Penalized Sample Average Approximation Methods for Stochastic Mathematical Programs with Complementarity Constraints

Authors:
Yongchao Liu;Huifu Xu;Jane J. Ye
Affiliations:
Department of Mathematics, Dalian Maritime University, 116026 Dalian, China;School of Mathematics, University of Southampton, Southampton SO17 1BJ, United Kingdom;Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
Venue:
Mathematics of Operations Research
Year:
2011

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Abstract

This paper considers a one-stage stochastic mathematical program with a complementarity constraint (SMPCC), where uncertainties appear in both the objective function and the complementarity constraint, and an optimal decision on both upper-and lower-level decision variables must be made before the realization of the uncertainties. A partially exactly penalized sample average approximation (SAA) scheme is proposed to solve the problem. Asymptotic convergence of optimal solutions and stationary points of the penalized SAA problem is carried out. It is shown under some moderate conditions that the statistical estimators obtained from solving the penalized SAA problems converge almost surely to its true counterpart as the sample size increases. Exponential rate of convergence of estimators is also established under some additional conditions.