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Markov chain analysis of genetic algorithms applied to fitness functions perturbed concurrently by additive and multiplicative noise

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Abstract

We analyze the transition and convergence properties of genetic algorithms (GAs) applied to fitness functions perturbed concurrently by additive and multiplicative noise. Both additive noise and multiplicative noise are assumed to take on finitely many values. We explicitly construct a Markov chain that models the evolution of GAs in this noisy environment and analyze it to investigate the algorithms. Our analysis shows that this Markov chain is indecomposable; it has only one positive recurrent communication class. Using this property, we establish a condition that is both necessary and sufficient for GAs to eventually (i.e., as the number of iterations goes to infinity) find a globally optimal solution with probability 1. Similarly, we identify a condition that is both necessary and sufficient for the algorithms to eventually with probability 1 fail to find any globally optimal solution. Our analysis also shows that the chain has a stationary distribution that is also its steady-state distribution. Based on this property and the transition probabilities of the chain, we compute the exact probability that a GA is guaranteed to select a globally optimal solution upon completion of each iteration.

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References

  1. Arnold, D.V.: Noisy Optimization with Evolution Strategies. Kluwer Academic, Boston (2002)

    Book  MATH  Google Scholar 

  2. Beyer, H.G.: Evolutionary algorithms in noisy environments: theoretical issues and guidelines for practice. Comput. Methods Appl. Mech. Eng. 186, 239–267 (2000)

    Article  MATH  Google Scholar 

  3. Chen, A., Subprasom, K., Ji, Z.: A simulation-based multi-objective genetic algorithm (SMOGA) procedure for BOT network design problem. Optim. Eng. 7, 225–247 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Davis, T.E., Principe, J.: A Markov chain framework for the simple genetic algorithm. Evol. Comput. 1, 269–288 (1993)

    Article  Google Scholar 

  5. Di Pietro, A., White, L., Barone, L.: Applying evolutionary algorithms to problems with noisy, time-consuming fitness functions. In: Proceedings of the 2004 Congress on Evolutionary Computation, vol. 2, pp. 1254–1261 (2004)

    Google Scholar 

  6. Goldberg, D.E., Rudnick, M.W.: Genetic algorithms and the variance of fitness. Complex Syst. 5, 265–278 (1991)

    MATH  Google Scholar 

  7. Hopkins, W.E.: Optimal stabilization of families of linear stochastic differential equations with jump coefficients and multiplicative noise. SIAM J. Control Optim. 25, 1587–1600 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jebalia, M., Auger, A.: On multiplicative noise models for stochastic search. In: Proceedings of the 2008 International Conference on Parallel Problem Solving from Nature, pp. 52–61 (2008)

    Google Scholar 

  9. Jin, Y., Branke, J.: Evolutionary optimization in uncertain environments—a survey. IEEE Trans. Evol. Comput. 3, 303–317 (2005)

    Article  Google Scholar 

  10. Karlin, S., Taylor, H.M.: A First Course in Stochastic Processes. Academic Press, New York (1975)

    MATH  Google Scholar 

  11. Karlin, S., Taylor, H.M.: A Second Course in Stochastic Processes. Academic Press, New York (1981)

    MATH  Google Scholar 

  12. Leung, K.S., Duan, Q.H., Xu, Z.B., Wong, C.K.: A new model of simulated evolutionary computation—convergence analysis and specifications. IEEE Trans. Evol. Comput. 5, 3–16 (2001)

    Article  Google Scholar 

  13. Miller, B.L., Goldberg, D.E.: Genetic algorithms, selection schemes, and the varying effects of noise. Evol. Comput. 4, 113–131 (1996)

    Article  Google Scholar 

  14. Nakama, T.: Markov chain analysis of genetic algorithms applied to fitness functions perturbed by multiple sources of additive noise. Stud. Comput. Intell. 149, 123–136 (2008)

    Article  Google Scholar 

  15. Nakama, T.: Theoretical analysis of genetic algorithms in noisy environments based on a Markov model. In: Proceedings of the 2008 Genetic and Evolutionary Computation Conference, pp. 1001–1008 (2008)

    Chapter  Google Scholar 

  16. Nissen, V., Propach, J.: On the robustness of population-based versus point-based optimization in the presence of noise. IEEE Trans. Evol. Comput. 2, 107–119 (1998)

    Article  Google Scholar 

  17. Nix, A., Vose, M.D.: Modeling genetic algorithm with Markov chains. Ann. Math. Artif. Intell. 5, 27–34 (1992)

    Article  MathSciNet  Google Scholar 

  18. Primbs, J.A.: Portfolio optimization applications of stochastic receding horizon control. In: Proceedings of the 2007 American Control Conference, vol. 1, pp. 1811–1816 (2007)

    Chapter  Google Scholar 

  19. Rudin, L.I., Osher, S.: Total variation based image restoration with free local constraints. In: Proceedings of the 2004 International Conference on Image Processing, vol. 1, pp. 31–35 (2004)

    Google Scholar 

  20. Rudolph, G.: Convergence analysis of canonical genetic algorithms. IEEE Trans. Neural Netw. 5, 96–101 (1994)

    Article  Google Scholar 

  21. Rudolph, G.: Partial order approach to noisy fitness functions. In: Proceedings of the 2001 IEEE Congress on Evolutionary Computation, pp. 318–325 (2001)

    Google Scholar 

  22. Suzuki, J.: A Markov chain analysis on simple genetic algorithms. IEEE Trans. Syst. Man Cybern. 25, 655–659 (1995)

    Article  Google Scholar 

  23. Vose, M.D.: The Simple Genetic Algorithm. MIT Press, Cambridge (1999)

    MATH  Google Scholar 

  24. Vose, M.D., Liepins, G.E.: Punctuated equilibria in genetic search. Complex Syst. 5, 31–44 (1991)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, The Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD, 21218, USA

    Takehiko Nakama

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  1. Takehiko Nakama
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Correspondence to Takehiko Nakama.

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Supported by the Acheson J. Duncan Fund for the Advancement of Research in Statistics.

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Nakama, T. Markov chain analysis of genetic algorithms applied to fitness functions perturbed concurrently by additive and multiplicative noise. Comput Optim Appl 51, 601–622 (2012). https://doi.org/10.1007/s10589-010-9371-1

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  • DOI: https://doi.org/10.1007/s10589-010-9371-1

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