The Dynamics of Cumulative Step Size Adaptation on the Ellipsoid Model.

Area of Science:

• Optimization algorithms
• Computational intelligence
• Evolutionary computation

Background:

• The [Formula: see text]-Evolution Strategy (ES) is a population-based metaheuristic optimization algorithm.
• Cumulative Step Size Adaptation (CSA) is a control rule used to adapt the step size in ES algorithms.
• The ellipsoid model is a common benchmark function in optimization used to evaluate algorithm performance.

Purpose of the Study:

• To investigate the behavior of the [Formula: see text]-Evolution Strategy (ES) with cumulative step size adaptation (CSA) on the ellipsoid model.
• To analyze the convergence properties and steady-state behavior of the ES-CSA algorithm.
• To determine the impact of CSA on mutation strength and expected running time.

Main Methods:

• Dynamic systems analysis was employed to study the ES-CSA algorithm.
• A nonlinear system of difference equations was derived to model the mean value evolution.
• The system was simplified to obtain closed-form solutions for steady-state behavior in the asymptotic limit.
• Mutation strength and expected running time were calculated.

Main Results:

• The ES-CSA algorithm exhibits linear convergence order on the ellipsoid model.
• Closed-form solutions for steady-state behavior were derived for large search space dimensions.
• The CSA control rule allows for a significantly larger steady-state mutation strength compared to standard settings.
• A formula for the expected running time was derived.

Conclusions:

• The CSA control rule enhances the performance of ES algorithms in non-noisy environments due to increased mutation strength.
• The study provides insights into the selection of the cumulation parameter (c) and damping constant (D).
• The findings contribute to a better understanding of adaptive parameter control in evolutionary computation.