Non-isotropic Stable Mutation

Abstract

The most popular mutation operation in evolutionary algorithms based on a real representation of individuals is addition to each component of the parent individual a normally distributed random value. Mutations in phenotypic evolutionary algorithms (see Chap. 2), like evolutionary programming (Table 2.2), evolutionary strategies (2.12) and evolutionary search of soft selection (2.13), are typical realizations of the above approach. Therefore, the most natural application of $\alpha$-stable distributions to the mutation operation is exchanging the additive Gaussian disturbance of each component $x_i$ of a parent vector by a random variable of the symmetric $\alpha$-stable distribution $X_i\sim S\alpha S$ (one can calculate such a random variable based on Theorem 3.10), which is controlled by the scale parameter $\sigma _i$.