## Properties of expected residual minimization model for a class of stochastic complementarity problems.(English)Zbl 1266.65103

Summary: Expected residual minimization (ERM) model which minimizes an expected residual function defined by an NCP function has been studied in the literature for solving stochastic complementarity problems. In this paper, we first give the definitions of stochastic $$P$$-function, stochastic $$P_0$$-function, and stochastic uniformly $$P$$-function. Furthermore, the conditions such that the function is a stochastic $$P(P_0)$$-function are considered. We then study the boundedness of solution set and global error bounds of the expected residual functions defined by the “Fischer-Burmeister” (FB) function and “min” function. The conclusion indicates that solutions of the ERM model are robust in the sense that they may have a minimum sensitivity with respect to random parameter variations in stochastic complementarity problems. On the other hand, we employ quasi-Monte Carlo methods and derivative-free methods to solve ERM model.

### MSC:

 65K10 Numerical optimization and variational techniques
Full Text:

### References:

 [1] F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, NY, USA, 2003. · Zbl 1062.90002 [2] R. W. Cottle, J.-S. Pang, and R. E. Stone, The Linear Complementarity Problem, Computer Science and Scientific Computing, Academic Press, Boston, Mass, USA, 1992. · Zbl 0795.90071 [3] G.-H. Lin and M. Fukushima, “New reformulations for stochastic nonlinear complementarity problems,” Optimization Methods & Software, vol. 21, no. 4, pp. 551-564, 2006. · Zbl 1113.90110 [4] G. Gürkan, A. Y. Özge, and S. M. Robinson, “Sample-path solution of stochastic variational inequalities,” Mathematical Programming, vol. 84, no. 2, pp. 313-333, 1999. · Zbl 0972.90079 [5] X. Chen and M. Fukushima, “Expected residual minimization method for stochastic linear complementarity problems,” Mathematics of Operations Research, vol. 30, no. 4, pp. 1022-1038, 2005. · Zbl 1162.90527 [6] X. Chen, C. Zhang, and M. Fukushima, “Robust solution of monotone stochastic linear complementarity problems,” Mathematical Programming, vol. 117, no. 1-2, pp. 51-80, 2009. · Zbl 1165.90012 [7] P. Tseng, “Growth behavior of a class of merit functions for the nonlinear complementarity problem,” Journal of Optimization Theory and Applications, vol. 89, no. 1, pp. 17-37, 1996. · Zbl 0866.90127 [8] H. Fang, X. Chen, and M. Fukushima, “Stochastic R0 matrix linear complementarity problems,” SIAM Journal on Optimization, vol. 18, no. 2, pp. 482-506, 2007. · Zbl 1151.90052 [9] G.-H. Lin, X. Chen, and M. Fukushima, “New restricted NCP functions and their applications to stochastic NCP and stochastic MPEC,” Optimization, vol. 56, no. 5-6, pp. 641-953, 2007. · Zbl 1172.90455 [10] C. Ling, L. Qi, G. Zhou, and L. Caccetta, “The SC1 property of an expected residual function arising from stochastic complementarity problems,” Operations Research Letters, vol. 36, no. 4, pp. 456-460, 2008. · Zbl 1155.90461 [11] C. Zhang and X. Chen, “Stochastic nonlinear complementarity problem and applications to traffic equilibrium under uncertainty,” Journal of Optimization Theory and Applications, vol. 137, no. 2, pp. 277-295, 2008. · Zbl 1163.90034 [12] C. Zhang and X. Chen, “Smoothing projected gradient method and its application to stochastic linear complementarity problems,” SIAM Journal on Optimization, vol. 20, no. 2, pp. 627-649, 2009. · Zbl 1204.65073 [13] G. L. Zhou and L. Caccetta, “Feasible semismooth Newton method for a class of stochastic linear complementarity problems,” Journal of Optimization Theory and Applications, vol. 139, no. 2, pp. 379-392, 2008. · Zbl 1191.90085 [14] X. L. Li, H. W. Liu, and Y. K. Huang, “Stochastic P matrix and P0 matrix linear complementarity problem,” Journal of Systems Science and Mathematical Sciences, vol. 31, no. 1, pp. 123-128, 2011. · Zbl 1249.15041 [15] K. L. Chung, A Course in Probability Theory, Academic Press, New York, NY, USA, 2nd edition, 1974. · Zbl 0345.60003 [16] B. Chen and P. T. Harker, “Smooth approximations to nonlinear complementarity problems,” SIAM Journal on Optimization, vol. 7, no. 2, pp. 403-420, 1997. · Zbl 0879.90177
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.