Convergence Properties of the (μ/μ_I, λ)-ES on the Rastrigin Function

Area of Science:

• Computational Intelligence
• Optimization Algorithms
• Mathematical Analysis

Background:

• The Rastrigin function is a highly multimodal benchmark for global optimization.
• Understanding convergence properties is crucial for effective optimization strategies.

Purpose of the Study:

• To derive a novel aggregated progress rate measure for analyzing the Rastrigin function.
• To investigate convergence properties related to mutation strength and population size.

Main Methods:

• Derivation of an aggregated progress rate measure based on residual distance and normally distributed coordinates.
• Analysis of Rastrigin noise floor for moderate mutation strengths.
• Investigation of local attraction and escape conditions for small mutation strengths.

Main Results:

• A characteristic distance-dependent Rastrigin noise floor was derived for moderate mutation strengths.
• An escape condition was established for small mutation strengths, highlighting optimization challenges.
• A population scaling relation was derived, showing good agreement with experimental data for high global convergence rates.

Conclusions:

• The derived measure facilitates deeper insights into Rastrigin function convergence.
• Optimizing the Rastrigin function requires careful consideration of mutation strength and population size.
• Increasing population size is essential for achieving high global convergence rates.