OneMax is not the Easiest Function for Fitness Improvements.

We study the (1:s+1) success rule for controlling the population size of the (1,λ)- EA. It was shown by Hevia Fajardo and Sudholt that this parameter control mechanism can run into problems for large s if the fitness landscape is too easy. They conjectured that this problem is worst for the ONEMAX benchmark, since in some well-established sense ONEMAX is known to be the easiest fitness landscape. In this paper we disprove this conjecture. We show that there exist s and ɛ such that the self-adjusting (1,λ)-EA with the (1:s+1)-rule optimizes ONEMAX efficiently when started with ɛn zero-bits, but does not find the optimum in polynomial time on DYNAMIC BINVAL. Hence, we show that there are landscapes where the problem of the (1:s+1)-rule for controlling the population size of the (1,λ)-EA is more severe than for ONEMAX. The key insight is that, while ONEMAX is the easiest function for decreasing the distance to the optimum, it is not the easiest fitness landscape with respect to finding fitness-improving steps.

Probabilistic Analysis of the (1+1)-Evolutionary Algorithm.

We give a detailed analysis of the optimization time of the [Formula: see text]-Evolutionary Algorithm under two simple fitness functions (OneMax and LeadingOnes). The problem has been approached in the evolutionary algorithm literature in various ways and with different degrees of rigor. Our asymptotic approximations for the mean and the variance represent the strongest of their kind. The approach we develop is based on an asymptotic resolution of the underlying recurrences and can also be extended to characterize the corresponding limiting distributions. While most of our approximations can be derived by simple heuristic calculations based on the idea of matched asymptotics, the rigorous justifications are challenging and require a delicate error analysis.