Search for point in interval, with high–low feedback

Abstract

A point H is known to lie on a given bounded interval ℋ. A searcher wishes to locate H by making successive guesses g1, g2, …, each with the knowledge of whether the previous guesses were too high or low, or exactly right. Under these circumstances it is easy to devise a search strategy which ensures the convergence of the gi to H. One such strategy is the ‘halving’ strategy which always guesses the midpoint of the interval on which H is currently known to lie. The problem becomes well defined, and more difficult, if the searcher has to minimize a given cost function which in some way measures the speed of convergence of the gi to H.