Optimum search for the location of the maximum of a unimodal function

This paper exhibits an optimum strategy for the sequential estimation of, or search for, the location of the maximum of a unimodal function, when is initially uniformly distributed over some interval. The explicit search strategy which is found is valid for a variety of expected cost functions that add the expected cost of observation to the expected cost of terminal decision. The search problem possesses some of the features of problems in the areas of sequential analysis and stochastic approximation. The search stopping time can be determined as the search proceeds as in problems of sequential analysis. However, unlike many sequential analysis problems, the observational outcomes are somewhat within our control by a choice of observation or trial points. In common with problems of stochastic approximation, we attempt to determine the maximum of an unknown regression function. Contrary to many problems in stochastic approximation, though, the observations are noiseless, and the regression function is not required to be smooth or regular in the neighborhood of . The main result is that the strategy minimizing the expected cost, drawn from the class of randomized, optional stopping strategies, is nonrandomized and of a size that can be fixed in advance of observation.