Performance of the $(\mu /\mu _{I},\lambda )\hbox {-}\sigma {\rm SA}$ -ES on a Class of PDQFs

This paper investigates the behavior of (μ/μ_I,λ)-σSA-ES on a class of positive definite quadratic forms. After introducing the fitness environment and the strategy, the self-adaptation mechanism is analyzed with the help of the self-adaptation response function. Afterward, the steady state of the strategy is analyzed. The dynamical equations for the expectation of the mutation strength σ and the localization parameter ζ will be derived. Building on that, the progress rate ϕ is analyzed and tuned by means of the learning parameter τ. An approximate formula for τ_opt, yielding locally maximal progress, is presented. Finally, the performance of the σSA-rule is compared with the performance of the cumulative step size adaptation rule, and a rough approximation for the expected runtime is presented.