Interactive search for the global maximum of a function of many variables

A new algorithm to locate the global maximum of a function defined in a multidimensional rectangular domain, which is the transformation of the object function in such a way that its global maximum corresponds to infinity while other secondary maxima are reduced to zero.

Abstract

This paper discusses a new algorithm to locate the global maximum of a function defined in a multidimensional rectangular domain. The number of dimensions is as large as, or even more than, 5 or 10. There are two important elements in this algorithm. One is the transformation of the object function in such a way that its global maximum corresponds to infinity while other secondary maxima are reduced to zero. Actually there is some departure from the ideal transformation because of possible overflow on the computer. This portion of the algorithm precedes the interactive (or conversational) use of a graphic display system. This interactive part makes the other element of the algorithm. A multidimensional point is represented as a curve on the display screen. By projecting numerous points in the multidimensional space to similarly numerous curves on the screen of the graphic display device, the human eye can make overall recognition much more efficiently than computers. This fact is exploited to reduce the problem to that of a set of unimodal peaks. Once the supporting domain for each of these peaks is separated by visual aid, one may leave the computer to handle the rest of the problem for itself. A number of numerical experiments are done and discussed to provide evidence regarding the feasibility of the proposed algorithm.