A General dichotomy of evolutionary algorithms on monotone functions

Abstract

It is known that the (1 + 1)-EA with mutation rate c/n optimises every monotone function efficiently if c < 1, and needs exponential time on some monotone functions (HotTopic functions) if c > c0 = 2.13692… We study the same question for a large variety of algorithms, particularly for (1 + λ)-EA, (μ + 1)-EA, (μ + 1)-GA, their fast counterparts like fast (1 + 1)-EA, and for (1 + (λ, λ))-GA. We prove that all considered mutation-based algorithms show a similar dichotomy for HotTopic functions, or even for all monotone functions. For the (1 + (λ, λ))-GA, this dichotomy is in the parameter cγ, which is the expected number of bit flips in an individual after mutation and crossover, neglecting selection. For the fast algorithms, the dichotomy is in m2/m1, where m1 and m2 are the first and second falling moment of the number of bit flips. Surprisingly, the range of efficient parameters is not affected by either population size μ nor by the offspring population size λ. The picture changes completely if crossover is allowed. The genetic algorithms (μ + 1)-GA and (μ + 1)-fGA are efficient for arbitrary mutations strengths if μ is large enough.