On strong homogeneity of a class of global optimization algorithms working with infinite and infinitesimal scales

TL;DR

This paper investigates the property of strong homogeneity in global optimization algorithms, analyzing its theoretical and numerical aspects, especially when dealing with infinite or infinitesimal scales, and introduces new problem classes and computational paradigms.

Contribution

It introduces a new class of problems with infinite or infinitesimal Lipschitz constants and studies the strong homogeneity of optimization algorithms within the Infinity Computing framework.

Findings

Strong homogeneity can lead to ill-conditioning with extreme scaling constants.

A new class of problems with infinite or infinitesimal Lipschitz constants is proposed.

Numerical experiments support theoretical insights.

Abstract

The necessity to find the global optimum of multiextremal functions arises in many applied problems where finding local solutions is insufficient. One of the desirable properties of global optimization methods is \emph{strong homogeneity} meaning that a method produces the same sequences of points where the objective function is evaluated independently both of multiplication of the function by a scaling constant and of adding a shifting constant. In this paper, several aspects of global optimization using strongly homogeneous methods are considered. First, it is shown that even if a method possesses this property theoretically, numerically very small and large scaling constants can lead to ill-conditioning of the scaled problem. Second, a new class of global optimization problems where the objective function can have not only finite but also infinite or infinitesimal Lipschitz constants is introduced. Third, the strong homogeneity of several Lipschitz global optimization algorithms is studied in the framework of the Infinity Computing paradigm allowing one to work \emph{numerically} with a variety of infinities and infinitesimals. Fourth, it is proved that a class of efficient univariate methods enjoys this property for finite, infinite and infinitesimal scaling and shifting constants. Finally, it is shown that in certain cases the usage of numerical infinities and infinitesimals can avoid ill-conditioning produced by scaling. Numerical experiments illustrating theoretical results are described.