Tight Runtime Guarantees From Understanding the Population Dynamics of the GSEMO Multi-Objective Evolutionary Algorithm

TL;DR

This paper provides a detailed analysis of the GSEMO multi-objective evolutionary algorithm, establishing tight runtime bounds and improving understanding of its population dynamics on classic benchmarks.

Contribution

It offers the first matching lower bounds for GSEMO's runtime on COCZ and extends analysis to other benchmarks, advancing theoretical understanding of MOEAs.

Findings

Proves a lower bound of Ω(n^2 log n) for COCZ.

Shows GSEMO finds a constant fraction of the Pareto front in O(n^2) time.

Establishes an Ω(n^{k+1}) lower bound for the OJZJ benchmark with gap parameter k.

Abstract

The global simple evolutionary multi-objective optimizer (GSEMO) is a simple, yet often effective multi-objective evolutionary algorithm (MOEA). By only maintaining non-dominated solutions, it has a variable population size that automatically adjusts to the needs of the optimization process. The downside of the dynamic population size is that the population dynamics of this algorithm are harder to understand, resulting, e.g., in the fact that only sporadic tight runtime analyses exist. In this work, we significantly enhance our understanding of the dynamics of the GSEMO, in particular, for the classic CountingOnesCountingZeros (COCZ) benchmark. From this, we prove a lower bound of order $\Omega(n^2 \log n)$, for the first time matching the seminal upper bounds known for over twenty years. We also show that the GSEMO finds any constant fraction of the Pareto front in time $O(n^2)$, improving over the previous estimate of $O(n^2 \log n)$ for the time to find the first Pareto optimum. Our methods extend to other classic benchmarks and yield, e.g., the first $\Omega(n^{k+1})$ lower bound for the OJZJ benchmark in the case that the gap parameter is $k \in \{2,3\}$. We are therefore optimistic that our new methods will be useful in future mathematical analyses of MOEAs.