The First Theoretical Approximation Guarantees for the Non-Dominated Sorting Genetic Algorithm III (NSGA-III)

Abstract

This work conducts a first theoretical analysis studying how well the NSGA-III approximates the Pareto front when the population size N is less than the Pareto front size. We show that when N is at least the number Nr of reference points, then the approximation quality, measured by the maximum empty interval (MEI) indicator, on the OneMinMax benchmark is such that there is no empty interval longer than ⌈(5-2√2)n/(Nr-1)⌉. This bound is independent of N, which suggests that further increasing the population size does not increase the quality of approximation when Nr is fixed. This is a notable difference to the NSGA-II with sequential survival selection, where increasing the population size improves the quality of the approximations. We also prove two results indicating approximation difficulties when N<Nr. These theoretical results suggest that the best setting to approximate the Pareto front is Nr=N. In our experiments, we observe that with this setting the NSGA-III computes optimal approximations, very different from the NSGA-II, for which optimal approximations have not been observed so far.