From Understanding the Population Dynamics of the NSGA-II to the First Proven Lower Bounds

TL;DR

This paper provides the first proven lower bounds on the runtime of NSGA-II for specific problems, using a new mathematical understanding of its population dynamics, showing that larger populations do not reduce the asymptotic number of evaluations needed.

Contribution

It introduces a novel mathematical analysis of NSGA-II's population dynamics and establishes tight lower bounds for its runtime on certain benchmark problems.

Findings

NSGA-II requires (Nn\u2212log n) evaluations for OneMinMax.

It needs (Nn^k) evaluations for OneJumpZeroJump with jump size k.

Larger populations do not improve the asymptotic runtime bounds.

Abstract

Due to the more complicated population dynamics of the NSGA-II, none of the existing runtime guarantees for this algorithm is accompanied by a non-trivial lower bound. Via a first mathematical understanding of the population dynamics of the NSGA-II, that is, by estimating the expected number of individuals having a certain objective value, we prove that the NSGA-II with suitable population size needs $\Omega(Nn\log n)$ function evaluations to find the Pareto front of the OneMinMax problem and $\Omega(Nn^k)$ evaluations on the OneJumpZeroJump problem with jump size $k$. These bounds are asymptotically tight (that is, they match previously shown upper bounds) and show that the NSGA-II here does not even in terms of the parallel runtime (number of iterations) profit from larger population sizes. For the OneJumpZeroJump problem and when the same sorting is used for the computation of the crowding distance contributions of the two objectives, we even obtain a runtime estimate that is tight including the leading constant.