Mathematical Runtime Analysis for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II)

TL;DR

This paper provides the first mathematical runtime analysis of NSGA-II, demonstrating its efficiency with larger populations and limitations with smaller ones on benchmark problems, supported by experimental validation.

Contribution

It introduces the first rigorous mathematical analysis of NSGA-II's runtime, comparing its performance to other algorithms and identifying conditions for efficiency and limitations.

Findings

NSGA-II with larger populations matches SEMO and GSEMO runtime guarantees.

Small populations cause NSGA-II to miss parts of the Pareto front.

Experimental results confirm theoretical predictions.

Abstract

The non-dominated sorting genetic algorithm II (NSGA-II) is the most intensively used multi-objective evolutionary algorithm (MOEA) in real-world applications. However, in contrast to several simple MOEAs analyzed also via mathematical means, no such study exists for the NSGA-II so far. In this work, we show that mathematical runtime analyses are feasible also for the NSGA-II. As particular results, we prove that with a population size four times larger than the size of the Pareto front, the NSGA-II with two classic mutation operators and four different ways to select the parents satisfies the same asymptotic runtime guarantees as the SEMO and GSEMO algorithms on the basic OneMinMax and LeadingOnesTrailingZeros benchmarks. However, if the population size is only equal to the size of the Pareto front, then the NSGA-II cannot efficiently compute the full Pareto front: for an exponential number of iterations, the population will always miss a constant fraction of the Pareto front. Our experiments confirm the above findings.