Sampling Laws for Stochastically Constrained Simulation Optimization on Finite Sets

Consider the context of selecting an optimal system from among a finite set of competing systems, based on a "stochastic" objective function and subject to multiple "stochastic" constraints. In this context, we characterize the asymptotically optimal sample allocation that maximizes the rate at which the probability of false selection tends to zero in two scenarios: first in the context of general light-tailed distributions, and second in the specific context in which the objective function and constraints may be observed together as multivariate normal random variates.

In the context of general light-tailed distributions, we present the optimal allocation as the result of a concave maximization problem for which the optimal solution is the result of solving one of two nonlinear systems of equations. The first result of its kind, the optimal allocation is particularly easy to obtain in contexts where the underlying distributions are known or can be assumed, e.g., normal, Bernoulli. A consistent estimator for the optimal allocation and a corresponding sequential algorithm for implementation are provided. Various numerical examples demonstrate where and to what extent the proposed allocation differs from competing algorithms.

In the context of multivariate normal distributions, we present an exact, asymptotically optimal allocation. This allocation is the result of a concave maximization problem in which there are at least as many constraints as there are suboptimal systems. Each constraint corresponding to a suboptimal system is a convex optimization problem. Thus the optimal allocation may easily be obtained in the context of a "small" number of systems, where the quantifier "small" depends on the available computing resources. A consistent estimator for the optimal allocation and a fully sequential algorithm, fit for implementation, are provided. The sequential algorithm performs significantly better than equal allocation in finite time across a variety of randomly generated problems.

The results presented in the general and multivariate normal context provide the first foundation of exact asymptotically optimal sampling methods in the context of "stochastically" constrained simulation optimization on finite sets. Particularly, the general optimal allocation model is likely to be most useful when correlation between the objective and constraint estimators is low, but the data are non-normal. The multivariate normal optimal allocation model is likely to be useful when the multivariate normal assumption is reasonable or the correlation is high.