Kinetic description and convergence analysis of genetic algorithms for global optimization

TL;DR

This paper develops a kinetic framework for understanding genetic algorithms, proving their convergence to global optima and deriving continuous models, supported by numerical experiments on benchmark problems.

Contribution

It introduces a kinetic and a Boltzmann-like PDE model for GAs, providing the first rigorous convergence analysis and linking them to mean-field dynamics.

Findings

Proves convergence of GAs under mild assumptions.

Derives a continuous kinetic model for GAs.

Numerical experiments validate the models and analyze configurations.

Abstract

Genetic Algorithms (GA) are a class of metaheuristic global optimization methods inspired by the process of natural selection among individuals in a population. Despite their widespread use, a comprehensive theoretical analysis of these methods remains challenging due to the complexity of the heuristic mechanisms involved. In this work, relying on the tools of statistical physics, we take a first step towards a mathematical understanding of GA by showing how their behavior for a large number of individuals can be approximated through a time-discrete kinetic model. This allows us to prove the convergence of the algorithm towards a global minimum under mild assumptions on the objective function for a popular choice of selection mechanism. Furthermore, we derive a time-continuous model of GA, represented by a Boltzmann-like partial differential equation, and establish relations with other kinetic and mean-field dynamics in optimization. Numerical experiments support the validity of the proposed kinetic approximation and investigate the asymptotic configurations of the GA particle system for different selection mechanisms and benchmark problems.