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Neural network smoothing approximation method for stochastic variational inequality problems
1.  School of Economics, Southwest University for Nationalities, Chengdu, Sichuan 610041, China 
2.  Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064 
References:
[1] 
R. P. Agdeppa, N. Yamashita and M. Fukushima, Convex expected residual models for stochastic affine variational inequality problems and its application to the traffic equilibrium problem, Pacific Journal of Optimization, 6 (2010), 319. 
[2] 
R. J. Aumann, Integrals of setvalue function, Journal of Mathematical Analysis and Applications, 12 (1965), 112. doi: 10.1016/0022247X(65)900491. 
[3] 
B. T. Chen and P. T. Harker, Smooth approximations to nonlinear complementarity problems, SIAM Journal on Optimization, 7 (1997), 403420. doi: 10.1137/S1052623495280615. 
[4] 
X. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems, Mathematics of Operations Research, 30 (2005), 10221038. doi: 10.1287/moor.1050.0160. 
[5] 
X. Chen and G. H. Lin, CVaRbased formulation and approximation method for Stochastic variational inequalities, Numerical Algebra, Control and Optimization, 1 (2011), 3548. doi: 10.3934/naco.2011.1.35. 
[6] 
X. Chen, C. Zhang and M. Fukushima, Robust solution of monotone stochastic linear complementarity problems, Mathematical Programming, 117 (2009), 5180. doi: 10.1007/s101070070163z. 
[7] 
F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. 
[8] 
M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Mathematical Programming, 53 (1992), 99110. doi: 10.1007/BF01585696. 
[9] 
F. Facchinei and J. S. Pang, FiniteDimensional Variational Inequalities and Complementarity Problems, Springer, New York, 2003. doi: 10.1007/b97544. 
[10] 
H. Fang, X. Chen and M. Fukushima, Stochastic $R_0$ matrix linear complementarity problems, SIAM Journal on Optimization, 18 (2007), 482506. doi: 10.1137/050630805. 
[11] 
P. T. Harker and J. S. Pang, Finitedimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Mathematical Programming, 48 (1990), 161220. doi: 10.1007/BF01582255. 
[12] 
W. W. Hogan, Energy policy models for project independence, Computers and Operations Research, 2 (1975), 251271. doi: 10.1016/03050548(75)900088. 
[13] 
H. Jiang and H. F. Xu, Stochastic approximation approaches to the stochastic variational inequality problem, IEEE Transactions on Automatic Control, 53 (2008), 14621475. doi: 10.1109/TAC.2008.925853. 
[14] 
D. Kinderlehre and G. Stampacchia, An Intruduction to Variational Inequalities and Their Aplications, Academic Press, New York, 1980. 
[15] 
G. H. Lin, X. Chen and M. Fukushima, New restricted NCP function and their applications to stochastic NCP and stochastic MPEC, Optimization, 56 (2007), 641953. doi: 10.1080/02331930701617320. 
[16] 
G. H. Lin and M. Fukushima, Stochastic equilibrium problems and stochastic mathematical programs with equilibrium constraints: A survey, Pacific Journal of Optimization, 6 (2010), 455482. 
[17] 
G. H. Lin and M. Fukushima, New reformulations for stochastic nonlinear complementarity peoblems, Optimization Methods and Software, 21 (2006), 551564. doi: 10.1080/10556780600627610. 
[18] 
C. Ling, L. Qi, G. Zhou and L. Caccetta, The SC' property of an expected residual function arising from stochastic complementarity problems, Operations Research Letters, 36 (2008), 456460. doi: 10.1016/j.orl.2008.01.010. 
[19] 
M. J. Luo and G. H. Lin, Expected residual minimization method for stochastic variational inequality problems, Journal of Optimization Theory and Applications, 140 (2009), 103116. doi: 10.1007/s1095700894396. 
[20] 
M. J. Luo and G. H. Lin, Convergence results of the ERM method for nonlinear stochastic variational inequality problems, Journal of Optimization Theory and Applications, 142 (2009), 569581. doi: 10.1007/s1095700995343. 
[21] 
F. W. Meng, J. Sun and M. Goh, Stochastic optimization problems with CVaR risk measure and their sample average approximation, Journal of Optimization Theory and Applications, 146 (2010), 399418. doi: 10.1007/s1095701096763. 
[22] 
L. Q. Qi, D. F. Sun and G. L. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Mathematical Programming, 87 (2000), 135. 
[23] 
R. T. Rockafellar and S. Uryasev, Optimization of conditional valueatrisk, Journal of Risk, 2 (2000), 493517. 
[24] 
A. Ruszczynski and A. Shapiro, Stochastic Programming, Elsevier, Amsterdam, 2003. 
[25] 
A. Shapiro, Stochastic Programming by Monte Carlo Simulation Methods, Stochastic Programming EPrint Series, 2000. 
[26] 
M. Z. Wang, M. M. Ali and G. H. Lin, Sample average approximation method for stochastic complementarity problems with applications to supply chain supernetworks, Journal of Industrial and Management Optimization, 7 (2011), 317345. doi: 10.3934/jimo.2011.7.317. 
[27] 
D. De Wolf and Y. Smeers, A stochastic version of a StackelbergNashCournot equilibrium model, Management Science, 43 (1997), 190197. 
[28] 
H. Xu, Sample average approximation methods for a class of stochastic variational inequality problems, AsiaPacific Journal of Operational Research, 27 (2010), 103119. doi: 10.1142/S0217595910002569. 
[29] 
H. Xu and D. Zhang, Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications, Mathematical Programming, 119 (2009), 371401. doi: 10.1007/s1010700802140. 
[30] 
C. Zhang and X. Chen, Stochastic nonlinear complementarity problem and applications to traffic equilibrium under uncertainty, Journal of Optimization Theory and Applications, 137 (2008), 277295. doi: 10.1007/s1095700893586. 
show all references
References:
[1] 
R. P. Agdeppa, N. Yamashita and M. Fukushima, Convex expected residual models for stochastic affine variational inequality problems and its application to the traffic equilibrium problem, Pacific Journal of Optimization, 6 (2010), 319. 
[2] 
R. J. Aumann, Integrals of setvalue function, Journal of Mathematical Analysis and Applications, 12 (1965), 112. doi: 10.1016/0022247X(65)900491. 
[3] 
B. T. Chen and P. T. Harker, Smooth approximations to nonlinear complementarity problems, SIAM Journal on Optimization, 7 (1997), 403420. doi: 10.1137/S1052623495280615. 
[4] 
X. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems, Mathematics of Operations Research, 30 (2005), 10221038. doi: 10.1287/moor.1050.0160. 
[5] 
X. Chen and G. H. Lin, CVaRbased formulation and approximation method for Stochastic variational inequalities, Numerical Algebra, Control and Optimization, 1 (2011), 3548. doi: 10.3934/naco.2011.1.35. 
[6] 
X. Chen, C. Zhang and M. Fukushima, Robust solution of monotone stochastic linear complementarity problems, Mathematical Programming, 117 (2009), 5180. doi: 10.1007/s101070070163z. 
[7] 
F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. 
[8] 
M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Mathematical Programming, 53 (1992), 99110. doi: 10.1007/BF01585696. 
[9] 
F. Facchinei and J. S. Pang, FiniteDimensional Variational Inequalities and Complementarity Problems, Springer, New York, 2003. doi: 10.1007/b97544. 
[10] 
H. Fang, X. Chen and M. Fukushima, Stochastic $R_0$ matrix linear complementarity problems, SIAM Journal on Optimization, 18 (2007), 482506. doi: 10.1137/050630805. 
[11] 
P. T. Harker and J. S. Pang, Finitedimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Mathematical Programming, 48 (1990), 161220. doi: 10.1007/BF01582255. 
[12] 
W. W. Hogan, Energy policy models for project independence, Computers and Operations Research, 2 (1975), 251271. doi: 10.1016/03050548(75)900088. 
[13] 
H. Jiang and H. F. Xu, Stochastic approximation approaches to the stochastic variational inequality problem, IEEE Transactions on Automatic Control, 53 (2008), 14621475. doi: 10.1109/TAC.2008.925853. 
[14] 
D. Kinderlehre and G. Stampacchia, An Intruduction to Variational Inequalities and Their Aplications, Academic Press, New York, 1980. 
[15] 
G. H. Lin, X. Chen and M. Fukushima, New restricted NCP function and their applications to stochastic NCP and stochastic MPEC, Optimization, 56 (2007), 641953. doi: 10.1080/02331930701617320. 
[16] 
G. H. Lin and M. Fukushima, Stochastic equilibrium problems and stochastic mathematical programs with equilibrium constraints: A survey, Pacific Journal of Optimization, 6 (2010), 455482. 
[17] 
G. H. Lin and M. Fukushima, New reformulations for stochastic nonlinear complementarity peoblems, Optimization Methods and Software, 21 (2006), 551564. doi: 10.1080/10556780600627610. 
[18] 
C. Ling, L. Qi, G. Zhou and L. Caccetta, The SC' property of an expected residual function arising from stochastic complementarity problems, Operations Research Letters, 36 (2008), 456460. doi: 10.1016/j.orl.2008.01.010. 
[19] 
M. J. Luo and G. H. Lin, Expected residual minimization method for stochastic variational inequality problems, Journal of Optimization Theory and Applications, 140 (2009), 103116. doi: 10.1007/s1095700894396. 
[20] 
M. J. Luo and G. H. Lin, Convergence results of the ERM method for nonlinear stochastic variational inequality problems, Journal of Optimization Theory and Applications, 142 (2009), 569581. doi: 10.1007/s1095700995343. 
[21] 
F. W. Meng, J. Sun and M. Goh, Stochastic optimization problems with CVaR risk measure and their sample average approximation, Journal of Optimization Theory and Applications, 146 (2010), 399418. doi: 10.1007/s1095701096763. 
[22] 
L. Q. Qi, D. F. Sun and G. L. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Mathematical Programming, 87 (2000), 135. 
[23] 
R. T. Rockafellar and S. Uryasev, Optimization of conditional valueatrisk, Journal of Risk, 2 (2000), 493517. 
[24] 
A. Ruszczynski and A. Shapiro, Stochastic Programming, Elsevier, Amsterdam, 2003. 
[25] 
A. Shapiro, Stochastic Programming by Monte Carlo Simulation Methods, Stochastic Programming EPrint Series, 2000. 
[26] 
M. Z. Wang, M. M. Ali and G. H. Lin, Sample average approximation method for stochastic complementarity problems with applications to supply chain supernetworks, Journal of Industrial and Management Optimization, 7 (2011), 317345. doi: 10.3934/jimo.2011.7.317. 
[27] 
D. De Wolf and Y. Smeers, A stochastic version of a StackelbergNashCournot equilibrium model, Management Science, 43 (1997), 190197. 
[28] 
H. Xu, Sample average approximation methods for a class of stochastic variational inequality problems, AsiaPacific Journal of Operational Research, 27 (2010), 103119. doi: 10.1142/S0217595910002569. 
[29] 
H. Xu and D. Zhang, Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications, Mathematical Programming, 119 (2009), 371401. doi: 10.1007/s1010700802140. 
[30] 
C. Zhang and X. Chen, Stochastic nonlinear complementarity problem and applications to traffic equilibrium under uncertainty, Journal of Optimization Theory and Applications, 137 (2008), 277295. doi: 10.1007/s1095700893586. 
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