American Institute of Mathematical Sciences

April  2015, 11(2): 645-660. doi: 10.3934/jimo.2015.11.645

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

Received  June 2012 Revised  May 2014 Published  September 2014

This paper is concerned with solving a stochastic variational inequality problem (for short, SVIP) from a viewpoint of minimization of mixed conditional value-at-risk (CVaR). The regularized gap function for SVIP is used to define a loss function for the SVIP and mixed CVaR to measure the loss. In this setting, SVIP can be reformulated as a deterministic minimization problem. We show that the reformulation is a convex program for a huge class of SVIP under suitable conditions. Since mixed CVaR involves the plus function and mathematical expectation, the neural network smoothing function and Monte Carlo method are employed to get an approximation problem of the minimization reformulation. Finally, we consider the convergence of optimal solutions and stationary points of the approximation.
Citation: Hui-Qiang Ma, Nan-Jing Huang. Neural network smoothing approximation method for stochastic variational inequality problems. Journal of Industrial & Management Optimization, 2015, 11 (2) : 645-660. doi: 10.3934/jimo.2015.11.645
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), 3.   Google Scholar [2] R. J. Aumann, Integrals of set-value function,, Journal of Mathematical Analysis and Applications, 12 (1965), 1.  doi: 10.1016/0022-247X(65)90049-1.  Google Scholar [3] B. T. Chen and P. T. Harker, Smooth approximations to nonlinear complementarity problems,, SIAM Journal on Optimization, 7 (1997), 403.  doi: 10.1137/S1052623495280615.  Google Scholar [4] X. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems,, Mathematics of Operations Research, 30 (2005), 1022.  doi: 10.1287/moor.1050.0160.  Google Scholar [5] X. Chen and G. H. Lin, CVaR-based formulation and approximation method for Stochastic variational inequalities,, Numerical Algebra, 1 (2011), 35.  doi: 10.3934/naco.2011.1.35.  Google Scholar [6] X. Chen, C. Zhang and M. Fukushima, Robust solution of monotone stochastic linear complementarity problems,, Mathematical Programming, 117 (2009), 51.  doi: 10.1007/s10107-007-0163-z.  Google Scholar [7] F. H. Clarke, Optimization and Nonsmooth Analysis,, Wiley, (1983).   Google Scholar [8] M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems,, Mathematical Programming, 53 (1992), 99.  doi: 10.1007/BF01585696.  Google Scholar [9] F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems,, Springer, (2003).  doi: 10.1007/b97544.  Google Scholar [10] H. Fang, X. Chen and M. Fukushima, Stochastic $R_0$ matrix linear complementarity problems,, SIAM Journal on Optimization, 18 (2007), 482.  doi: 10.1137/050630805.  Google Scholar [11] P. T. Harker and J. S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications,, Mathematical Programming, 48 (1990), 161.  doi: 10.1007/BF01582255.  Google Scholar [12] W. W. Hogan, Energy policy models for project independence,, Computers and Operations Research, 2 (1975), 251.  doi: 10.1016/0305-0548(75)90008-8.  Google Scholar [13] H. Jiang and H. F. Xu, Stochastic approximation approaches to the stochastic variational inequality problem,, IEEE Transactions on Automatic Control, 53 (2008), 1462.  doi: 10.1109/TAC.2008.925853.  Google Scholar [14] D. Kinderlehre and G. Stampacchia, An Intruduction to Variational Inequalities and Their Aplications,, Academic Press, (1980).   Google Scholar [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), 641.  doi: 10.1080/02331930701617320.  Google Scholar [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), 455.   Google Scholar [17] G. H. Lin and M. Fukushima, New reformulations for stochastic nonlinear complementarity peoblems,, Optimization Methods and Software, 21 (2006), 551.  doi: 10.1080/10556780600627610.  Google Scholar [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), 456.  doi: 10.1016/j.orl.2008.01.010.  Google Scholar [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), 103.  doi: 10.1007/s10957-008-9439-6.  Google Scholar [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), 569.  doi: 10.1007/s10957-009-9534-3.  Google Scholar [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), 399.  doi: 10.1007/s10957-010-9676-3.  Google Scholar [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), 1.   Google Scholar [23] R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk,, Journal of Risk, 2 (2000), 493.   Google Scholar [24] A. Ruszczynski and A. Shapiro, Stochastic Programming,, Elsevier, (2003).   Google Scholar [25] A. Shapiro, Stochastic Programming by Monte Carlo Simulation Methods,, Stochastic Programming E-Print Series, (2000).   Google Scholar [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), 317.  doi: 10.3934/jimo.2011.7.317.  Google Scholar [27] D. De Wolf and Y. Smeers, A stochastic version of a Stackelberg-Nash-Cournot equilibrium model,, Management Science, 43 (1997), 190.   Google Scholar [28] H. Xu, Sample average approximation methods for a class of stochastic variational inequality problems,, Asia-Pacific Journal of Operational Research, 27 (2010), 103.  doi: 10.1142/S0217595910002569.  Google Scholar [29] H. Xu and D. Zhang, Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications,, Mathematical Programming, 119 (2009), 371.  doi: 10.1007/s10107-008-0214-0.  Google Scholar [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), 277.  doi: 10.1007/s10957-008-9358-6.  Google Scholar

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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), 3.   Google Scholar [2] R. J. Aumann, Integrals of set-value function,, Journal of Mathematical Analysis and Applications, 12 (1965), 1.  doi: 10.1016/0022-247X(65)90049-1.  Google Scholar [3] B. T. Chen and P. T. Harker, Smooth approximations to nonlinear complementarity problems,, SIAM Journal on Optimization, 7 (1997), 403.  doi: 10.1137/S1052623495280615.  Google Scholar [4] X. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems,, Mathematics of Operations Research, 30 (2005), 1022.  doi: 10.1287/moor.1050.0160.  Google Scholar [5] X. Chen and G. H. Lin, CVaR-based formulation and approximation method for Stochastic variational inequalities,, Numerical Algebra, 1 (2011), 35.  doi: 10.3934/naco.2011.1.35.  Google Scholar [6] X. Chen, C. Zhang and M. Fukushima, Robust solution of monotone stochastic linear complementarity problems,, Mathematical Programming, 117 (2009), 51.  doi: 10.1007/s10107-007-0163-z.  Google Scholar [7] F. H. Clarke, Optimization and Nonsmooth Analysis,, Wiley, (1983).   Google Scholar [8] M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems,, Mathematical Programming, 53 (1992), 99.  doi: 10.1007/BF01585696.  Google Scholar [9] F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems,, Springer, (2003).  doi: 10.1007/b97544.  Google Scholar [10] H. Fang, X. Chen and M. Fukushima, Stochastic $R_0$ matrix linear complementarity problems,, SIAM Journal on Optimization, 18 (2007), 482.  doi: 10.1137/050630805.  Google Scholar [11] P. T. Harker and J. S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications,, Mathematical Programming, 48 (1990), 161.  doi: 10.1007/BF01582255.  Google Scholar [12] W. W. Hogan, Energy policy models for project independence,, Computers and Operations Research, 2 (1975), 251.  doi: 10.1016/0305-0548(75)90008-8.  Google Scholar [13] H. Jiang and H. F. Xu, Stochastic approximation approaches to the stochastic variational inequality problem,, IEEE Transactions on Automatic Control, 53 (2008), 1462.  doi: 10.1109/TAC.2008.925853.  Google Scholar [14] D. Kinderlehre and G. Stampacchia, An Intruduction to Variational Inequalities and Their Aplications,, Academic Press, (1980).   Google Scholar [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), 641.  doi: 10.1080/02331930701617320.  Google Scholar [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), 455.   Google Scholar [17] G. H. Lin and M. Fukushima, New reformulations for stochastic nonlinear complementarity peoblems,, Optimization Methods and Software, 21 (2006), 551.  doi: 10.1080/10556780600627610.  Google Scholar [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), 456.  doi: 10.1016/j.orl.2008.01.010.  Google Scholar [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), 103.  doi: 10.1007/s10957-008-9439-6.  Google Scholar [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), 569.  doi: 10.1007/s10957-009-9534-3.  Google Scholar [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), 399.  doi: 10.1007/s10957-010-9676-3.  Google Scholar [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), 1.   Google Scholar [23] R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk,, Journal of Risk, 2 (2000), 493.   Google Scholar [24] A. Ruszczynski and A. Shapiro, Stochastic Programming,, Elsevier, (2003).   Google Scholar [25] A. Shapiro, Stochastic Programming by Monte Carlo Simulation Methods,, Stochastic Programming E-Print Series, (2000).   Google Scholar [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), 317.  doi: 10.3934/jimo.2011.7.317.  Google Scholar [27] D. De Wolf and Y. Smeers, A stochastic version of a Stackelberg-Nash-Cournot equilibrium model,, Management Science, 43 (1997), 190.   Google Scholar [28] H. Xu, Sample average approximation methods for a class of stochastic variational inequality problems,, Asia-Pacific Journal of Operational Research, 27 (2010), 103.  doi: 10.1142/S0217595910002569.  Google Scholar [29] H. Xu and D. Zhang, Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications,, Mathematical Programming, 119 (2009), 371.  doi: 10.1007/s10107-008-0214-0.  Google Scholar [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), 277.  doi: 10.1007/s10957-008-9358-6.  Google Scholar
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