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The necessity to find the global optimum of multiextremal functions arises inmany applied problems where finding local solutions is insufficient. One of thedesirable properties of global optimization methods is \emph{stronghomogeneity} meaning that a method produces the same sequences of points wherethe objective function is evaluated independently both of multiplication of thefunction by a scaling constant and of adding a shifting constant. In thispaper, several aspects of global optimization using strongly homogeneousmethods are considered. First, it is shown that even if a method possesses thisproperty theoretically, numerically very small and large scaling constants canlead to ill-conditioning of the scaled problem. Second, a new class of globaloptimization problems where the objective function can have not only finite butalso infinite or infinitesimal Lipschitz constants is introduced. Third, thestrong homogeneity of several Lipschitz global optimization algorithms isstudied in the framework of the Infinity Computing paradigm allowing one towork \emph{numerically} with a variety of infinities and infinitesimals.Fourth, it is proved that a class of efficient univariate methods enjoys thisproperty for finite, infinite and infinitesimal scaling and shifting constants.Finally, it is shown that in certain cases the usage of numerical infinitiesand infinitesimals can avoid ill-conditioning produced by scaling. Numericalexperiments illustrating theoretical results are described.

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