Rigorous Runtime Analysis of Diversity Optimization with GSEMO on OneMinMax

TL;DR

This paper provides a rigorous runtime analysis of the GSEMO algorithm with a diversity heuristic on the OneMinMax problem, showing it efficiently achieves optimal diversity in expected quadratic time.

Contribution

It offers the first theoretical analysis of the last optimization step of GSEMO with diversity heuristics on a bi-objective problem, proving expected quadratic runtime.

Findings

GSEMO reaches optimal diversity in expected O(n^2) time for odd n.

Analysis of the population's random walk explains the convergence behavior.

The study advances understanding of diversity optimization in evolutionary algorithms.

Abstract

The evolutionary diversity optimization aims at finding a diverse set of solutions which satisfy some constraint on their fitness. In the context of multi-objective optimization this constraint can require solutions to be Pareto-optimal. In this paper we study how the GSEMO algorithm with additional diversity-enhancing heuristic optimizes a diversity of its population on a bi-objective benchmark problem OneMinMax, for which all solutions are Pareto-optimal. We provide a rigorous runtime analysis of the last step of the optimization, when the algorithm starts with a population with a second-best diversity, and prove that it finds a population with optimal diversity in expected time $O(n^2)$, when the problem size $n$ is odd. For reaching our goal, we analyse the random walk of the population, which reflects the frequency of changes in the population and their outcomes.