Analysis of a Simple Evolutionary Algorithm for Minimization in Euclidean Spaces

Abstract

Although evolutionary algorithms (EAs) are widely used in practical optimization, their theoretical analysis is stillin its infancy. Up to now results on the (expected) runtime are limited to discrete search spaces, yet EAs are mostly applied to continuous optimization problems. So far results on the runtime of EAs for continuous search spaces rely on validation by experiments/simulations since merely a simplifying model of the respective stochastic process is investigated.

Here a first algorithmic analysis of the expected runtime of a simple, but fundamental EA for the search space IRⁿ is presented. Namely, the so-called (1+1) Evolution Strategy ((1+1) ES) is investigated on unimodal functions that are monotone with respect to the distance between search point and optimum. A lower bound on the expected runtime is proven under the only assumption that isotropic distributions are used to generate the random mutation vectors. Consequently, this bound holds for any mutation adaptation mechanism. Finally, we prove that the commonly used “Gauss mutations” in combination with the socalled 1/5-rule for the mutation adaptation do achieve asymptotically optimal expected runtime.

supported by the Deutsche Forschungsgemeinschaft (DFG) as part of the collaborative research center “Computational Intelligence” (SFB 531)