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Stochastic Mathematical Programs with Equilibrium Constraints, Modeling and Sample Average Approximation

Alexander Shapiro (ashapiro***at***isye.gatech.edu)
Huifu Xu (h.xu***at***maths.soton.ac.uk)

Abstract: In this paper, we discuss the sample average approximation (SAA) method applied to a class of stochastic mathematical programs with variational (equilibrium) constraints. To this end, we briefly investigate piecewise structure and directional differentiability of both -- the lower level equilibrium solution and objective integrant. We show almost sure convergence of optimal values, optimal solutions (both local and global) and generalized Karush-Kohn-Tucker points of the SAA program to their true counterparts. We also study uniform exponential convergence of the sample average approximations, and as a consequence derive estimates of the sample size required to solve the true problem with a given accuracy. Finally we present some preliminary numerical test results.

Keywords: Stochastic programming, equilibrium constraints, Stackelberg-Nash-Cournot Equilibrium, variational inequality, sample average approximation, exponential convergence, smoothing.

Category 1: Complementarity and Variational Inequalities

Category 2: Stochastic Programming

Citation: Preprint, School of Industrial and Systems Engineering, Georgia Institute of Technology, Antalanta, Georgia 30332-0205, USA

Download: [Postscript][PDF]

Entry Submitted: 01/17/2005
Entry Accepted: 01/17/2005
Entry Last Modified: 06/25/2005

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