Runtime Analysis of Functions Where Widely-Used Evolutionary Multi-Objective Algorithms Beat Simple Ones

Abstract

In evolutionary multi-objective optimisation, runtime analysis examines the (expected) time for Evolutionary Multi-Objective Algorithms (EMOAs) to cover the Pareto front. It has recently been applied to NSGA-II, NSGA-III, and SMS-EMOA. However, most analyses showed that these widely-used algorithms have the same runtime guarantee as the simplest EMOA, (G)SEMO. To our knowledge, no runtime analyses demonstrate an advantage of a popular EMOA over (G)SEMO for deterministic problems. We propose such problems to illustrate the superiority of popular EMOAs over (G)SEMO. We introduce a classification of multi-objective problems and identify the so-called ( \(a\) , \(b\) )-Pareto-sparse problems that are difficult for (G)SEMO as Pareto-optimal points are separated by large genotypic distances. A general lower bound on the expected number of fitness evaluations for (G)SEMO to solve any ( \(a\) , \(b\) )-Pareto-sparse problem is proven. On many example problems, this bound is \(n^{\Omega(n)}\) : OneTrapZeroTrap , a generalisation of Trap function to two objectives, the OneJumpZeroJump class with a large gap parameter, and a class of bi-objective MaxSat instances called OneZeroCountSat . Therefore, (G)SEMO performs poorly on all these problems. Conversely, we prove that the three popular EMO algorithms—NSGA-II, NSGA-III and SMS-EMOA—enhanced with a mild diversity mechanism of avoiding genotype duplication, are highly efficient as they optimise OneTrapZeroTrap in only \(O(n\log{n})\) fitness evaluations in expectation. Experimental results on OneTrapZeroTrap and OneZeroCountSat match our theoretical prediction that (G)SEMO always fails, while the other algorithms always succeed. Our analysis reveals the importance of the key components in these sophisticated algorithms and contributes to a better understanding of their capabilities.