Unified description of evolutionary strategies over continuous parameter spaces

Area of Science:

• Computational Physics
• Optimization Theory
• Mathematical Modeling

Background:

• Standard optimization techniques like simulated annealing and biological self-reproduction are widely used.
• These methods often face challenges in complex, high-dimensional landscapes.
• Understanding the underlying mathematical framework can lead to improved strategies.

Purpose of the Study:

• To unify thermodynamical and biological self-reproduction strategies under a common mathematical framework.
• To analyze the spectral properties of the Hamiltonian for both strategies.
• To introduce a novel hybrid optimization strategy by mixing these approaches.

Main Methods:

• Reduction of optimization processes to Schrödinger eigenvalue problems.
• Analysis of the Hamiltonian spectrum for different optimization landscapes.
• Investigation of model cases with exact analytical solutions.
• Comparative analysis of thermodynamical and self-reproduction strategies.

Main Results:

• Both simulated annealing and self-reproduction strategies can be mapped to Schrödinger eigenvalue problems.
• The spectral properties of the Hamiltonian differ between the two strategies, reflecting landscape and dynamics.
• A new optimization tool is proposed by combining elements of both strategies.

Conclusions:

• The Schrödinger eigenvalue problem provides a unifying perspective for diverse optimization strategies.
• The spectral analysis offers insights into the behavior of optimization dynamics.
• Mixing thermodynamical and biologically motivated strategies presents a promising avenue for enhanced optimization performance.