Benchmarking a (μ+λ)(\mu+\lambda) Genetic Algorithm with Configurable Crossover Probability

We investigate a family of $(\mu+\lambda)$ Genetic Algorithms (GAs) which creates offspring either from mutation or by recombining two randomly chosen parents. By scaling the crossover probability, we can thus interpolate from a fully mutation-only algorithm towards a fully crossover-based GA. We analyze, by empirical means, how the performance depends on the interplay of population size and the crossover probability. Our comparison on 25 pseudo-Boolean optimization problems reveals an advantage of crossover-based configurations on several easy optimization tasks, whereas the picture for more complex optimization problems is rather mixed. Moreover, we observe that the ``fast'' mutation scheme with its are power-law distributed mutation strengths outperforms standard bit mutation on complex optimization tasks when it is combined with crossover, but performs worse in the absence of crossover. We then take a closer look at the surprisingly good performance of the crossover-based $(\mu+\lambda)$ GAs on the well-known LeadingOnes benchmark problem. We observe that the optimal crossover probability increases with increasing population size $\mu$. At the same time, it decreases with increasing problem dimension, indicating that the advantages of the crossover are not visible in the asymptotic view classically applied in runtime analysis. We therefore argue that a mathematical investigation for fixed dimensions might help us observe effects which are not visible when focusing exclusively on asymptotic performance bounds

Tight bounds on the expected runtime of a standard steady state genetic algorithm

Recent progress in the runtime analysis of evolutionary algorithms (EAs) has allowed the derivation of upper bounds on the expected runtime of standard steady-state genetic algorithms (GAs). These upper bounds have shown speed-ups of the GAs using crossover and mutation over the same algorithms that only use mutation operators (i.e., steady-state EAs) both for standard unimodal (i.e., ONEMAX) and multimodal (i.e., JUMP) benchmark functions. The bounds suggest that populations are beneficial to the GA as well as higher mutation rates than the default 1/n rate. However, making rigorous claims was not possible because matching lower bounds were not available. Proving lower bounds on crossover-based EAs is a notoriously difficult task as it is hard to capture the progress that a diverse population can make. We use a potential function approach to prove a tight lower bound on the expected runtime of the (2+1) GA for ONEMAX for all mutation rates c/n with c<1.422. This provides the last piece of the puzzle that completes the proof that larger population sizes improve the performance of the standard steady-state GA for ONEMAX for various mutation rates, and it proves that the optimal mutation rate for the (2+1) GA on ONEMAX is (97−−√−5)/(4n)≈1.2122/n