An entropic approach to the optimum search for the optimum of a unimodal function

Abstract

Elementary properties of the discrete entropy from information theory show how to select the shortest strategy to perform a series of auxiliary experiments in order to eliminate, step by step, the existing uncertainty on the possible outcomes of a main experiment. Let us take the following main experiment: locate a subinterval of length 2ε of the initial interval [ a, b] where a unimodal function f attains its optimum. The auxiliary experiments are defined in two ways: 1. Case 1: At each step compare a pair of observations corresponding to two inner points of the current interval and select adequately a subinterval where the unique optimum is located. 2. Case 2: At each step compare a triad of observations corresponding to three inner points of the current interval and select adequately a subinterval where the optimum is located. The entropic approach shows, in a simple way, that the optimum search is obtained when the search points are consecutive Fibonacci numbers in case 1 and when the search points are equidistant in case 2.