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2011, 1(1): 35-48. doi: 10.3934/naco.2011.1.35

CVaR-based formulation and approximation method for stochastic variational inequalities

1. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

2. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Received  August 2010 Revised  November 2010 Published  February 2011

In this paper, we study the stochastic variational inequality problem (SVIP) from a viewpoint of minimization of conditional value-at-risk. We employ the D-gap residual function for VIPs to define a loss function for SVIPs. In order to reduce the risk of high losses in applications of SVIPs, we use the D-gap function and conditional value-at-risk to present a deterministic minimization reformulation for SVIPs. We show that the new reformulation is a convex program under suitable conditions. Furthermore, by using the smoothing techniques and the Monte Carlo methods, we propose a smoothing approximation method for finding a solution of the new reformulation and show that this method is globally convergent with probability one.
Citation: Xiaojun Chen, Guihua Lin. CVaR-based formulation and approximation method for stochastic variational inequalities. Numerical Algebra, Control and Optimization, 2011, 1 (1) : 35-48. doi: 10.3934/naco.2011.1.35
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]

P. Artzner, F. Delbaen, J. M. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228. doi: 10.1111/1467-9965.00068.

[3]

S. Boyd and L. Vandenberghe, "Convex Optimization," Cambridge University Press, Cambridge, 2004.

[4]

C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Computational Optimization and Applications, 5 (1996), 97-138. doi: 10.1007/BF00249052.

[5]

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.

[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]

R. W. Cottle, J. S. Pang and R. E. Stone, "The Linear Complementarity Problem," Academic Press, New York, 1992.

[8]

F. Facchinei and J. S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems," Springer-Verlag, New York, 2003.

[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]

M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Mathematical Programming, 53 (1992), 99-110. doi: 10.1007/BF01585696.

[11]

M. Fukushima, Merit functions for variational inequality and complementarity problems, in "Nonlinear Optimization and Applications"(eds. G. Di Pillo and F. Giannessi), Plenum Press, New York, 1996, 155-170.

[12]

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.

[13]

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.

[14]

W. W. Hogan, Point-to-set maps in mathematical programming, SIAM Review, 15 (1973), 591-603. doi: 10.1137/1015073.

[15]

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.

[16]

I. V. Konnov, "Equilibrium Models and Variational Inequalities," Elsevier, Amsterdam, 2007.

[17]

D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications," Academic Press, New York, 1980.

[18]

T. Larsson and M. Patriksson, A class of gap functions for variational inequalities, Mathematical Programming, 64 (1994), 53-79. doi: 10.1007/BF01582565.

[19]

G. H. Lin, Combined Monte Carlo sampling and penalty method for stochastic nonlinear complementarity problems, Mathematics of Computation, 78 (2009), 1671-1686. doi: 10.1090/S0025-5718-09-02206-6.

[20]

G. H. Lin, X. Chen and M. Fukushima, New restricted NCP function and their applications to stochastic NCP and stochastic MPEC, Optimization, 56 (2007), 641-753. doi: 10.1080/02331930701617320.

[21]

G. H. Lin and M. Fukushima, New reformulations for stochastic complementarity problems, Optimization Methods and Software, 21 (2006), 551-564. doi: 10.1080/10556780600627610.

[22]

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.

[23]

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.

[24]

M. J. Luo and G. H. Lin, Expected residual minimization method for stochastic variational inequality problems, Journal of Optimization Theory and Application, 140 (2009), 103-116. doi: 10.1007/s10957-008-9439-6.

[25]

M. J. Luo and G. H. Lin, Convergence results of the ERM method for nonlinear stochastic variational inequality problems, Journal of Optimization Theory and Application, 142 (2009), 569-581. doi: 10.1007/s10957-009-9534-3.

[26]

M. J. Luo and G. H. Lin, Stochastic variational inequality problems with additional constraints and their applications in supply chain network equilibria,, Pacific Journal of Optimization, (). 

[27]

J. S. Pang, Complementarity problems, in "Handbook in Global Optimization"}(eds. R. Horst and P. Pardalos, Kluwer Academic Publishers, Boston, 1994.

[28]

J. M. Peng, Convexity of the implicit Lagrangian, Journal of Optimization Theory and Applications, 92 (1997), 331-341. doi: 10.1023/A:1022607213765.

[29]

R. T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distributions, Journal of Banking and Finance, 26 (2002), 1443-1471. doi: 10.1016/S0378-4266(02)00271-6.

[30]

R. T. Rockafellar and R. J. B. Wets, "Variational Analysis," Springer, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.

[31]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 493-517.

[32]

A. Ruszczynski and A. Shapiro, "Stochastic Programming, Handbooks in Operations Research and Management Science," Elsevier, 2003.

[33]

H. Xu, Sample average approximation methods for a class of stochastic variational inequality problems, Asia-Pacific Journal of Operations Research, 27 (2010), 103-119.

[34]

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.

[35]

N. Yamashita, K. Taji and M. Fukushima, Unconstrained optimization reformulations of variational inequality problems, Journal of Optimization Theory and Applications, 92 (1997), 439-456. doi: 10.1023/A:1022660704427.

[36]

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.

[37]

C. Zhang, X. Chen and A. Sumalee, Robust Wardrops user equilibrium assignment under stochastic demand and supply: expected residual minimization approach, Transportation Research Part B, 2010 (Online first).

[38]

G. L. Zhou and L. Caccetta, Feasible semismooth Newton method for a class of stochastic linear complementarity problems, Journal of Optimization Theory and Applications, 139 (2008), 379-392. doi: 10.1007/s10957-008-9406-2.

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]

P. Artzner, F. Delbaen, J. M. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228. doi: 10.1111/1467-9965.00068.

[3]

S. Boyd and L. Vandenberghe, "Convex Optimization," Cambridge University Press, Cambridge, 2004.

[4]

C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Computational Optimization and Applications, 5 (1996), 97-138. doi: 10.1007/BF00249052.

[5]

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.

[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]

R. W. Cottle, J. S. Pang and R. E. Stone, "The Linear Complementarity Problem," Academic Press, New York, 1992.

[8]

F. Facchinei and J. S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems," Springer-Verlag, New York, 2003.

[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]

M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Mathematical Programming, 53 (1992), 99-110. doi: 10.1007/BF01585696.

[11]

M. Fukushima, Merit functions for variational inequality and complementarity problems, in "Nonlinear Optimization and Applications"(eds. G. Di Pillo and F. Giannessi), Plenum Press, New York, 1996, 155-170.

[12]

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.

[13]

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.

[14]

W. W. Hogan, Point-to-set maps in mathematical programming, SIAM Review, 15 (1973), 591-603. doi: 10.1137/1015073.

[15]

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.

[16]

I. V. Konnov, "Equilibrium Models and Variational Inequalities," Elsevier, Amsterdam, 2007.

[17]

D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications," Academic Press, New York, 1980.

[18]

T. Larsson and M. Patriksson, A class of gap functions for variational inequalities, Mathematical Programming, 64 (1994), 53-79. doi: 10.1007/BF01582565.

[19]

G. H. Lin, Combined Monte Carlo sampling and penalty method for stochastic nonlinear complementarity problems, Mathematics of Computation, 78 (2009), 1671-1686. doi: 10.1090/S0025-5718-09-02206-6.

[20]

G. H. Lin, X. Chen and M. Fukushima, New restricted NCP function and their applications to stochastic NCP and stochastic MPEC, Optimization, 56 (2007), 641-753. doi: 10.1080/02331930701617320.

[21]

G. H. Lin and M. Fukushima, New reformulations for stochastic complementarity problems, Optimization Methods and Software, 21 (2006), 551-564. doi: 10.1080/10556780600627610.

[22]

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.

[23]

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.

[24]

M. J. Luo and G. H. Lin, Expected residual minimization method for stochastic variational inequality problems, Journal of Optimization Theory and Application, 140 (2009), 103-116. doi: 10.1007/s10957-008-9439-6.

[25]

M. J. Luo and G. H. Lin, Convergence results of the ERM method for nonlinear stochastic variational inequality problems, Journal of Optimization Theory and Application, 142 (2009), 569-581. doi: 10.1007/s10957-009-9534-3.

[26]

M. J. Luo and G. H. Lin, Stochastic variational inequality problems with additional constraints and their applications in supply chain network equilibria,, Pacific Journal of Optimization, (). 

[27]

J. S. Pang, Complementarity problems, in "Handbook in Global Optimization"}(eds. R. Horst and P. Pardalos, Kluwer Academic Publishers, Boston, 1994.

[28]

J. M. Peng, Convexity of the implicit Lagrangian, Journal of Optimization Theory and Applications, 92 (1997), 331-341. doi: 10.1023/A:1022607213765.

[29]

R. T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distributions, Journal of Banking and Finance, 26 (2002), 1443-1471. doi: 10.1016/S0378-4266(02)00271-6.

[30]

R. T. Rockafellar and R. J. B. Wets, "Variational Analysis," Springer, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.

[31]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 493-517.

[32]

A. Ruszczynski and A. Shapiro, "Stochastic Programming, Handbooks in Operations Research and Management Science," Elsevier, 2003.

[33]

H. Xu, Sample average approximation methods for a class of stochastic variational inequality problems, Asia-Pacific Journal of Operations Research, 27 (2010), 103-119.

[34]

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.

[35]

N. Yamashita, K. Taji and M. Fukushima, Unconstrained optimization reformulations of variational inequality problems, Journal of Optimization Theory and Applications, 92 (1997), 439-456. doi: 10.1023/A:1022660704427.

[36]

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.

[37]

C. Zhang, X. Chen and A. Sumalee, Robust Wardrops user equilibrium assignment under stochastic demand and supply: expected residual minimization approach, Transportation Research Part B, 2010 (Online first).

[38]

G. L. Zhou and L. Caccetta, Feasible semismooth Newton method for a class of stochastic linear complementarity problems, Journal of Optimization Theory and Applications, 139 (2008), 379-392. doi: 10.1007/s10957-008-9406-2.

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