Expected Residual Minimization Method for Stochastic Variational Inequality Problems



This paper considers a stochastic variational inequality problem (SVIP). We first formulate SVIP as an optimization problem (ERM problem) that minimizes the expected residual of the so-called regularized gap function. Then, we focus on a SVIP subclass in which the function involved is assumed to be affine. We study the properties of the ERM problem and propose a quasi-Monte Carlo method for solving the problem. Comprehensive convergence analysis is included as well.


Stochastic variational inequalities Level sets Quasi-Monte Carlo methods Convergence 


  1. 1.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003) Google Scholar
  2. 2.
    Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53, 99–110 (1992) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Fukushima, M.: Merit functions for variational inequality and complementarity problems. In: Di Pillo, G., Giannessi, F. (eds.) Nonlinear Optimization and Applications, pp. 155–170. Plenum, New York (1996) Google Scholar
  4. 4.
    Chen, X., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30, 1022–1038 (2005) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chen, X., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117, 51–80 (2009) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    De Wolf, D., Smeers, Y.: A stochastic version of a Stackelberg-Nash-Cournot equilibrium model. Manag. Sci. 43, 190–197 (1997) MATHCrossRefGoogle Scholar
  7. 7.
    Fang, H., Chen, X., Fukushima, M.: Stochastic R 0 matrix linear complementarity problems. SIAM J. Optim. 18, 482–506 (2007) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gürkan, G., Özge, A.Y., Robinson, S.M.: Sample-path solution of stochastic variational inequalities. Math. Program. 84, 313–333 (1999) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Lin, G.H., Chen, X., Fukushima, M.: New restricted NCP function and their applications to stochastic NCP and stochastic MPEC. Optimization 56, 641–753 (2007) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lin, G.H., Fukushima, M.: New reformulations for stochastic nonlinear complementarity problems. Optim. Methods Softw. 21, 551–564 (2006) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Ling, C., Qi, L., Zhou, G., Caccetta, L.: The SC’ property of an expected residual function arising from stochastic complementarity problems. Oper. Res. Lett. 36, 456–460 (2008) CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Zhang, C., Chen, X.: Stochastic nonlinear complementarity problem and applications to traffic equilibrium under uncertainty. J. Optim. Theory Appl. 137, 277–295 (2008) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Patrick, B.: Probability and Measure. Wiley-Interscience, New York (1995) MATHGoogle Scholar
  14. 14.
    Harker, P.T., Pang, J.S.: Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Math. Program. 48, 161–220 (1990) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992) MATHGoogle Scholar
  16. 16.
    Birge, J.R.: Quasi-Monte Carlo approaches to option pricing. Technical Report 94-19, Department of Industrial and Operations Engineering, University of Michigan (1994) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Applied MathematicsDalian University of TechnologyDalianChina

Personalised recommendations