Skip to main content
SpringerLink
Log in
Book cover

Theoretical Aspects of Evolutionary Computing pp 439–493Cite as

Asymptotic Results for Genetic Algorithms with Applications to Nonlinear Estimation

  • Chapter

Part of the book series: Natural Computing Series ((NCS))

Abstract

Genetic algorithms (GAs) are stochastic search methods based on natural evolution processes. They are defined as a system of particles (or individuals) evolving randomly and undergoing adaptation in a time non-necessarily homogeneous environment represented by a collection of fitness functions. The purpose of this work is to study the long-time behavior as well as large population asymptotic of GAs. Another side topic is to discuss the applications of GAs in numerical function analysis, Feynman—Kac formulae approximations, and in nonlinear filtering problems. Several variations and refinements will also be presented including continuous-time and branching particle models with random population size.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Abela, D. Abramson, M. Krishnamoorthy, A. De Silval, and G. Mills. Computing Optimal Schedules for Landing Aircraft. Technical Report, Department of Computer Systems Eng. R.M.I.T., Melbourne, May 25, 1993.

    Google Scholar 

  2. J. Baker. Adaptive selection methods for genetic algorithms. In J. Grefenstette, editor, Proceedings of the First International Conference on Genetic Algorithms and their Applications, pages 101–111. Lawrence Erlbaum, Hillsdale, NJ, 1985.

    Google Scholar 

  3. J. Baker. Reducing bias and inefficiency in the selection algorithm. In J. Grefenstette, editor, Proceedings of the Second International Conference on Genetic Algorithms and their Applications, pages 14–21. Lawrence Erlbaum, Hillsdale, NJ, 1987.

    Google Scholar 

  4. P. Barbe and P. Bertail. The Weighted Bootstrap. Lecture Notes in Statistics 98. Springer-Verlag, Berlin Heidelberg New York, 1995.

    Book  MATH  Google Scholar 

  5. R. Bott and J. P. Mayberry. Matrices and Trees. Economics Activity Analysis. Wiley, New York, 1954.

    Google Scholar 

  6. H. Carvalho. Filtrage Optimal Non Linéaire du Signal GPS NAVSTAR en Racalage de Centrales de Navigation. Thèse de L’Ecole Nationale Supérieure de l’Aéronautique et de l’Espace, September 1995.

    Google Scholar 

  7. H. Carvalho, P. Del Moral, A. Monin, and G. Salut. Optimal non-linear filtering in GPS/INS integration. IEEE Trans. on Aerospace and Electronic Systems, 33(3):835–850, 1997.

    Article  Google Scholar 

  8. R. Cerf. Asymptotic convergence of a genetic algorithm. C.R.Acad. Sci. Paris Sér. I Math., 319(3):271–276, 1994.

    MathSciNet  MATH  Google Scholar 

  9. R. Cerf. The dynamics of mutation-selection algorithms with large population sizes. Ann. Inst.H.Poincaré Probab. Statist., 32(4):455–508, 1996.

    MathSciNet  MATH  Google Scholar 

  10. R. Cerf. Asymptotic convergence of genetic algorithms. Adv. in Appl. Probab., 30(2):521–550, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  11. D. Crisan, J. Gaines, and T. J. Lyons. A particle approximation of the solution of the Kushner-Stratonovitch equation. SIAM J. Appl. Math., 58(5):1568–1590, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  12. D. Crisan, P. Del Moral, and T. J. Lyons. Discrete filtering using branching and interacting particle systems. Markov Processes and Related Fields, 5(3):293–318, 1999.

    MathSciNet  MATH  Google Scholar 

  13. D. Crisan, P. Del Moral, and T. J. Lyons. Interacting particle systems approximations of the Kushner-Stratonovitch equation. Advances in Applied Probability, 31(3):819–838, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  14. D. Crisan and M. Grunwald. Large deviation comparison of branching algorithms versus re-sampling algorithms. Preprint, Imperial College, London, 1998.

    Google Scholar 

  15. D. Dawson. Measure-valued Markov processes. In P. L. Hennequin, editor, Ecole d’Eté de Probabilités de Saint-Flour XXI-1991, Lecture Notes in Mathematics 1541. Springer-Verlag, Berlin Heidelberg New York, 1993.

    Google Scholar 

  16. D. Delahaye, J.-M. Alliot, M. Schoenauer, and J.-L. Farges. Genetic algorithms for automatic regrouping of air traffic control sectors. In J. R. McDonnell, R. G. Reynolds, and D. B. Fogel, editors, Proceedings of the 4th Annual Conference on Evolutionary Programming, pages 657–672. MIT Press, Cambridge, MA, March 1995.

    Google Scholar 

  17. P. Del Moral, J. Jacod, and Ph. Protter. The Monte-Carlo method for filtering with discrete-time observations. Publications du Laboratoire de Probabilités, 453, Paris VI, France, June 1998.

    Google Scholar 

  18. P. Del Moral and J. Jacod. The Monte-Carlo method for filtering with discrete-time observations. Central Limit Theorems. Publications du Laboratoire de Statistiques et Probabilités, 7, Toulouse III, France, 1999.

    Google Scholar 

  19. P. Del Moral and J. Jacod. Interacting particle filtering with discrete observations. Publications du Laboratoire de Statistiques et Probabilités, 8, Toulouse III, France, 1999.

    Google Scholar 

  20. P. Del Moral and L. Miclo. Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering. Publications du Laboratoire de Statistiques et Probabilités, 5, Toulouse III, France, 1999.

    Google Scholar 

  21. P. Del Moral and L. Miclo. On the convergence and the applications of the generalized simulated annealing. SIAM Control and Optimization, 37(4): 1222–1250.

    Google Scholar 

  22. P. Del Moral and L. Miclo. Asymptotic stability of nonlinear semigroups of Feynman-Kac type. Publications du Laboratoire de Statistiques et Probabilités, 4, Toulouse III, France, 1999.

    Google Scholar 

  23. P. Del Moral and L. Miclo. A Moran particle system approximation of Feynman-Kac formulae. Publications du Laboratoire de Statistiques et Probabilités, 11, Toulouse III, France, 1998.

    Google Scholar 

  24. P. Del Moral and A. Guionnet. Large deviations for interacting particle systems. Applications to nonlinear filtering problems. Stochastic Processes and their Applications, 78:69–95, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  25. P. Del Moral and A. Guionnet. On the stability of measure-valued processes. Applications to nonlinear filtering and interacting particle systems. Publications du Laboratoire de Statistiques et Probabilités, 3, Toulouse III, France, 1998.

    Google Scholar 

  26. P. Del Moral. Nonlinear filtering: interacting particle solution. Markov Processes and Related Fields, 2(4):555–580, 1996.

    MathSciNet  MATH  Google Scholar 

  27. P. Del Moral. Measure valued processes and interacting particle systems. Application to nonlinear filtering problems. Ann. Appl. Probab., 8(2):438–495, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  28. P. Del Moral, J. C. Noyer, and G. Salut. Résolution particulaire et traitement nonlinéaire du signal: application Radar/Sonar. Revue du Traitement du Signal, Septembre 1995.

    Google Scholar 

  29. P. Del Moral, G. Rigal, J. C. Noyer, and G. Salut. Traitement non-linéaire du signal par reseau particulaire: application radar. Quatorzième colloque GRETSI, Juan les Pins, 13–16 Septembre 1993.

    Google Scholar 

  30. P. Del Moral, G. Rigal, and G. Salut. Estimation et commande optimale non lineaire. Contract D.R.E.T.-DIGILOG-LAAS/CNRS, SM.MCY/685.92/A, 89.34.553.00.470.75.01, Report No. 2, 18 Mars 1992.

    Google Scholar 

  31. S. Ethier and T. Kurtz. Markov Processes, Characterization and Convergence. Wiley series in probability and mathematical statistics. John Wiley, New York, 1986.

    Book  MATH  Google Scholar 

  32. M. I. Freidlin and A. D. Wentzell. Random Perturbations of Dynamical Systems. Grundlehren der math. Wissenschaften, vol. 260. Springer-Verlag, Berlin Heidelberg New York, 1984.

    Book  MATH  Google Scholar 

  33. D. E. Goldberg. Genetic algorithms and rule learning in dynamic control systems. In J. Grefenstette, editor, Proceedings of the First International Conference on Genetic Algorithms, pages 8–15. Lawrence Erlbaum, Hillsdale, NJ, 1985.

    Google Scholar 

  34. D. E. Goldberg. Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Reading, MA, 1989.

    MATH  Google Scholar 

  35. N. J. Gordon, D. J. Salmon, and A. F. M. Smith. Novel Approach to Non-Linear/NonGaussian Bayesian State Estimation. IEEE, 1993.

    Google Scholar 

  36. C. Graham and S. Méléard. Stochastic particle approximations for generalized Boltzmann models and convergence estimates. The Annals of Probability, 25(1):115–132, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  37. J. H. Holland. Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor, 1975.

    Google Scholar 

  38. J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes. Grundlehren der math. Wissenschaften, vol. 288. Springer-Verlag, Berlin Heidelberg New York, 1987.

    Book  MATH  Google Scholar 

  39. J. M. Johnson and Y. Rahmat-Samii. Genetic algorithms in engineering electromagnetics. AP-S Magazine. 39:7–25, 1997.

    Article  Google Scholar 

  40. C. Kane and M. Schoenauer. Topological optimum design using genetic algorithms. Control and Cybernetics, 25(5), 1996.

    MathSciNet  Google Scholar 

  41. G. Kitagawa. Monte-Carlo filter and smoother for non-Gaussian nonlinear state space models. Comput. and Graphical Stat., 5(1):1–25, 1996.

    MathSciNet  Google Scholar 

  42. J. W. Kwiatkowski. Algorithms for Index Tracking. Department of Business Studies, University of Edinburgh, UK, 1991.

    Google Scholar 

  43. R. S. Liptser and A. N. Shiryayev. Theory of Martingales. Kluwer Academic, Dordrecht, 1989.

    Book  MATH  Google Scholar 

  44. S. Méléard. Asymptotic behaviour of some interacting particle systems; McKeanVlasov and Boltzmann models. In D. Talay and L. Tubaro, editors, Probabilistic Models for Nonlinear Partial Differential Equations, Montecatini Terme, 1995, Lecture Notes in Mathematics 1627. Springer-Verlag, Berlin Heidelberg New York, 1996.

    Google Scholar 

  45. C. Musso and N. Oudjane. Regularized Particle Schemes applied to the Tracking Problem. Preprint, ONERA Chatillon, 1998.

    Google Scholar 

  46. C. Musso and N. Oudjane. Regularization schemes for branching particle systems as a numerical solving method of the nonlinear filtering problem. Preprint, ONERA Chatillon, 1998.

    Google Scholar 

  47. Y. Nishiyama. Some central limit theorems for ∞-valued semimartingales and their applications. Probability Theory and Related Fields, 108:459–494, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  48. D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Grundlehren der math. Wissenschaften, vol. 293. Springer-Verlag, Berlin Heidelberg New York, 1991.

    MATH  Google Scholar 

  49. J. Shapcott. Index Tracking: Genetic Algorithms for Investment Portfolio Selection. EPCC—SS92–24, September 1992.

    Google Scholar 

  50. T. Shiga and H. Tanaka. Central limit theorem for a system of Markovian particles with mean field interaction. Z.f. Wahrscheinlichkeitstheorie verw. Geb., 69:439–459, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  51. M. Schoenauer, L. Kallel, and F. Jouve. Mechanical inclusions identification by evolutionary computation. Finite Elem., 5(5–6):619–648, 1996.

    MathSciNet  MATH  Google Scholar 

  52. M. Schoenauer, F. Jouve, and L. Kallel. Identification of mechanical inclusions. In D. Dasgupta and Z. Michalewicz, editors, Evolutionary Algorithms in Engineering Applications, pages 479–496. Springer-Verlag, Berlin Heidelberg New York, 1997.

    Google Scholar 

  53. D. Treyer, D. S. Weile, and E. Michielsen. The application of novel genetic algorithms to electromagnetic problems. Applied Computational Electromagnetics Symnposiumn Digest, 2:1382–1386. Monterey, CA, 1997.

    Google Scholar 

  54. A. Trouvé. Parallélisation massive du recuit simulé. Thèse de Doctorat, Université Paris XI, January 1993.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Statistiques et Probabilités CNRS-UMR C5583, Université Paul Sabatier, 118, route de Narbonne, 31062, Toulouse cedex, France

    P. Del Moral & L. Miclo

Authors
  1. P. Del Moral
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Miclo
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Ecole Polytechnique, CMAP, 91128, Palaiseau Cedex, France

    Leila Kallel

  2. Departement Wiskunde & Informatica, Universiteit Antwerpen (RUCA), Groenenborgerlaan 171, 2020, Antwerpen, Belgium

    Bart Naudts

  3. Department of Electronics and Computer Science Highfield, University of Southampton, SO17 1BJ, Southampton, UK

    Alex Rogers

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Del Moral, P., Miclo, L. (2001). Asymptotic Results for Genetic Algorithms with Applications to Nonlinear Estimation. In: Kallel, L., Naudts, B., Rogers, A. (eds) Theoretical Aspects of Evolutionary Computing. Natural Computing Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04448-3_22

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-04448-3_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08676-2

  • Online ISBN: 978-3-662-04448-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation