# American Institute of Mathematical Sciences

January  2016, 12(1): 1-15. doi: 10.3934/jimo.2016.12.1

## Smoothing and sample average approximation methods for solving stochastic generalized Nash equilibrium problems

 1 School of Mathematics, Liaoning University, Liaoning 110031, China 2 School of Management, Shanghai University, Shanghai 200444, China

Received  January 2014 Revised  October 2014 Published  April 2015

In this paper, we consider the class of stochastic generalized Nash equilibrium problems (SGNEP). Such problems have a wide range of applications and have attracted significant attention recently. First, using the first order optimality condition of SGNEP and the nonlinear complementary function, we present an expected residual minimization (ERM) model for the case when the involved functions are not continuously differentiable. Then, we introduce a smoothing function, depending on a smoothing parameter, to yield a smooth approximate ERM model. We further show that the solutions of this smooth ERM model converge to the solutions of the original ERM model as the smoothing parameter tends to zero. Since the ERM formulation contains an expectation, we further propose a sample average approximate problem for the ERM model. Moreover, we show that the global optimal solutions of these approximate problems converge to the global optimal solutions of the ERM problem with probability one. Here, convergence can be achieved in two ways. One is to fix the smoothing parameter, the other is to let the smoothing parameter tend to zero as the sample increases.
Citation: Mei Ju Luo, Yi Zeng Chen. Smoothing and sample average approximation methods for solving stochastic generalized Nash equilibrium problems. Journal of Industrial and Management Optimization, 2016, 12 (1) : 1-15. doi: 10.3934/jimo.2016.12.1
##### References:
 [1] J. R. Birge, Quasi-Monte Carlo Approaches to Option Pricing, Technical Report 94-19, Department of Industrial and Operations Engineering, University of Michigan, 1994. [2] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. [3] X. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems, Mathematics of Operations Research, 30 (2005), 1022-1038. doi: 10.1287/moor.1050.0160. [4] X. Chen, C. Zhang and M. Fukushima, Robust solution of monotone stochastic linear complementarity problems, Mathematical Programming, 117 (2009), 51-80. doi: 10.1007/s10107-007-0163-z. [5] D. De Wolf and Y. Smeers, A stochastic version of a Stackelberg-Nash-Cournot equilibrium model, Management Science, 43 (1997), 190-197. [6] A. Fischer, A special Newton-type optimization method, Optimization, 24 (1992), 269-284. doi: 10.1080/02331939208843795. [7] F. Facchinei and C. Kanzow, Generalized Nash equilibrium problems, A Quarterly Journal of Operations Research, 5 (2007), 173-210. doi: 10.1007/s10288-007-0054-4. [8] F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer-Verlag, New York, 2003. doi: 10.1007/b97544. [9] H. Fang, X. Chen and M. Fukushima, Stochastic $R_0$ matrix linear complementarity problems, SIAM Journal on Optimization, 18 (2007), 482-506. doi: 10.1137/050630805. [10] G. Gürkan, A. Y. Özge and S. M. Robinson, Sample-path solution of stochastic variational inequalities, Mathematical Programming, 84 (1999), 313-333. doi: 10.1007/s101070050024. [11] J. Gao and Y. Liu, Stochastic Nash equilibrium with a numerical solution method, Computer Science, 3496 (2005), 811-816. doi: 10.1007/11427391_130. [12] J. B. Krawczyk, Numerical solutions to coupled-constraint (or generalised Nash) equilibrium problems, Computational Management Science, 4 (2007), 183-204. doi: 10.1007/s10287-006-0033-9. [13] C. Ling, L. Qi, G. Zhou and L. Caccetta, The $SC^1$ property of an expected residual function arising from stochastic complementarity problems, Operations Research Letters, 36 (2008), 456-460. doi: 10.1016/j.orl.2008.01.010. [14] G. H. Lin, X. Chen and M. Fukushima, New restricted NCP function and their applications to stochastic NCP and stochastic MPEC, Optimization, 56 (2007), 641-953. doi: 10.1080/02331930701617320. [15] G. H. Lin and M. Fukushima, New reformulations for stochastic nonlinear complementarity problems, Optimization Methods and Software, 21 (2006), 551-564. doi: 10.1080/10556780600627610. [16] P. Y. Li, Z. F. He and G. H. Lin, Sampling average approximation method for a class of stochastic Nash equilibrium problems, Optimization Methods and Software, 28 (2013), 785-795. doi: 10.1080/10556788.2012.750321. [17] H. Mukaidani, Stochastic Nash equilibrium seeking for games with general nonlinear payoffs, SIAM Journal on Control and Optimization, 49 (2011), 1659-1679. doi: 10.1137/100811738. [18] H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, Philadelphia, SIAM, 1992. doi: 10.1137/1.9781611970081. [19] J. F. Nash, Non-Cooperative games, Annals of Mathematics, 54 (1951), 286-295. doi: 10.2307/1969529. [20] R. T. Rockafellar and R. J. B.wets, Variational Analysis, Springer-Verlag, Berlin Heidelberg, 1998. doi: 10.1007/978-3-642-02431-3. [21] A. Shapiro, Monte Carlo sampling approch to stochastic programming, European Series of Applied and Industrial Mathematics: Proceeding, 13 (2003), 65-73. [22] A. Shapiro, Monte carlo sampling methods, stochastic programming, Handbooks in Operations Research and Management Science, 10 (2003), 353-425. doi: 10.1016/S0927-0507(03)10006-0. [23] A. Shapiro and H. F. Xu, Stochasic mathematical programs with equiblbrium constraints, modelling and sample average approximation, Optimization, 57 (2008), 395-418. doi: 10.1080/02331930801954177. [24] P. Tseng, Growth behavior of a class of merit functions for the nonlinear complementarity problem, Journal of Optimization Theory and Applications, 89 (1996), 17-37. doi: 10.1007/BF02192639. [25] H. F. Xu and D. L. Zhang, Stochastic Nash equilibrium problems: Sample average approximation and applications, Computational Optimization and Applications, 55 (2013), 597-645. doi: 10.1007/s10589-013-9538-7. [26] H. F. Xu and D. L. Zhang, Smooth sample average appproximation of stationary points in nonsmooth stochastic optimization and applications, Mathematical Programming Series A, 119 (2009), 371-401. doi: 10.1007/s10107-008-0214-0. [27] Y. H. Yuan, L. W. Zhang and Y. Wu, A smoothing Newton method based on sample average approximation for a class of stochastic generalized Nash equilibrium problems,, Pacific Journal of Optimization, (). [28] C. Zhang and X. Chen, Stochastic nonlinear complementarity problem and applications to traffic equilibrium under uncertainty, Journal of Optimization Theory and Applications, 137 (2008), 277-295. doi: 10.1007/s10957-008-9358-6.

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##### References:
 [1] J. R. Birge, Quasi-Monte Carlo Approaches to Option Pricing, Technical Report 94-19, Department of Industrial and Operations Engineering, University of Michigan, 1994. [2] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. [3] X. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems, Mathematics of Operations Research, 30 (2005), 1022-1038. doi: 10.1287/moor.1050.0160. [4] X. Chen, C. Zhang and M. Fukushima, Robust solution of monotone stochastic linear complementarity problems, Mathematical Programming, 117 (2009), 51-80. doi: 10.1007/s10107-007-0163-z. [5] D. De Wolf and Y. Smeers, A stochastic version of a Stackelberg-Nash-Cournot equilibrium model, Management Science, 43 (1997), 190-197. [6] A. Fischer, A special Newton-type optimization method, Optimization, 24 (1992), 269-284. doi: 10.1080/02331939208843795. [7] F. Facchinei and C. Kanzow, Generalized Nash equilibrium problems, A Quarterly Journal of Operations Research, 5 (2007), 173-210. doi: 10.1007/s10288-007-0054-4. [8] F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer-Verlag, New York, 2003. doi: 10.1007/b97544. [9] H. Fang, X. Chen and M. Fukushima, Stochastic $R_0$ matrix linear complementarity problems, SIAM Journal on Optimization, 18 (2007), 482-506. doi: 10.1137/050630805. [10] G. Gürkan, A. Y. Özge and S. M. Robinson, Sample-path solution of stochastic variational inequalities, Mathematical Programming, 84 (1999), 313-333. doi: 10.1007/s101070050024. [11] J. Gao and Y. Liu, Stochastic Nash equilibrium with a numerical solution method, Computer Science, 3496 (2005), 811-816. doi: 10.1007/11427391_130. [12] J. B. Krawczyk, Numerical solutions to coupled-constraint (or generalised Nash) equilibrium problems, Computational Management Science, 4 (2007), 183-204. doi: 10.1007/s10287-006-0033-9. [13] C. Ling, L. Qi, G. Zhou and L. Caccetta, The $SC^1$ property of an expected residual function arising from stochastic complementarity problems, Operations Research Letters, 36 (2008), 456-460. doi: 10.1016/j.orl.2008.01.010. [14] G. H. Lin, X. Chen and M. Fukushima, New restricted NCP function and their applications to stochastic NCP and stochastic MPEC, Optimization, 56 (2007), 641-953. doi: 10.1080/02331930701617320. [15] G. H. Lin and M. Fukushima, New reformulations for stochastic nonlinear complementarity problems, Optimization Methods and Software, 21 (2006), 551-564. doi: 10.1080/10556780600627610. [16] P. Y. Li, Z. F. He and G. H. Lin, Sampling average approximation method for a class of stochastic Nash equilibrium problems, Optimization Methods and Software, 28 (2013), 785-795. doi: 10.1080/10556788.2012.750321. [17] H. Mukaidani, Stochastic Nash equilibrium seeking for games with general nonlinear payoffs, SIAM Journal on Control and Optimization, 49 (2011), 1659-1679. doi: 10.1137/100811738. [18] H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, Philadelphia, SIAM, 1992. doi: 10.1137/1.9781611970081. [19] J. F. Nash, Non-Cooperative games, Annals of Mathematics, 54 (1951), 286-295. doi: 10.2307/1969529. [20] R. T. Rockafellar and R. J. B.wets, Variational Analysis, Springer-Verlag, Berlin Heidelberg, 1998. doi: 10.1007/978-3-642-02431-3. [21] A. Shapiro, Monte Carlo sampling approch to stochastic programming, European Series of Applied and Industrial Mathematics: Proceeding, 13 (2003), 65-73. [22] A. Shapiro, Monte carlo sampling methods, stochastic programming, Handbooks in Operations Research and Management Science, 10 (2003), 353-425. doi: 10.1016/S0927-0507(03)10006-0. [23] A. Shapiro and H. F. Xu, Stochasic mathematical programs with equiblbrium constraints, modelling and sample average approximation, Optimization, 57 (2008), 395-418. doi: 10.1080/02331930801954177. [24] P. Tseng, Growth behavior of a class of merit functions for the nonlinear complementarity problem, Journal of Optimization Theory and Applications, 89 (1996), 17-37. doi: 10.1007/BF02192639. [25] H. F. Xu and D. L. Zhang, Stochastic Nash equilibrium problems: Sample average approximation and applications, Computational Optimization and Applications, 55 (2013), 597-645. doi: 10.1007/s10589-013-9538-7. [26] H. F. Xu and D. L. Zhang, Smooth sample average appproximation of stationary points in nonsmooth stochastic optimization and applications, Mathematical Programming Series A, 119 (2009), 371-401. doi: 10.1007/s10107-008-0214-0. [27] Y. H. Yuan, L. W. Zhang and Y. Wu, A smoothing Newton method based on sample average approximation for a class of stochastic generalized Nash equilibrium problems,, Pacific Journal of Optimization, (). [28] C. Zhang and X. Chen, Stochastic nonlinear complementarity problem and applications to traffic equilibrium under uncertainty, Journal of Optimization Theory and Applications, 137 (2008), 277-295. doi: 10.1007/s10957-008-9358-6.
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