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

Andre Opris

Abstract: Evolutionary algorithms are widely used for multi-objective optimization, with NSGA-III being particularly effective for problems with more than three objectives, unlike NSGA-II. Despite its empirical success, its theoretical understanding remains limited, especially regarding 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. We show that, for population sizes n+1 ≤ µ = O(log(n)^c (n+1)) where c<1 is a constant, NSGA-III requires Ω(n^2 \log n / \mu) generations in expectation for covering the Pareto front, providing one of the first lower bounds for NSGA-III on a classical benchmark. Complementing this, we also improve the best known upper bound for 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 constant m and (2n/m + 1)^(m/2) ≤ µ ∈ O((log n)^(1/2) (2n/m + 1)^(m/2)). This yields tight runtime bounds for m=2, and the surprising result that NSGA-III outperforms NSGA-II by a factor of µ/n in the expected runtime.

External IDs:dblp:conf/aaai/Opris26