Monte Carlo optimization by stochastic approximation (with application to harvesting of Atlantic menhaden)

Stochastic approximation can be viewed as a methodology for designing a sequence of response surface experiments. Although stochastic approximation has not been employed widely by statisticians, several authors agree that it is potentially very useful for a variety of statistical problems. In a recent study of the Atlantic menhaden, a commercially important fish in the herring family (Clupeidae), we made extended use of stochastic approximation and were quite pleased with the results. This paper is intended to introduce stochastic approximation to those statisticians unfamiliar with the area. A stochastic simulation model of the menhaden population is used as an example, but the paper is not addressed to only those working in fisheries. In this model, two variables are used to define the harvesting policy. For any values of these ~ variables, the model will produce a random catch, and for a specified utility function the objective is to find the values of the variables which maximize the expected utility of the catch. Therefore, this is a classical response surface problem. However, nonsequential response surface methods would be extremely expensive to apply here. We used stochastic approximation to estimate the policy maximizing the expected utility of the catch. Much of the paper duscusses the application of known results, but there are also some new results. In particular, we show how the use of common random numbers, a standard variance reduction technique, can be applied to stochastic approximation.