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On the Impact of Systematic Noise on the Evolutionary Optimization Performance—A Sphere Model Analysis

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Abstract

Quality evaluations in optimization processes are frequently noisy. In particular evolutionary algorithms have been shown to cope with such stochastic variations better than other optimization algorithms. So far mostly additive noise models have been assumed for the analysis. However, we will argue in this paper that this restriction must be relaxed for a large class of applied optimization problems. We suggest “systematic noise” as an alternative scenario, where the noise term is added to the objective parameters or to environmental parameters inside the fitness function. We thoroughly analyze the sphere function with systematic noise for the evolution strategy with global intermediate recombination. The progress rate formula and a measure for the efficiency of the evolutionary progress lead to a recommended ratio between μ and λ. Furthermore, analysis of the dynamics identifies limited regions of convergence dependent on the normalized noise strength and the normalized mutation strength. A residual localization error R ∞ can be quantified and a second μ to λ ratio is derived by minimizing R ∞.

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References

  1. M. Abramowitz and I. A. Stegun, Pocketbook of Mathematical Functions, Verlag Harri Deutsch, Thun, 1984.

    Google Scholar 

  2. B. C. Arnold, N. Balakrishnan, and H. N. Nagaraja, A First Course in Order Statistics, Wiley: New York, 1992.

    Google Scholar 

  3. D. V. Arnold, Noisy Optimization with Evolution Strategies, Kluwer Academic Publishers: Dordrecht, 2002.

    Google Scholar 

  4. D. V. Arnold and H.-G. Beyer, “Local performace of the (µ /µI λ)-ES in a noisy environment,” in Foundations of Genetic Algorithms, W. Martin and W. Spears (Eds.), Morgan Kaufmann: San Francisco, CA, 2001, vol. 6, pp. 127–141.

    Google Scholar 

  5. D. V. Arnold and H.-G. Beyer, “Performance analysis of evolution strategies with multi-recombination in high-dimensional ℝN-search spaces disturbed by noise,” Theoretical Computer Science, vol. 289, pp. 629–647, 2002.

    Google Scholar 

  6. H.-G. Beyer, “Toward a theory of evolution strategies: Some asymptotical results from the (1 + λ)-theory,” Evolutionary Computation, vol. 1, no. 2, pp. 165–188, 1993.

    Google Scholar 

  7. H.-G. Beyer, “Toward a theory of evolution strategies: Self-adaptation,” Evolutionary Computation, vol. 3, no. 3, pp 311–347, 1996.

    Google Scholar 

  8. H.-G. Beyer, “Evolutionary algorithms in noisy environments: Theoretical issues and guidelines for practice,” Computer Methods in Applied Mechanics and Engineering, vol. 186, nos. 2–4, pp. 239–267, 2000.

    Google Scholar 

  9. H.-G. Beyer, The Theory of Evolution Strategies, Natural Computing Series. Springer: Heidelberg, 2001.

    Google Scholar 

  10. H.-G. Beyer and D. V. Arnold, “The steady state behavior of (µ / µI, λ)-ES on ellipsoidal fitness models disturbed by noise,” in GECCO-2003: Proceedings of the Genetic and Evolutionary Computation Conference, E. CantÚ-Paz, J. A. Foster, L. D. K. Deb, Davis, R. Roy, U.-M. O'Reilly, H.-G. Beyer, R. Standish, G. Kendall, S. Wilson, M. Harman, J. Wegener, D. Dasgupta, M. A. Potter, A. C. Schultz, K. A. Dowsland, N. Jonoska, and J. Miller (Eds.) Springer: Berlin, Germany, 2003, pp. 525–536.

    Google Scholar 

  11. H.-G. Beyer, M. Olhofer, and B. Sendhoff, “On the behavior of (µ /µI λ)-ES optimizing functions disturbed by generalized noise,” in Foundations of Genetic Algorithms, K. De Jong, R. Poli, and J. Rowe (Eds.), Morgan Kaufmann: San Francisco, CA, 2003, vol. 7, pp. 307–328.

    Google Scholar 

  12. J. Branke, “Efficient evolutionary algorithms for searching robust solutions,” in Adaptive Computing in Design and Manufacture (ACDM), I. C. Parmee (Ed.), Springer Verlag, 2000, pp. 275–285.

  13. J. Branke, Evolutionary Optimization in Dynamic Environments, Kluwer Academic Publishers: Dordrecht, 2002.

    Google Scholar 

  14. D. Floreano and F. Mondada, “Evolution of homing navigation in a real mobile robot,” IEEE Transactions on Systeme, Man and Cybernetics: Part B Cybernetics, vol. 26, no. 3, pp. 396–407, 1996.

    Google Scholar 

  15. N. Hansen and A. Ostermeier, “Adapting arbitrary normal mutation distributions in evolution strategies: The covariance matrix adaptation,” in Proceedings of 1996 IEEE Int'l Conf. on Evolutionary Computation (ICEC '96), IEEE Press: NY, 1996, pp. 312–317.

    Google Scholar 

  16. N. Hansen and A. Ostermeier, “Completely derandomized self-adaptation in evolution strategies,” Evolutionary Computation, vol. 9, no. 2, pp. 159–195, 2001.

    Google Scholar 

  17. Y. Jin and B. Sendhoff, “Trade-off between optimality and robustness: An evolutionary multiobjective approach,” in Evolutionary Multi-Criterion Optimization, C. M. Fonseca, P. J. Fleming, E. Zitzler, K. Deb, and L. Thiele, (Eds.), Springer Verlag, 2003, pp. 237–251.

  18. S. Obayashi, Y. Yamaguchi, and T. Nakamura, “Multiobjective genetic algorithm for multidisciplinary design of transonic wing planform,” Journal of Aircraft, vol. 34, no. 5, pp. 690–693, 1997.

    Google Scholar 

  19. M. Olhofer, T. Arima, Y. Jin, T. Sonoda, and B. Sendhoff, “Optimisation of transonic gas turbine blades with evolution strategies,” Honda Technical Reviews, April 2002, pp. 203–216.

  20. M. Olhofer, T. Arima, T. Sonoda, M. Fischer, and B. Sendhoff, “Aerodynamic shape optimisation using evolution strategies,” in Optimisation in Industry, I. Parmee and P. Hajela (Eds.), Springer Verlag: Berlin, 2002, pp. 83–94.

    Google Scholar 

  21. M. Olhofer, Y. Jin, and B. Sendhoff, “Adaptive encoding for aerodynamic shape optimization using evolution strategies,” in Congress on Evolutionary Computation (CEC), Seoul, Korea, May 2001, vol. 2, pp. 576–583.

    Google Scholar 

  22. I. Rechenberg, Evolutionsstrategie '94, Frommann-Holzboog Verlag: Stuttgart, 1994.

    Google Scholar 

  23. H.-P. Schwefel, Numerical Optimization of Computer Models, Wiley: Chichester, 1981.

    Google Scholar 

  24. B. Sendhoff, H.-G. Beyer, and M. Olhofer, “On noise induced multi-modality in evolutionary algorithms,” in Proceedings of the 4th Asia-Pacific Conference on Simulated Evolution And Learning-SEAL, L. Wang, K. C. Tan, T. Furuhashi, J.-H. Kim, and X. Yao (Eds.), 2002, vol. 1, pp. 219–224.

  25. T. Sonoda, Y. Yamaguchi, T. Arima, M. Olhofer, B. Sendhoff, and H.-A. Schreiber, “Advanced high turning compressor airfoils for low reynolds number condition, Part 1: Design and optimization,” in Proceedings of the ASME Turbo Expo 2003: Power for Land, Sea, and Air, June 16–19, 2003, Atlanta, Georgia, USA.

  26. P. Stagge, “Averaging efficiently in the presence of noise,” in Parallel Problem Solving from Nature-PPSN V, A. E. Eiben, T. Bäck, M. Schoenauer, and H.-P. Schwefel (Eds.), Springer Verlag, 1998, pp. 188–197.

  27. A. Thompson and P. Layzell, “Evolution of robustness in an electronics design,” in Proc. 3rd Int. Conf. on Evolvable Systems (ICES2000): From Biology to Hardware, J. Miller, A. Thompson, P. Thomson, and T. Fogarty (Eds.), vol. 1801 of LNCS, Springer-Verlag, 2000, pp. 218–228.

  28. S. Tsutsui and A. Gosh, “Genetic algorithms with a robust solution searching scheme,” IEEE Transactions on Evolutionary Computation, vol. 1, no. 3, pp. 201–208, 1997.

    Google Scholar 

  29. D. Wiesmann, U. Hammel, and T. B¨ ack, “Robust design of multilayer optical coatings by means of evolutionary algorithms,” IEEE Transactions on Evolutionary Computation, vol. 2, no. 4, pp. 162–167, 1998.

    Google Scholar 

  30. Y. Yamaguchi and T. Arima, “Multiobjective optimization for transonic compressor stator blade,” in 8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, 2000.

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Authors and Affiliations

  1. Department of Computer Science XI, University of Dortmund, D-44221 Dortmund, Germany

    Hans-Georg Beyer

  2. Honda Research Institute Europe GmbH, Carl-Legien-Str. 30, D-63073 Offenbach/Main, Germany

    Markus Olhofer & Bernhard Sendhoff

Authors
  1. Hans-Georg Beyer
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  2. Markus Olhofer
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  3. Bernhard Sendhoff
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Beyer, HG., Olhofer, M. & Sendhoff, B. On the Impact of Systematic Noise on the Evolutionary Optimization Performance—A Sphere Model Analysis. Genetic Programming and Evolvable Machines 5, 327–360 (2004). https://doi.org/10.1023/B:GENP.0000036020.79188.a0

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