An Archive Can Bring Provable Speed-ups in Multi-Objective Evolutionary Algorithms
Chao Bian, Shengjie Ren, Miqing Li, Chao Qian
TL;DR
This work addresses the practical issue that MOEAs may lose Pareto-optimal solutions and that maintaining a population matching the Pareto front size is often impractical. It analyzes NSGA-II and SMS-EMOA with an archive on two standard bi-objective problems, OneMinMax and LeadingOnesTrailingZeroes, and proves polynomial speed-ups in expected running time by reducing the needed population size to a constant. The core results show running-time bounds of $O(μ n log n)$ for OneMinMax and $O(μ n^2 + μ^2 n log n)$ for LeadingOnesTrailingZeroes for both algorithms (with modest values of $μ$), implying speed-ups by a factor of Θ(n) compared to non-archive settings. These theoretical findings are supported by experiments that demonstrate empirical acceleration and highlight the practical impact for designing more efficient archive-enabled MOEAs.
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.