A 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^-nδn, where {δ_n} (a parameter of the algorithm) is a positive sequence converging to zero at an arbitrarily slow rate.