Runtime Analysis of Probabilistic Crowding and Restricted Tournament Selection for Bimodal Optimisation

TL;DR

This paper rigorously analyzes the runtime of probabilistic crowding and restricted tournament selection niching methods in bimodal optimization, revealing conditions under which they succeed or fail in finding multiple optima.

Contribution

It provides the first theoretical runtime analysis of probabilistic crowding and RTS niching methods on bimodal functions, identifying parameter conditions for success or failure.

Findings

Probabilistic crowding fails to find good solutions in exponential time.

RTS fails with small window sizes, leading to exponential runtime.

Large window sizes enable RTS to find both optima efficiently.

Abstract

Many real optimisation problems lead to multimodal domains and so require the identification of multiple optima. Niching methods have been developed to maintain the population diversity, to investigate many peaks in parallel and to reduce the effect of genetic drift. Using rigorous runtime analysis, we analyse for the first time two well known niching methods: probabilistic crowding and restricted tournament selection (RTS). We incorporate both methods into a $(\mu+1)~EA$ on the bimodal function Twomax where the goal is to find two optima at opposite ends of the search space. In probabilistic crowding, the offspring compete with their parents and the survivor is chosen proportionally to its fitness. On Twomax probabilistic crowding fails to find any reasonable solution quality even in exponential time. In RTS the offspring compete against the closest individual amongst $w$ (window size) individuals. We prove that RTS fails if $w$ is too small, leading to exponential times with high probability. However, if w is chosen large enough, it finds both optima for Twomax in time $O(\mu n \log{n})$ with high probability. Our theoretical results are accompanied by experimental studies that match the theoretical results and also shed light on parameters not covered by the theoretical results.