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

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Abstract

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.

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References

  1. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)

    Google Scholar 

  2. Chen, X., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30, 1022–1038 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen, X., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117, 51–80 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Fang, H., Chen, X., Fukushima, M.: Stochastic R 0 matrix linear complementarity problems. SIAM J. Optim. 18, 482–506 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  6. Lin, G.H., Fukushima, M.: New reformulations for stochastic complementarity problems. Optim. Methods Softw. 21, 551–564 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Luo, M.J., Lin, G.H.: Expected residual minimization method for stochastic variational inequality problems. J. Optim. Theory Appl. 140, 103–116 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Zhang, C., Chen, X.: Stochastic nonlinear complementarity problem and applications to traffic equilibrium under uncertainty. J. Optim. Theory Appl. 137, 277–295 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53, 99–110 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  10. Patrick, B.: Probability and Measure. A Wiley-Interscience Publication. Wiley, New York (1995)

    MATH  Google Scholar 

  11. Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Dalian University of Technology, Dalian, 116024, China

    M. J. Luo & G. H. Lin

Authors
  1. M. J. Luo
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  2. G. H. Lin
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Corresponding author

Correspondence to G. H. Lin.

Additional information

Communicated by M. Fukushima.

This work was supported in part by Project 10771025 supported by NSFC and SRFDP 20070141063 of China.

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Luo, M.J., Lin, G.H. Convergence Results of the ERM Method for Nonlinear Stochastic Variational Inequality Problems. J Optim Theory Appl 142, 569–581 (2009). https://doi.org/10.1007/s10957-009-9534-3

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  • DOI: https://doi.org/10.1007/s10957-009-9534-3

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