A Property of the Krolak-Cooper Extension of Fibonaccian Search

Abstract

Introduction. In a previous paper, a search was discussed which represents a possible extension of the Fibonaccian search [1] to several variables. At that time no claim was made about this search other than the fact that given a strictly unimodal function, it would reduce the uncertainty in the location of the maximum to a prescribed hypervolume provided that a sufficient number of functional evaluations be made along any ridgeline in order to make the ridgeline's true maximum value and its calculated maximum value indistinguishable. No claim was made to its being optimal in the same sense as that of the one-dimensional search. This paper, however, will make a very restricted claim on its being an optimal procedure when compared to other types of searches seeking to find the exact maximum of a finite n-dimensional rectangular lattice over which a strictly unimodal function is defined, and comparing maximum values of two (n 1)-dimensional hyperplanes oriented perpendicularly to the axis of one of the variables.