Towards a Rigorous Understanding of the Population Dynamics of the NSGA-III: Tight Runtime Bounds

TL;DR

This paper provides tight theoretical runtime bounds for NSGA-III on bi-objective and multi-objective OneMinMax problems, advancing understanding of its population dynamics and demonstrating its efficiency over NSGA-II.

Contribution

It establishes the first proven lower bound and improves upper bounds for NSGA-III's runtime on classical benchmarks, revealing its competitive advantage.

Findings

Proven $ ext{Omega}(n^2 ext{log}(n)/ ext{mu})$ lower bound for bi-objective NSGA-III.

Improved upper bounds for multi-objective NSGA-III on OneMinMax problems.

NSGA-III outperforms NSGA-II by a factor of $ ext{mu}/n$ in expected runtime.

Abstract

Evolutionary algorithms are widely used for solving multi-objective optimization problems. A prominent example is NSGA-III, which is particularly well suited for solving problems involving more than three objectives, distinguishing it from the classical NSGA-II. Despite its empirical success, the theoretical understanding of NSGA III remains very limited, especially with respect to runtime analysis. A central open problem concerns its population dynamics, which involve controlling the maximum number of individuals sharing the same fitness value during the exploration process. In this paper, we make a significant step towards such an understanding by proving tight runtime bounds for NSGA-III on the bi-objective OneMinMax ($2$-OMM) problem. Firstly, we prove that NSGA-III requires $\Omega(n^2 \log(n) / \mu)$ generations in expectation to optimize $2$-OMM assuming the population size $\mu$ satisfies $n+1 \leq \mu =O(\log(n)^c(n+1))$ where $n$ denotes the problem size and $c<1$ is a constant. Apart from~\cite{opris2025multimodal}, this is the first proven lower runtime bound for NSGA-III on a classical benchmark problem. Complementing this, we secondly improve the best known upper bound of NSGA-III on the $m$-objective OneMinMax problem ($m$-OMM) of $O(n \log(n))$ generations by a factor of $\mu /(2n/m + 1)^{m/2}$ for a constant number $m$ of objectives and population size $(2n/m + 1)^{m/2} \leq \mu \in O(\sqrt{\log(n)} (2n/m + 1)^{m/2})$. This yields tight runtime bounds in the case $m = 2$, and the surprising result that NSGA-III beats NSGA-II by a factor of $\mu/n$ in the expected runtime.