Closed-form sampling laws for stochastically constrained simulation optimization on large finite sets

Consider the context of constrained simulation optimization (SO), that is, optimization problems where the objective function and constraints are known through a Monte Carlo simulation, with corresponding estimators possibly dependent. We identify the nature of sampling plans that characterize efficient algorithms, particularly in large countable spaces. We show that in a certain asymptotic sense, the optimal sampling characterization, that is, the sampling budget for each system that guarantees optimal convergence rates, depends on a single easily estimable quantity called the score. This result provides a useful and easily implementable sampling allocation that approximates the optimal allocation, which is otherwise intractable due to it being the solution to a difficult bilevel optimization problem. Our results point to a simple sequential algorithm for efficiently solving large-scale constrained simulation optimization problems on finite sets.