Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T17:10:52.327Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic convergence of genetic algorithms

Part of: Limit theorems Markov 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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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 deviations 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 92D15: Problems related to evolution
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 2 , June 1998 , pp. 521 - 550
Copyright
Copyright © Applied Probability Trust 1998 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Ben Arous, G. and Cerf, R. (1996). Metastability of the three dimensional Ising model on a torus at very low temperatures. Electronic J. Prob. 1, 1–55.Google Scholar
[2] Cerf, R. (1996). The dynamics of mutation–selection algorithms with large population sizes. Ann. Inst. Henri Poincaré 32, 455–508.Google Scholar
[3] Deuschel, J. D. and Mazza, C. (1994). L2 convergence of time nonhomogeneous Markov processes. I. Spectral estimates. Ann. Appl. Prob. 4, 1012–1056.CrossRefGoogle Scholar
[4] Freidlin, M. I. and Wentzell, A. D. (1984). Random Perturbations of Dynamical Systems. Springer, New York.CrossRefGoogle Scholar
[5] Goldberg, D. (1989). Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley.Google Scholar
[6] Holland, J. H. (1975). Adaptation in Natural and Artificial Systems. Ann Arbor, The University of Michigan Press.Google Scholar
[7] Kifer, Y. (1988). Random Perturbations of Dynamical Systems. Boston, Basel, Birk-häuser.Google Scholar
[8] Neves, E. J. (1995). A discrete variational problem related to Ising droplets at low temperatures. J. Stat. Phys. 80, 103–123.Google Scholar
[9] Neves, E. J. and Schonmann, R. H. (1991). Critical droplets and metastability for a Glauber dynamics at very low temperatures. Commun. Math. Phys. 137, 209–230.Google Scholar
[10] Pólya, G. and Szegö, G. (1972). Problems and Theorems in Analysis. I. Berlin, Heidelberg, Springer.Google Scholar
[11] Seneta, E. (1981). Non-Negative Matrices and Markov Chains. New York, Springer.CrossRefGoogle Scholar
[12] Trouvé, A., (1995). Asymptotical behaviour of several interacting annealing processes. Prob. Theory Rel. Fields 102, 123–143.CrossRefGoogle Scholar