# American Institute of Mathematical Sciences

2012, 2(1): 145-156. doi: 10.3934/naco.2012.2.145

## A class of smoothing SAA methods for a stochastic linear complementarity problem

 1 School of Mathematics, Liaoning Normal University, Dalian, 116029, China 2 School of Management, University of Southampton, Highfield Southampton SO17 1BJ, United Kingdom 3 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024

Received  January 2011 Revised  October 2011 Published  March 2012

A class of smoothing sample average approximation (SAA) methods is proposed for solving a stochastic linear complementarity problem, where the underlying function is the expected value of stochastic function. Existence and convergence results to the proposed methods are provided and some numerical results are reported to show the efficiency of the methods proposed.
Citation: Jie Zhang, Yue Wu, Liwei Zhang. A class of smoothing SAA methods for a stochastic linear complementarity problem. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 145-156. doi: 10.3934/naco.2012.2.145
##### References:
 [1] B. Chen and P. T. Harker, A non-interior point continuation method for linear complementarity problems, SIAM Journal on Matrix Analysis and Applications, 14 (1993), 1168-1190. doi: 10.1137/0614081. [2] C. Chen and O. L. Mangasarian, Smoothing methods for convex inequalities and linear complementarity problems, Mathematical Programming, 71 (1995), 51-69. doi: 10.1007/BF01592244. [3] C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Comp. Optim. and Appl., 5 (1996), 97-138. doi: 10.1007/BF00249052. [4] X. J. Chen, Smoothing methods for complementarity problems and their applications: a survey, J. Oper. Res. Soc. Japan, 43 (2000), 32-47. doi: 10.1016/S0453-4514(00)88750-5. [5] X. J. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems, Math. Oper. Res., 30 (2005), 1022-1038. doi: 10.1287/moor.1050.0160. [6] F. Facchinei and J. S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems, Volumes I and II," Springer-Verlag, New York, 2003. [7] 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. [8] H. Jiang and H. Xu, Stochastic approximation approaches to the stochastic variational inequality problem, IEEE Transactions on Automatic Control, 53 (2008), 1462-1475. doi: 10.1109/TAC.2008.925853. [9] C. Kanzow, "Some Tools Allowing Interior-Point Methods to Become Noninterior," Technical Report, Institute of Applied Mathematics, University of Hamburg, Germany, 1994. [10] J. S. Pang, Error bounds in mathematical programming, Mathematical Programming, 79 (1997), 299-332. doi: 10.1007/BF02614322. [11] S. M. Robinson, Some continuity properties of polyhedral multifunctions, Mathematical Programming Study, 14 (1981), 206-214. [12] R. T. Rockafellar and R. J. B. Wets, "Variational Analysis," Berlin Heidelberg, 1998. [13] A. Shapiro, D. Dentcheva and A. Ruszczynski, "Lectures on Stochastic Programming: Modeling and Theory," SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898718751. [14] S. Smale, Algorithms for solving equations, in "Proceedings of the International Congress of Mathematicians," Ameri. Math. Sot., Providence, 1987. [15] J. Takaki and N. Yamashita, A derivative-free trust-region algorithm for unconstrained optimization with controlled error, Numerical Algebra, Control and Optimization, 1 (2011), 117-145. doi: 10.3934/naco.2011.1.117. [16] H. Xu and D. Zhang, Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications, Mathematical Programming, 119 (2009), 371-401. doi: 10.1007/s10107-008-0214-0. [17] H. Xu, Sample average approximation methods for a class of stochastic variational inequality problems, Asian Pacific Journal of Operations Research, 27 (2010), 103-119. doi: 10.1142/S0217595910002569. [18] Y. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least squares, Numerical Algebra, Control and Optimization, 1 (2011), 15-34. doi: 10.3934/naco.2011.1.15. [19] L. Zhang, J. Zhang and Y. Wu, On the convergence of coderivative of SAA solution mapping for a parametric stochastic generalized equation, Set-valued Anal., 19 (2011), 107-134. doi: 10.1007/s11228-010-0141-0.

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##### References:
 [1] B. Chen and P. T. Harker, A non-interior point continuation method for linear complementarity problems, SIAM Journal on Matrix Analysis and Applications, 14 (1993), 1168-1190. doi: 10.1137/0614081. [2] C. Chen and O. L. Mangasarian, Smoothing methods for convex inequalities and linear complementarity problems, Mathematical Programming, 71 (1995), 51-69. doi: 10.1007/BF01592244. [3] C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Comp. Optim. and Appl., 5 (1996), 97-138. doi: 10.1007/BF00249052. [4] X. J. Chen, Smoothing methods for complementarity problems and their applications: a survey, J. Oper. Res. Soc. Japan, 43 (2000), 32-47. doi: 10.1016/S0453-4514(00)88750-5. [5] X. J. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems, Math. Oper. Res., 30 (2005), 1022-1038. doi: 10.1287/moor.1050.0160. [6] F. Facchinei and J. S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems, Volumes I and II," Springer-Verlag, New York, 2003. [7] 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. [8] H. Jiang and H. Xu, Stochastic approximation approaches to the stochastic variational inequality problem, IEEE Transactions on Automatic Control, 53 (2008), 1462-1475. doi: 10.1109/TAC.2008.925853. [9] C. Kanzow, "Some Tools Allowing Interior-Point Methods to Become Noninterior," Technical Report, Institute of Applied Mathematics, University of Hamburg, Germany, 1994. [10] J. S. Pang, Error bounds in mathematical programming, Mathematical Programming, 79 (1997), 299-332. doi: 10.1007/BF02614322. [11] S. M. Robinson, Some continuity properties of polyhedral multifunctions, Mathematical Programming Study, 14 (1981), 206-214. [12] R. T. Rockafellar and R. J. B. Wets, "Variational Analysis," Berlin Heidelberg, 1998. [13] A. Shapiro, D. Dentcheva and A. Ruszczynski, "Lectures on Stochastic Programming: Modeling and Theory," SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898718751. [14] S. Smale, Algorithms for solving equations, in "Proceedings of the International Congress of Mathematicians," Ameri. Math. Sot., Providence, 1987. [15] J. Takaki and N. Yamashita, A derivative-free trust-region algorithm for unconstrained optimization with controlled error, Numerical Algebra, Control and Optimization, 1 (2011), 117-145. doi: 10.3934/naco.2011.1.117. [16] H. Xu and D. Zhang, Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications, Mathematical Programming, 119 (2009), 371-401. doi: 10.1007/s10107-008-0214-0. [17] H. Xu, Sample average approximation methods for a class of stochastic variational inequality problems, Asian Pacific Journal of Operations Research, 27 (2010), 103-119. doi: 10.1142/S0217595910002569. [18] Y. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least squares, Numerical Algebra, Control and Optimization, 1 (2011), 15-34. doi: 10.3934/naco.2011.1.15. [19] L. Zhang, J. Zhang and Y. Wu, On the convergence of coderivative of SAA solution mapping for a parametric stochastic generalized equation, Set-valued Anal., 19 (2011), 107-134. doi: 10.1007/s11228-010-0141-0.
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