Order Statistics for Convergence Velocity Analysis of Simplified Evolutionary Algorithms

Abstract The theory of order statistics is utilized in this paper to analyze the convergence velocity ϕ of simplified evolutionary algorithms based on mutation and (μ + , λ)-selection. A general, representation-independent way of theoretical analysis is outlined and put in concrete terms for two specific objective functions: The binary counting ones objective function and the continuous sphere model. This way, the same method of theoretical analysis describes the behavior of a search strategy similar to a genetic algorithm as well as a search strategy similar to an evolution strategy. The resulting convergence velocity graphics exhibit a striking similarity, such that the underlying principles of evolutionary search seem to be valid independently of the particular search space. The expectations E ( Z ν:λ ) of the ν-th order statistics turn out to be of paramount importance for determining the convergence velocity ϕ. The expectation E ( Z λ:λ ) for the continuous normal case is identical to Rechenberg's progress coefficient c 1, λ , and the averages of the μ upper order statistics are known as the selection differential in the literature. For the case of an underlying normal distribution (as in evolution strategies) many of the results from order statistics can be utilized e.g. to determine the asymptotic behavior of an (1, λ)-strategy, while in the discrete case no such result is available. The order statistics approach turns out to be a very useful theoretical framework for the derivation of quite general results (including upper bounds) on the convergence velocity of evolutionary algorithms.