Towards a Rigorous Understanding of the Population Dynamics of the NSGA-III: Tight Runtime Bounds
Andre Opris
TL;DR
This work delivers the first tight runtime characterizations for NSGA-III on a classical benchmark by analyzing its population dynamics through the lens of the maximum cover number $\beta$. It proves a tight expected lower bound of $\Omega(n^2 \log n / \mu)$ for the bi-objective OneMinMax problem under sub-polynomial population sizes and derives an improved upper bound for $m$-objective OneMinMax that scales with the reference-point mechanism and population size, achieving 3–5 sentence tightness for $m=2$. The results show NSGA-III can outperform NSGA-II by a factor of $\mu/n$ in expectation on these problems, via more uniform front distribution. Together, the lower and upper bounds illuminate the algorithm’s exploration and exploitation balance, providing a solid foundation for understanding and improving NSGA-III on rugged, many-objective landscapes and guiding practical choices of population size $\mu$.
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 $Ω(n^2 \log(n) / μ)$ generations in expectation to optimize $2$-OMM assuming the population size $μ$ satisfies $n+1 \leq μ=O(\log(n)^c(n+1))$ 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 $μ/(2n/m + 1)^{m/2}$ for a constant number $m$ of objectives and population size $(2n/m + 1)^{m/2} \leq μ\in O(\sqrt{\log(n)} (2n/m + 1)^{m/2})$. This yields tight runtime bounds in the case $m = 2$, and the surprising result that NSGA-III beats NSGA-II by a factor of $μ/n$ in the expected runtime.