Effects of 1-greedy S-metric-selection on innumerably large Pareto fronts

Abstract

Evolutionary multi-objective algorithms (EMOA) using performance indicators for the selection of individuals have turned out to be a successful technique for multi-objective problems. Especially, the selection based on the S-metric, as implemented in the SMS-EMOA, seems to be effective. A special feature of this EMOA is the greedy (μ + 1) selection. Based on a pathological example for a population of size two and a discrete Pareto front it has been proven that a (μ + 1)- (or 1-greedy) EMOA may fail in finding a population maximizing the S-metric. This work investigates the performance of (μ + 1)-EMOA with small fixed-size populations on Pareto fronts of innumerable size. We prove that an optimal distribution of points can always be achieved on linear Pareto fronts. Empirical studies support the conjecture that this also holds for convex and concave Pareto fronts, but not for continuous shapes in general. Furthermore, the pathological example is generalized to a continuous objective space and it is demonstrated that also (μ + k)-EMOA are not able to robustly detect the globally optimal distribution. © Springer-Verlag 2009.