Variable-Sample Methods and Simulated Annealing for Discrete Stochastic Optimization

In this paper we discuss the application of a certain class of Monte Carlo methods to stochastic optimization problems. Particularly, we study variable-sample techniques, in which the objective function is replaced, at each iteration, by a sample average approximation. We first provide general results on the schedule of sample sizes, under which variable-sample methods yield consistent estimators as well as bounds on the estimation error. Because the convergence analysis is performed sample-path wise, we are able to obtain our results in a flexible setting, which includes the possibility of using different sampling distributions along the algorithm, without making strong assumptions on the underlying distributions. In particular, we allow the distributions to depend on the decision variables x. We illustrate these ideas by studying a modification of the wellknown simulated annealing method, adapting it to the variable-sample scheme, and show conditions for convergence of the algorithm.