An Archive Can Bring Provable Speed-ups in Multi-Objective Evolutionary Algorithms

TL;DR

This paper proves that using an archive in multi-objective evolutionary algorithms (MOEAs) can theoretically guarantee polynomial speed-ups by reducing the population size, confirming a popular practical approach with formal analysis.

Contribution

The paper provides the first theoretical proof that archives can accelerate MOEAs, showing polynomial speed-ups for NSGA-II and SMS-EMOA on standard problems.

Findings

Using an archive reduces expected running time polynomially.

Population size can be minimized to a small constant with an archive.

Archives confirm practical benefits with theoretical backing.

Abstract

In the area of multi-objective evolutionary algorithms (MOEAs), there is a trend of using an archive to store non-dominated solutions generated during the search. This is because 1) MOEAs may easily end up with the final population containing inferior solutions that are dominated by other solutions discarded during the search process and 2) the population that has a commensurable size of the problem's Pareto front is often not practical. In this paper, we theoretically show, for the first time, that using an archive can guarantee speed-ups for MOEAs. Specifically, we prove that for two well-established MOEAs (NSGA-II and SMS-EMOA) on two commonly studied problems (OneMinMax and LeadingOnesTrailingZeroes), using an archive brings a polynomial acceleration on the expected running time. The reason is that with an archive, the size of the population can reduce to a small constant; there is no need for the population to keep all the Pareto optimal solutions found. This contrasts existing theoretical studies for MOEAs where a population with a commensurable size of the problem's Pareto front is needed. The findings in this paper not only provide a theoretical confirmation for an increasingly popular practice in the design of MOEAs, but can also be beneficial to the theory community towards studying more practical MOEAs.