Runtime Analysis of Restricted Tournament Selection for Bimodal Optimisation

Niching methods have been developed to maintain the population diversity, to investigate many peaks in parallel and to reduce the effect of genetic drift. We present the first rigorous runtime analyses of restricted tournament selection (RTS), embedded in a ($\mu$+1) EA, and analyse its effectiveness at finding both optima of the bimodal function ${\rm T{\small WO}M{\small AX}}$. In RTS, an offspring competes against the closest individual, with respect to some distance measure, amongst $w$ (window size) population members (chosen uniformly at random with replacement), to encourage competition within the same niche. We prove that RTS finds both optima on ${\rm T{\small WO}M{\small AX}}$ efficiently if the window size $w$ is large enough. However, if $w$ is too small, RTS fails to find both optima even in exponential time, with high probability. We further consider a variant of RTS selecting individuals for the tournament \emph{without} replacement. It yields a more diverse tournament and is more effective at preventing one niche from taking over the other. However, this comes at the expense of a slower progress towards optima when a niche collapses to a single individual. Our theoretical results are accompanied by experimental studies that shed light on parameters not covered by the theoretical results and support a conjectured lower runtime bound.