A First Runtime Analysis of the PAES-25: An Enhanced Variant of the Pareto Archived Evolution Strategy

TL;DR

This paper provides the first mathematical runtime analysis of PAES-25, an enhanced MOEA, deriving tight bounds for finding Pareto fronts on m-LOTZ problems and comparing its efficiency to existing algorithms.

Contribution

It introduces the first tight runtime bounds for PAES-25 on m-LOTZ, outperforming previous bounds for MOEAs on many-objective problems, and analyzes the impact of archivers on solution distribution.

Findings

Tight expected runtime bounds for PAES-25 on m-LOTZ for various m.

PAES-25 with bit mutation optimizes bi-objective LOTZ in O(n^4) iterations.

Archivers improve the distribution of solutions across the Pareto front.

Abstract

This paper presents a first mathematical runtime analysis of PAES-25, an enhanced version of the original Pareto Archived Evolution Strategy (PAES) coming from the study of telecommunication problems over two decades ago to understand the dynamics of local search of MOEAs on many-objective fitness landscapes. We derive tight expected runtime bounds of PAES-25 with one-bit mutation on $m$-LOTZ until the entire Pareto front is found: $\Theta(n^3)$ iterations if $m=2$, $\Theta(n^3 \log^2(n))$ iterations if $m=4$ and $\Theta(n(2n/m)^{m/2} \log(n/m))$ iterations if $m>4$ where $n$ is the problem size and $m$ the number of objectives. To the best of our knowledge, these are the first known tight runtime bounds for an MOEA outperforming the best known upper bound of $O(n^{m+1})$ for (G)SEMO on $m$-LOTZ when $m$ is at least $4$. We also show that archivers, such as the Adaptive Grid Archiver (AGA), Hypervolume Archiver (HVA) or Multi-Level Grid Archiver (MGA), help to distribute the set of solutions across the Pareto front of $m$-LOTZ efficiently. We also show that PAES-25 with standard bit mutation optimizes the bi-objective LOTZ benchmark in expected $O(n^4)$ iterations, and we discuss its limitations on other benchmarks such as OMM or COCZ.