Simulation-based optimization – convergence analysis and statistical inference

In this paper we consider optimization problems where the objective function is not given explicitly and should be estimated, say by a Monte Carlo simulation. We discuss convergence properties and statistical inference of the so-called ‘stochastic counterpart’ approach to such simulation-based optimization. This method is based on generation of a large sample and subsequent solution of a constructed approximation of the ‘true’ program by deterministic methods of nonlinear programming. The required ‘stochastic counterpart’ approximation can be constructed, for example, by the likelihood ratio-score function techniques. We discuss relative efficiency of the stochastic counterpart and stochastic approximation methods and show that for smooth unconstrained problems both methods produce asymptotically equivalent estimators provided the stochastic approximation estimators are calculated for an optimal choice of the corresponding stepsizes. We also discuss a few examples of nonsmooth and constrained problems and argue that in such cases the stochastic counterpart and stochastic approximation estimators may converge at asymptotically different rates.