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Convergence results of the ERM method for nonlinear stochastic variational inequality problems. (English) Zbl 1187.90295
The authors consider the expected residual minimization (ERM) method proposed by {\it M. J. Luo} and {\it G. H. Lin} [J. Optim. Theory Appl. 140, 103--116 (2009; Zbl 1190.90112)] and continue to study the proposed method for a stochastic variational inequality problem. The function involved is assumed to be nonlinear in this paper. The authors 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, a compact approximation approach is presented for the case where the sample space is noncompact.

90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
Full Text: DOI
[1] Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003) · Zbl 1062.90002
[2] Chen, X., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30, 1022--1038 (2005) · Zbl 1162.90527 · doi:10.1287/moor.1050.0160
[3] Chen, X., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117, 51--80 (2009) · Zbl 1165.90012 · doi:10.1007/s10107-007-0163-z
[4] Fang, H., Chen, X., Fukushima, M.: Stochastic R 0 matrix linear complementarity problems. SIAM J. Optim. 18, 482--506 (2007) · Zbl 1151.90052 · doi:10.1137/050630805
[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) · Zbl 1172.90455 · doi:10.1080/02331930701617320
[6] Lin, G.H., Fukushima, M.: New reformulations for stochastic complementarity problems. Optim. Methods Softw. 21, 551--564 (2006) · Zbl 1113.90110 · doi:10.1080/10556780600627610
[7] Luo, M.J., Lin, G.H.: Expected residual minimization method for stochastic variational inequality problems. J. Optim. Theory Appl. 140, 103--116 (2009) · Zbl 1190.90112 · doi:10.1007/s10957-008-9439-6
[8] Zhang, C., Chen, X.: Stochastic nonlinear complementarity problem and applications to traffic equilibrium under uncertainty. J. Optim. Theory Appl. 137, 277--295 (2008) · Zbl 1163.90034 · doi:10.1007/s10957-008-9358-6
[9] Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53, 99--110 (1992) · Zbl 0756.90081 · doi:10.1007/BF01585696
[10] Patrick, B.: Probability and Measure. A Wiley-Interscience Publication. Wiley, New York (1995)
[11] Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992) · Zbl 0761.65002