Journal of Optimization Theory and Applications

, Volume 142, Issue 3, pp 569–581 | Cite as

Convergence Results of the ERM Method for Nonlinear Stochastic Variational Inequality Problems



This paper considers the expected residual minimization (ERM) method proposed by Luo and Lin (J. Optim. Theory Appl. 140:103–116, 2009) for a class of stochastic variational inequality problems. Different from the work mentioned above, the function involved is assumed to be nonlinear in this paper. We first consider a quasi-Monte Carlo method for the case where the underlying sample space is compact and show that the ERM method is convergent under very mild conditions. Then, we suggest a compact approximation approach for the case where the sample space is noncompact.


Stochastic variational inequalities Residual functions Quasi-Monte Carlo methods Compact approximations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003) Google Scholar
  2. 2.
    Chen, X., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30, 1022–1038 (2005) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chen, X., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117, 51–80 (2009) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Fang, H., Chen, X., Fukushima, M.: Stochastic R 0 matrix linear complementarity problems. SIAM J. Optim. 18, 482–506 (2007) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    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
  6. 6.
    Lin, G.H., Fukushima, M.: New reformulations for stochastic complementarity problems. Optim. Methods Softw. 21, 551–564 (2006) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Luo, M.J., Lin, G.H.: Expected residual minimization method for stochastic variational inequality problems. J. Optim. Theory Appl. 140, 103–116 (2009) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    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
  9. 9.
    Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53, 99–110 (1992) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Patrick, B.: Probability and Measure. A Wiley-Interscience Publication. Wiley, New York (1995) MATHGoogle Scholar
  11. 11.
    Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Applied MathematicsDalian University of TechnologyDalianChina

Personalised recommendations