A General Dichotomy of Evolutionary Algorithms on Monotone Functions

It is known that the (1 + 1)-EA with mutation rate c/n optimizes every monotone function efficiently if c < 1, and needs exponential time on some monotone functions (HOTTOPIC functions) if c >= 2.2. We study the same question for a large variety of algorithms, particularly for the (1 + lambda)-EA, (mu + 1)-EA, (mu + 1)-GA, their "fast" counterparts, and for the (1 + (lambda, lambda))-GA. We find that all considered mutation-based algorithms show a similar dichotomy for HOTTOPIC functions, or even for all monotone functions. For the (1 + (lambda, lambda))-GA, this dichotomy is in the parameter c gamma, 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 m(2)/m(1), where m(1) and m(2) 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 mu nor by the offspring population size lambda. The picture changes completely if crossover is allowed. The genetic algorithms (mu + 1)-GA and (mu+1)-fGA are efficient for arbitrary mutations strengths if mu is large enough.