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Asymptotic convergence of genetic algorithms

Part of:Limit theoremsMarkov processes

Published online by Cambridge University Press:  01 July 2016

Raphaël Cerf
Raphaël Cerf*
Affiliation:
CNRS, Université d'Orsay
*
∗ Postal address: Université Paris sud, Mathématiques, bâtiment 425, 91405 Orsay Cedex, France. Email address: raphael.cerf@math.u-psud.fr
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Abstract

We study a markovian evolutionary process which encompasses the classical simple genetic algorithm. This process is obtained by randomly perturbing a very simple selection scheme. Using the Freidlin-Wentzell theory, we carry out a precise study of the asymptotic dynamics of the process as the perturbations disappear. We show how a delicate interaction between the perturbations and the selection pressure may force the convergence toward the global maxima of the fitness function. We put forward the existence of a critical population size, above which this kind of convergence can be achieved. We compute upper bounds of this critical population size for several examples. We derive several conditions to ensure convergence in the homogeneous case and these provide the first mathematically well-founded convergence results for genetic algorithms.

Keywords

Freidlin-Wentzell theorygenetic algorithmsstochastic optimization

MSC classification

Primary:60F10: Large deviations60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary:92D15: Problems related to evolution

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 2 , June 1998, pp. 521 - 550
Copyright
Copyright © Applied Probability Trust 1998 

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