On Scalability of Multi-Objective Evolutionary Algorithms on Combinatorial Optimisation Problems

TL;DR

This paper empirically investigates how multi-objective evolutionary algorithms scale on large combinatorial problems, revealing the impact of crossover mechanisms on their convergence speed and performance.

Contribution

It provides the first large-scale analysis of MOEAs on combinatorial problems, highlighting the importance of crossover in improving convergence speed.

Findings

SEMO's convergence slows as problem size increases.

Incorporating crossover accelerates SEMO's convergence.

Absence of crossover hampers SEMO's performance on large problems.

Abstract

Scalability of evolutionary algorithms refers to assessing how their performance changes as problem size increases. In the area of multi-objective optimisation, research on the scalability of multi-objective evolutionary algorithms (MOEAs) has predominantly focussed on continuous problems. However, multi-objective combinatorial optimisation problems (MOCOPs) differ from continuous ones. Their discrete and rigid structure often brings rugged landscape, numerous local optimal solutions and disjoint global optimal regions. This leads to different behaviour of MOEAs. For example, SEMO, a simple MOEA without mating selection and diversity maintenance mechanisms, has been shown to be highly competitive, and in many cases to outperform more sophisticated MOEAs on MOCOPs. Yet, it remains unclear whether such findings hold for large-scale cases. In this paper, we conduct an empirical investigation into the scalability of MOEAs on combinatorial problems, with problem size from 50 to 5,000. Our results show that SEMO experiences a decline in convergence speed as dimensionality increases, compared to other MOEAs such as NSGA-II, SMS-EMOA and MOEA/D. We further demonstrate that the absence of crossover is a major contributor to SEMO's underperformance in large-scale problems, and that incorporating crossover into SEMO can substantially accelerate convergence in general, despite being detrimental in spreading solutions over the Pareto front.