathematical Programming, 88 (2000), 211.  doi: 10.1007/PL00011375.  Google Scholar [4] F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley & Sons, (1983).   Google Scholar [5] G. Debreu, Definite and semidefinite quadratic forms,, Econometrica, 20 (1952), 295.  doi: 10.2307/1907852.  Google Scholar [6] F. Facchinei, A. Fischer and V. Piccialli, Generalized Nash equilibrium problems and Newton methods,, Mathematical Programming, 117 (2009), 163.  doi: 10.1007/s10107-007-0160-2.  Google Scholar [7] F. Facchinei and C. Kanzow, Penalty methods for the solution of generalized nash equilibrium problems,, SIAM Journal on Optimization, 20 (2010), 2228.  doi: 10.1137/090749499.  Google Scholar [8] F. Facchinei and C. Kanzow, Generalized Nash equilibrium problems,, Annals of Operations Research, 175 (2010), 177.  doi: 10.1007/s10479-009-0653-x.  Google Scholar [9] M. Fukushima, Restricted generalized Nash equilibria and controlled penalty algorithm,, Technical Report 2008-007, (2008), 2008.   Google Scholar [10] A. Heusinger and C. Kanzow, Optimization reformulations of the generalized Nash equilibrium problem using Nikaido-Isoda-type funcitons,, Computational Optimization and Applications, 43 (2009), 353.  doi: 10.1007/s10589-007-9145-6.  Google Scholar [11] A. Kesselman, S. Leonardi and V. Bonifaci, Game-theoretic analysis of internet switching with selfish users,, Lecture Notes in Computer Science, 3828 (2005), 236.  doi: 10.1007/11600930_23.  Google Scholar [12] R. Mifflin, Semismooth and semiconvex functions in constrained optimization,, SIAM Journal on Control and Optimization, 15 (1977), 959.  doi: 10.1137/0315061.  Google Scholar [13] L. W. McKenzie, On the existence of a general equilibrium for a competitive market,, Econometrica, 27 (1959), 54.  doi: 10.2307/1907777.  Google Scholar [14] J. F. Nash, Jr., Equilibrium points in $n$-person games,, Proceedings of the National Academy of Sciences of the USA, 36 (1950), 48.  doi: 10.1073/pnas.36.1.48.  Google Scholar [15] J. F. Nash, Non-cooperative games,, Annals of Mathematics (2), 54 (1951), 286.  doi: 10.2307/1969529.  Google Scholar [16] J. V. Neumann, Zur theorie der gesellschaftsspiele,, Mathematische Annalen, 100 (1928), 295.  doi: 10.1007/BF01448847.  Google Scholar [17] J. V. Neumann and O. Morgenstern, "Theory of Games and Economic Behavior,", Princeton University Press, (1953).   Google Scholar [18] J.-S. Pang and M. Fukushima, Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games,, Computational Management Science, 2 (2005), 21.  doi: 10.1007/s10287-004-0010-0.  Google Scholar [19] L. Qi and J. Sun, A nonsmooth version of Newton's method,, Mathematical Programming, 58 (1993), 353.  doi: 10.1007/BF01581275.  Google Scholar [20] L. Qi, D. Sun and G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities,, Mathematical Programming, 87 (2000), 1.   Google Scholar [21] J. B. Rosen, Existence and uniqueness of equilibrium points for concave $n$-person games,, Econometrica, 33 (1965), 520.  doi: 10.2307/1911749.  Google Scholar [22] D. Sun, The strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications,, Mathematics of Operations Research, 31 (2006), 761.  doi: 10.1287/moor.1060.0195.  Google Scholar [23] J. Sun, D. Sun and L. Qi, A squared smoothing Newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems,, SIAM Journal on Optimization, 14 (2004), 783.  doi: 10.1137/S1052623400379620.  Google Scholar

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##### References:
 [1] K. J. Arrow and G. Debreu, Existence of an equilibrium for a competitive economy,, Econometrica, 22 (1954), 265.  doi: 10.2307/1907353.  Google Scholar [2] D. P. Bertsekas, "Constrained Optimization and Lagrange Multiplier Methods,", Athena Scientific, (1996).   Google Scholar [3] B. Chen, X. Chen and C. Kanzow, A penalized Fischer-Burmeister NCP-function,, Mathematical Programming, 88 (2000), 211.  doi: 10.1007/PL00011375.  Google Scholar [4] F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley & Sons, (1983).   Google Scholar [5] G. Debreu, Definite and semidefinite quadratic forms,, Econometrica, 20 (1952), 295.  doi: 10.2307/1907852.  Google Scholar [6] F. Facchinei, A. Fischer and V. Piccialli, Generalized Nash equilibrium problems and Newton methods,, Mathematical Programming, 117 (2009), 163.  doi: 10.1007/s10107-007-0160-2.  Google Scholar [7] F. Facchinei and C. Kanzow, Penalty methods for the solution of generalized nash equilibrium problems,, SIAM Journal on Optimization, 20 (2010), 2228.  doi: 10.1137/090749499.  Google Scholar [8] F. Facchinei and C. Kanzow, Generalized Nash equilibrium problems,, Annals of Operations Research, 175 (2010), 177.  doi: 10.1007/s10479-009-0653-x.  Google Scholar [9] M. Fukushima, Restricted generalized Nash equilibria and controlled penalty algorithm,, Technical Report 2008-007, (2008), 2008.   Google Scholar [10] A. Heusinger and C. Kanzow, Optimization reformulations of the generalized Nash equilibrium problem using Nikaido-Isoda-type funcitons,, Computational Optimization and Applications, 43 (2009), 353.  doi: 10.1007/s10589-007-9145-6.  Google Scholar [11] A. Kesselman, S. Leonardi and V. Bonifaci, Game-theoretic analysis of internet switching with selfish users,, Lecture Notes in Computer Science, 3828 (2005), 236.  doi: 10.1007/11600930_23.  Google Scholar [12] R. Mifflin, Semismooth and semiconvex functions in constrained optimization,, SIAM Journal on Control and Optimization, 15 (1977), 959.  doi: 10.1137/0315061.  Google Scholar [13] L. W. McKenzie, On the existence of a general equilibrium for a competitive market,, Econometrica, 27 (1959), 54.  doi: 10.2307/1907777.  Google Scholar [14] J. F. Nash, Jr., Equilibrium points in $n$-person games,, Proceedings of the National Academy of Sciences of the USA, 36 (1950), 48.  doi: 10.1073/pnas.36.1.48.  Google Scholar [15] J. F. Nash, Non-cooperative games,, Annals of Mathematics (2), 54 (1951), 286.  doi: 10.2307/1969529.  Google Scholar [16] J. V. Neumann, Zur theorie der gesellschaftsspiele,, Mathematische Annalen, 100 (1928), 295.  doi: 10.1007/BF01448847.  Google Scholar [17] J. V. Neumann and O. Morgenstern, "Theory of Games and Economic Behavior,", Princeton University Press, (1953).   Google Scholar [18] J.-S. Pang and M. Fukushima, Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games,, Computational Management Science, 2 (2005), 21.  doi: 10.1007/s10287-004-0010-0.  Google Scholar [19] L. Qi and J. Sun, A nonsmooth version of Newton's method,, Mathematical Programming, 58 (1993), 353.  doi: 10.1007/BF01581275.  Google Scholar [20] L. Qi, D. Sun and G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities,, Mathematical Programming, 87 (2000), 1.   Google Scholar [21] J. B. Rosen, Existence and uniqueness of equilibrium points for concave $n$-person games,, Econometrica, 33 (1965), 520.  doi: 10.2307/1911749.  Google Scholar [22] D. Sun, The strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications,, Mathematics of Operations Research, 31 (2006), 761.  doi: 10.1287/moor.1060.0195.  Google Scholar [23] J. Sun, D. Sun and L. Qi, A squared smoothing Newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems,, SIAM Journal on Optimization, 14 (2004), 783.  doi: 10.1137/S1052623400379620.  Google Scholar
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