In this paper we study a Monte Carlo simulation-based approach to stochastic discrete optimization problems. The basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function. The obtained sample average optimization problem is solved, and the procedure is repeated several times until a stopping criterion is satisfied. We discuss convergence rates, stopping rules, and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem.

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In this paper we discuss Monte Carlo simulation based approximations of a stochastic programming problem. We show that if the corresponding random functions are convex piecewise smooth and the distribution is discrete, then (under mild additional assumptions) an optimal solution of the approximating problem provides an exact optimal solution of the true problem with probability one for sufficiently large sample size. Moreover, by using theory of Large Deviations, we show that the probability of such an event approaches one exponentially fast with increase of the sample size. In particular, this happens in the case of two stage stochastic programming with recourse if the corresponding distributions are discrete. The obtained results suggest that, in such cases, Monte Carlo simulation based methods could be very efficient. We present some numerical examples to illustrate the involved ideas.

In this article 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 pathwisely, we are able to obtain our results in a flexible setting, which requires mild assumptions on the distributions and which includes the possibility of using different sampling distributions along the algorithm. We illustrate these ideas by studying a modification of the well-known pure random search method, adapting it to the variable-sample scheme, and show conditions for convergence of the algorithm. Implementation issues are discussed and numerical results are presented to illustrate the ideas.

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