Self-adjusting population sizes for the (1,λ)-EA on monotone functions

Abstract

We study the (1,λ)-EA with mutation rate c/n for c≤1, where the population size is adaptively controlled with the (1:s+1)-success rule. Recently, Hevia Fajardo and Sudholt have shown that this setup with c=1 is efficient on OneMax for s<1, but inefficient if s≥18. Surprisingly, the hardest part is not close to the optimum, but rather at linear distance. We show that this behaviour is not specific to OneMax. If s is small, then the algorithm is efficient on all monotone functions, and if s is large, then it needs super-polynomial time on all monotone functions. In the former case, for c<1 we show a O(n) upper bound for the number of generations and O(nlog⁡n) for the number of function evaluations, and for c=1 we show O(nlog⁡n) generations and O(n2log⁡log⁡n) evaluations. We also show formally that optimization is always fast, regardless of s, if the algorithm starts in proximity of the optimum. All results also hold in a dynamic environment where the fitness function changes in each generation.An extended abstract, containing only the results without proofs, has been published at the PPSN conference [1].