Neural network smoothing approximation method for stochastic variational inequality problems

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

Neural network smoothing approximation method for stochastic variational inequality problems

Hui-Qiang Ma ^1, and Nan-Jing Huang ^2,

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

Abstract
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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.

Keywords: regularized gap function, neural network smoothing approximation, Stochastic variational inequality, Monte Carlo sampling approximation, convergence., mixed conditional value-at-risk.

Mathematics Subject Classification: Primary: 90C33, 90C15; Secondary: 91B3.

Citation: Hui-Qiang Ma, Nan-Jing Huang. Neural network smoothing approximation method for stochastic variational inequality problems. Journal of Industrial and 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), 3-19.
[2]	R. J. Aumann, Integrals of set-value function, Journal of Mathematical Analysis and Applications, 12 (1965), 1-12. doi: 10.1016/0022-247X(65)90049-1.
[3]	B. T. Chen and P. T. Harker, Smooth approximations to nonlinear complementarity problems, SIAM Journal on Optimization, 7 (1997), 403-420. 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), 1022-1038. doi: 10.1287/moor.1050.0160.
[5]	X. Chen and G. H. Lin, CVaR-based formulation and approximation method for Stochastic variational inequalities, Numerical Algebra, Control and Optimization, 1 (2011), 35-48. 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), 51-80. doi: 10.1007/s10107-007-0163-z.
[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), 99-110. doi: 10.1007/BF01585696.
[9]	F. Facchinei and J. S. Pang, Finite-Dimensional 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), 482-506. doi: 10.1137/050630805.
[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), 161-220. doi: 10.1007/BF01582255.
[12]	W. W. Hogan, Energy policy models for project independence, Computers and Operations Research, 2 (1975), 251-271. doi: 10.1016/0305-0548(75)90008-8.
[13]	H. Jiang and H. F. 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.
[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), 641-953. 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), 455-482.
[17]	G. H. Lin and M. Fukushima, New reformulations for stochastic nonlinear complementarity peoblems, Optimization Methods and Software, 21 (2006), 551-564. 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), 456-460. 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), 103-116. doi: 10.1007/s10957-008-9439-6.
[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), 569-581. doi: 10.1007/s10957-009-9534-3.
[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), 399-418. doi: 10.1007/s10957-010-9676-3.
[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), 1-35.
[23]	R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 493-517.
[24]	A. Ruszczynski and A. Shapiro, Stochastic Programming, Elsevier, Amsterdam, 2003.
[25]	A. Shapiro, Stochastic Programming by Monte Carlo Simulation Methods, Stochastic Programming E-Print 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), 317-345. doi: 10.3934/jimo.2011.7.317.
[27]	D. De Wolf and Y. Smeers, A stochastic version of a Stackelberg-Nash-Cournot equilibrium model, Management Science, 43 (1997), 190-197.
[28]	H. Xu, Sample average approximation methods for a class of stochastic variational inequality problems, Asia-Pacific Journal of Operational Research, 27 (2010), 103-119. 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), 371-401. doi: 10.1007/s10107-008-0214-0.
[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), 277-295. doi: 10.1007/s10957-008-9358-6.

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), 3-19.
[2]	R. J. Aumann, Integrals of set-value function, Journal of Mathematical Analysis and Applications, 12 (1965), 1-12. doi: 10.1016/0022-247X(65)90049-1.
[3]	B. T. Chen and P. T. Harker, Smooth approximations to nonlinear complementarity problems, SIAM Journal on Optimization, 7 (1997), 403-420. 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), 1022-1038. doi: 10.1287/moor.1050.0160.
[5]	X. Chen and G. H. Lin, CVaR-based formulation and approximation method for Stochastic variational inequalities, Numerical Algebra, Control and Optimization, 1 (2011), 35-48. 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), 51-80. doi: 10.1007/s10107-007-0163-z.
[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), 99-110. doi: 10.1007/BF01585696.
[9]	F. Facchinei and J. S. Pang, Finite-Dimensional 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), 482-506. doi: 10.1137/050630805.
[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), 161-220. doi: 10.1007/BF01582255.
[12]	W. W. Hogan, Energy policy models for project independence, Computers and Operations Research, 2 (1975), 251-271. doi: 10.1016/0305-0548(75)90008-8.
[13]	H. Jiang and H. F. 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.
[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), 641-953. 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), 455-482.
[17]	G. H. Lin and M. Fukushima, New reformulations for stochastic nonlinear complementarity peoblems, Optimization Methods and Software, 21 (2006), 551-564. 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), 456-460. 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), 103-116. doi: 10.1007/s10957-008-9439-6.
[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), 569-581. doi: 10.1007/s10957-009-9534-3.
[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), 399-418. doi: 10.1007/s10957-010-9676-3.
[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), 1-35.
[23]	R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 493-517.
[24]	A. Ruszczynski and A. Shapiro, Stochastic Programming, Elsevier, Amsterdam, 2003.
[25]	A. Shapiro, Stochastic Programming by Monte Carlo Simulation Methods, Stochastic Programming E-Print 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), 317-345. doi: 10.3934/jimo.2011.7.317.
[27]	D. De Wolf and Y. Smeers, A stochastic version of a Stackelberg-Nash-Cournot equilibrium model, Management Science, 43 (1997), 190-197.
[28]	H. Xu, Sample average approximation methods for a class of stochastic variational inequality problems, Asia-Pacific Journal of Operational Research, 27 (2010), 103-119. 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), 371-401. doi: 10.1007/s10107-008-0214-0.
[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), 277-295. doi: 10.1007/s10957-008-9358-6.

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