We propose and analyze a self-adaptive version of the (1, λ) evolutionary algorithm in which the current mutation rate is part of the individual and thus also subject to mutation. A rigorous runtime analysis on the OneMax benchmark function reveals that a simple local mutation scheme for the rate leads to an expected optimization time (number of fitness evaluations) of O(nλ/log λ + n log n). This time is asymptotically smaller than the optimization time of the classic (1, λ) EA and (1 + λ) EA for all static mutation rates and best possible among all λ-parallel mutation-based unbiased black-box algorithms.

We propose a new way to self-adjust the mutation rate in population-based evolutionary algorithms. Roughly speaking, it consists of creating half the offspring with a mutation rate that is twice the current mutation rate and the other half with half the current rate. The mutation rate is then updated to the rate used in that subpopulation which contains the best offspring.

In classical runtime analysis it has been observed that certain working principles of an evolutionary algorithm cannot be understood by only looking at the asymptotic order of the runtime, but that more precise estimates are needed. In this work we demonstrate that the same observation applies to black-box complexity analysis. We prove that the unary unbiased black-box complexity of the classic OneMax function class is n ln(n) -- cn ± o(n) for a constant c between 0.2539 and 0.2665. Our analysis yields a simple (1+1)-type algorithm achieving this runtime bound via a fitness-dependent mutation strength. When translated into a fixed-budget perspective, our algorithm with the same budget computes a solution that asymptotically is 13% closer to the optimum (given that the budget is at least 0.2675n).