One-dimensional optimization algorithm and its convergence rate under the Wiener measure

In this paper we describe an adaptive algorithm for approximating the global minimum of a continuous function on the unit interval, motivated by viewing the function as a sample path of a Wiener process. It operates by choosing the next observation point to maximize the probability that the objective function has a value at that point lower than an adaptively chosen threshold. The error converges to zero for any continuous function. Under the Wiener measure, the error converges to zero at rate e(-n deltan), where {delta (n)} (a parameter of the algorithm) is a positive sequence converging to zero at an arbitrarily slow rate. (C) 2001 Academic Press.