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Convergence Analysis of Sample Average Approximation Methods for a Class of Stochastic Mathematical Programs with Equality Constraints

School of Mathematics, University of Southampton, Highfield S017 1BJ, Southampton, United Kingdom
The Logistics Institute—Asia Pacific, National University of Singapore, 7 Engineering Drive 1, Singapore 117543
h.xu{at}soton.ac.uk, http://www.maths.soton.ac.uk/staff/Xu/
tlimf{at}nus.edu.sg

In this paper we discuss the sample average approximation (SAA) method for a class of stochastic programs with nonsmooth equality constraints. We derive a uniform Strong Law of Large Numbers for random compact set-valued mappings and use it to investigate the convergence of Karush-Kuhn-Tucker points of SAA programs as the sample size increases. We also study the exponential convergence of global minimizers of the SAA problems to their counterparts of the true problem. The convergence analysis is extended to a smoothed SAA program. Finally, we apply the established results to a class of stochastic mathematical programs with complementarity constraints and report some preliminary numerical test results.

Key Words: sample average approximations; strong law of large numbers; random set-valued mappings; stationary points
History: Received: April 11, 2005; revision received: May 15, 2006;