A First Runtime Analysis of the NSGA-II on a Multimodal Problem

TL;DR

This paper provides the first runtime analysis of NSGA-II on a multimodal benchmark, demonstrating its efficiency in handling local optima with different mutation strategies.

Contribution

It offers the first mathematical runtime analysis of NSGA-II on a multimodal problem, showing its effectiveness with various parent selection methods and mutation operators.

Findings

NSGA-II optimizes the benchmark in time O(N n^k) with population size N ≥ 4 times the Pareto front size.

Fast mutation improves runtime guarantees by a factor of k^{Ω(k)}.

NSGA-II handles local optima as well as the global SEMO algorithm.

Abstract

Very recently, the first mathematical runtime analyses of the multi-objective evolutionary optimizer NSGA-II have been conducted. We continue this line of research with a first runtime analysis of this algorithm on a benchmark problem consisting of two multimodal objectives. We prove that if the population size $N$ is at least four times the size of the Pareto front, then the NSGA-II with four different ways to select parents and bit-wise mutation optimizes the OneJumpZeroJump benchmark with jump size~$2 \le k \le n/4$ in time $O(N n^k)$. When using fast mutation, a recently proposed heavy-tailed mutation operator, this guarantee improves by a factor of $k^{\Omega(k)}$. Overall, this work shows that the NSGA-II copes with the local optima of the OneJumpZeroJump problem at least as well as the global SEMO algorithm.