An Analysis of Mutative σ-Self-Adaptation on Linear Fitness Functions

This paper investigates σ-self-adaptation for real valued evolutionary algorithms on linear fitness functions. We identify the step-size logarithm log σ as a key quantity to understand strategy behavior. Knowing the bias of mutation, recombination, and selection on log σ is sufficient to explain σ-dynamics and strategy behavior in many cases, even from previously reported results on non-linear and/or noisy fitness functions. On a linear fitness function, if intermediate multi-recombination is applied on the object parameters, the i-th best and the i-th worst individual have the same σ-distribution. Consequently, the correlation between fitness and step-size σ is zero. Assuming additionally that σ-changes due to mutation and recombination are unbiased, then σ-self-adaptation enlarges σ if and only if μ < λ/2, given (μ, λ)-truncation selection. Experiments show the relevance of the given assumptions.