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 $N_r$ 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 $\lceil\frac{(5-2\sqrt2)n}{N_r-1}\rceil$. This bound is independent of $N$, which suggests that further increasing the population size does not increase the quality of approximation when $N_r$ 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 $N<N_r$. These theoretical results suggest that the best setting to approximate the Pareto front is $N_r=N$. 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.