Monte Carlo sampling-based methods for stochastic optimization

Abstract

This paper surveys the use of Monte Carlo sampling-based methods for stochastic optimization problems. Such methods are required when—as it often happens in practice—the model involves quantities such as expectations and probabilities that cannot be evaluated exactly. While estimation procedures via sampling are well studied in statistics, the use of such methods in an optimization context creates new challenges such as ensuring convergence of optimal solutions and optimal values, testing optimality conditions, choosing appropriate sample sizes to balance the effort between optimization and estimation, and many other issues. Much work has been done in the literature to address these questions. The purpose of this paper is to give an overview of some of that work, with the goal of introducing the topic to students and researchers and providing a practical guide for someone who needs to solve a stochastic optimization problem with sampling.