Addressing ill-conditioning in global optimization using a software implementation of the Infinity Computer

The present chapter studies the impact of scaling on global optimization algorithms. In particular, the notion of strong homogeneity is under study. A method is strongly homogeneous if it produces the same sequences of evaluation points independently both of multiplication of the objective function by a scaling constant and of adding a shifting constant. 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. A new class of global optimization problems where the objective function can have not only finite but also infinite or infinitesimal Lipschitz constants is described. The strong homogeneity of several Lipschitz global optimization algorithms is then addressed within the Infinity Computing framework. It is finally shown that the usage of numerical infinities and infinitesimals can in certain cases avoid ill-conditioning produced by scaling.