Rigorous runtime analysis of a (μ+1)ES for the sphere function

Abstract

Evolutionary algorithms (EAs) are general, randomized search heuristics applied successfully to optimization problems both in discrete and in continuous search spaces. In recent years, substantial progress has been made in theoretical runtime analysis of EAs, in particular for pseudo-Boolean fitness functions f:(0,1)n → R. Compared to this, little is known about the runtime of simple and, in particular, more complex EAs for continuous functions f: Rn → R.In this paper, a first rigorous runtime analysis of a population-based EA in continuous search spaces is presented. A simple (μ+1) evolution strategy ((μ+1)ES) that uses Gaussian mutations adapted by the 1/5-rule as its search operator is studied on the well-known Sphere functionand the influence of μ and n on its runtime is examined. By generalizing the proof technique of randomized family trees, developed before w.r.t. discrete search spaces, asymptotically upper and lower bounds on the time for the population to make a predefined progress are derived. Furthermore, the utility of the 1/5-rule in population-based evolution strategies is shown. Finally, the behavior of the (μ+1)ES on multimodal functions is discussed.