zbMATH — the first resource for mathematics

Expected residual minimization method for stochastic variational inequality problems. (English) Zbl 1190.90112
Summary: 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.

90C15 Stochastic programming
Full Text: DOI
[1] Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003) · Zbl 1062.90002
[2] 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
[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) · Zbl 0996.90082
[4] 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
[5] 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
[6] De Wolf, D., Smeers, Y.: A stochastic version of a Stackelberg-Nash-Cournot equilibrium model. Manag. Sci. 43, 190–197 (1997) · Zbl 0889.90046 · doi:10.1287/mnsc.43.2.190
[7] 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
[8] Gürkan, G., Özge, A.Y., Robinson, S.M.: Sample-path solution of stochastic variational inequalities. Math. Program. 84, 313–333 (1999) · Zbl 0972.90079 · doi:10.1007/s101070050024
[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) · Zbl 1172.90455 · doi:10.1080/02331930701617320
[10] Lin, G.H., Fukushima, M.: New reformulations for stochastic nonlinear complementarity problems. Optim. Methods Softw. 21, 551–564 (2006) · Zbl 1113.90110 · doi:10.1080/10556780600627610
[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) · Zbl 1155.90461 · doi:10.1016/j.orl.2008.01.010
[12] 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
[13] Patrick, B.: Probability and Measure. Wiley-Interscience, New York (1995) · Zbl 0822.60002
[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) · Zbl 0734.90098 · doi:10.1007/BF01582255
[15] Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992) · Zbl 0761.65002
[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.