Isotropic Mutation Based on an $\alpha$-Stable Generator

Abstract

Application of the isotropic distribution based on an $\alpha$-stable generator $S\alpha SU(\sigma )$ (3.57) to a mutation operator in evolutionary algorithms seems to be closest to our purpose, which is the elimination of the dead surrounding effect. This conclusion stems from the fact that the random variable $\Vert X \Vert$, where $X\sim S\alpha SU(\sigma )$, depends only on the generating variable $\Vert R \Vert$ and not on the space dimension n (or a proof, see (3.59)). Also, we hope that this fact weakens the dead surrounding effect, so important in the case of $NS\alpha S(\varvec{\sigma })$ and $IS\alpha S(\sigma )$. Can it be eliminated completely? We try to answer this question using an experiment described in the first section. Next, like in Chap. 5, the local convergence analysis of the$(1+1)$ES$_{S;\alpha }$ and $(1+\lambda )$ES$_{S;\alpha }$ strategies, as well as simulation research on exploration and exploitation abilities of the ESTS$_{S;\alpha }$ algorithms, is presented. Finally simulation analysis of the ESSS$_{S;\alpha }$ algorithms’ effectiveness in chosen global optimization problems is presented.