No abstract

The stochastic root-finding problem is that of finding a zero of a vector-valued function known only through a stochastic simulation. The simulation-optimization problem is that of locating a real-valued function's minimum, again with only a stochastic simulation that generates function estimates. Retrospective approximation (RA) is a sample-path technique for solving such problems, where the solution to the underlying problem is approached via solutions to a sequence of approximate deterministic problems, each of which is generated using a specified sample size, and solved to a specified error tolerance. Our primary focus, in this paper, is providing guidance on choosing the sequence of sample sizes and error tolerances in RA algorithms. We first present an overview of the conditions that guarantee the correct convergence of RA's iterates. Then we characterize a class of error-tolerance and sample-size sequences that are superior to others in a certain precisely defined sense. We also identify and recommend members of this class and provide a numerical example illustrating the key results.

C onsider 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. Since the optimal allocation is the result of a concave maximization problem, its solution is particularly easy to obtain in contexts where the underlying distributions are known or can be assumed. We provide a consistent estimator for the optimal allocation and a corresponding sequential algorithm fit for implementation. Various numerical examples demonstrate how the proposed allocation differs from competing algorithms.

This article studies control-variate estimation where the control mean itself is estimated. Control-variate estimation in simulation experiments can significantly increase sampling efficiency and has traditionally been restricted to cases where the control has a known mean. In a previous paper the current authors generalized the idea of control variate estimation to the case where the control mean is only approximated. The result is a biased but possibly useful estimator. For that case, a mean square error optimal estimator was provided and its properties were discussed. This article generalizes classical control variate estimation to the case of Control Variates using Estimated Means (CVEMs). CVEMs replace the control mean with an estimated value for the control mean obtained from a prior simulation experiment. Although the resulting control-variate estimator is unbiased, it does introduce additional sampling error and so its properties are not the same as those of the standard control-variate estimator. A CVEM estimator is developed that minimizes the overall estimator variance. Both biased control variates and CVEMs can be used to improve the efficiency of stochastic simulation experiments. Their main appeal is that the restriction of having to know (deterministically) the exact value of the control mean is eliminated; thus, the space of possible controls is greatly increased.

We consider unconstrained optimization problems where only "stochastic" estimates of the objective function are observable as replicates from a Monte Carlo oracle. The Monte Carlo oracle is assumed to provide no direct observations of the function gradient. We present ASTRO-DF -a class of derivative-free trust-region algorithms, where a stochastic local interpolation model is constructed, optimized, and updated iteratively. Function estimation and model construction within ASTRO-DF is adaptive in the sense that the extent of Monte Carlo sampling is determined by continuously monitoring and balancing metrics of sampling error (or variance) and structural error (or model bias) within ASTRO-DF. Such balancing of errors is designed to ensure that Monte Carlo effort within ASTRO-DF is sensitive to algorithm trajectory, sampling more whenever an iterate is inferred to be close to a critical point and less when far away. We demonstrate the almost-sure convergence of ASTRO-DF's iterates to a first-order critical point when using linear or quadratic stochastic interpolation models. The question of using more complicated models, e.g., regression or stochastic kriging, in combination with adaptive sampling is worth further investigation and will benefit from the methods of proof presented here. We speculate that ASTRO-DF's iterates achieve the canonical Monte Carlo convergence rate, although a proof remains elusive.