On Maximizing Functions by Fibonacci Search

The search for a local maximum of a function f(x) involves a sequence of function evaluations, i.e.s observations of the value of f(x) for a fixed value of x. A sequential search scheme allows us to evaluate the function at different points, one after the other, using information from earlier evaluations to decide where to locate the next ones. At each stage, the smallest interval in which a maximum point of the function is known to lie is called the interval of uncertainty, Most of the theoretical search procedures terminate the search when either the interval of uncertainty is reduced to a specific size or two successive estimates of the maximum are closer than some predetermined value. However, an additional termination rule which surprisingly has not received much attention by theorists exists in most practical search codes, namely the number of function evaluations cannot exceed a predetermined number, which we denote by .N. A well-known procedure designed for a fixed number of function evaluations is the so-called Fibonacci search method. This method can be applied whenever the function is unimodal and the initial interval of undertainty is finite. In this paper, we propose a two-stage procedure which can be used whenever these requirements do not hold. In the first stage, the procedure tries to bracket the maximum point in a finite interval, and in the second it reduces this interval using the Fibonacci search method or a variation of it developed by Witzgall.