Monte Carlo Methods for Discrete Stochastic Optimization

In this paper we discuss the application of Monte Carlo techniques to discrete stochastic optimization problems. Particularly, we study two approaches: sample path methods, where the expectation in the objective function is replaced by a sample average approximation and the resulting deterministic problem is solved, and variable-sample techniques, in which the objective function is replaced, at each iteration, by a sample average approximation. For the former approach, we discuss convergence results based on large deviations theory and use those results to estimate a sample size that is sufficiently large to ensure that an e-optimal solution is obtained with specified probability. For the second approach — variable-sample techniques — we provide general results under which this type of method yields consistent estimators as well as bounds on the estimation error. Finally, we discuss an applications of this technique to a particular algorithm, the so-called simulated annealing method.