Statistical distribution of the convergence time of evolutionary algorithms for long-path problems

The behavior of a (1+1)-ES process on Rudolph´s binary long k paths is investigated extensively in the asymptotic framework with respect to string length l. First, the case of k=l^α is addressed. For α⩾1/2, we prove that the long k path is a long path for the (1+1)-ES in the sense that the process follows the entire path with no shortcuts, resulting in an exponential expected convergence time. For α<1/2, the expected convergence time is also exponential, but some shortcuts occur in the meantime that speed up the process. Next, in the case of constant k, the statistical distribution of convergence time is calculated, and the influence of population size is investigated for different (μ+λ)-ES. The histogram of the first hitting time of the solution shows an anomalous peak close to zero, which corresponds to an exceptional set of events that speed up the expected convergence time with a factor of l². A direct consequence of this exceptional set is that performing independent (1+1)-ES processes proves to be more advantageous than any population-based (μ+λ)-ES