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

TL;DR

This work presents the first mathematical runtime analysis of NSGA-II on a two-objective multimodal benchmark, OneJumpZeroJump, showing that with $N \ge 4(n-2k+3)$ and mutation rate $1/n$, NSGA-II achieves an expected runtime of at most $ (1+o(1)) K N n^k$ for jump sizes $k \in [2, \tfrac{1}{4}n]$; a heavy-tailed mutation further yields a speedup by a factor of $k^{\Omega(k)}$. The analysis distinguishes four parent-selection schemes and establishes stage-based arguments, with explicit constants $K$ depending on the selection method. Complementary experiments demonstrate substantial empirical gains from heavy-tailed mutation and notably from crossover in several settings, confirming practical relevance. The results indicate that NSGA-II can cope with local optima on multimodal MOEAs comparably to SEMO, and point to promising future work on lower bounds and the theoretical treatment of crossover and population dynamics.

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 $k^{Ω(k)}$. 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.

Theorems & Definitions (15)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof : Proof of Lemma \ref{['lem:stage2']}
• Lemma 5
• proof