Abstract
The multi-objective covariance matrix adaptation evolution strategy (MO-CMA-ES) combines a mutation operator that adapts its search distribution to the underlying optimization problem with multi-criteria selection. Here, a generational and two steady-state selection schemes for the MO-CMA-ES are compared. Further, a recently proposed method for computationally efficient adaptation of the search distribution is evaluated in the context of the MO-CMA-ES.
Keywords
- Covariance Matrix
- Pareto Front
- Multiobjective Optimization
- Cholesky Factor
- Evolution Strategy
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Rechenberg, I.: Evolutionsstrategie: Optimierung Technischer Systeme nach Prinzipien der Biologischen Evolution. Werkstatt Bionik und Evolutionstechnik. Frommann-Holzboog, Stuttgart (1973)
Schwefel, H.-P.: Evolution and Optimum Seeking. Sixth-Generation Computer Technology Series. John Wiley & Sons, Chichester (1995)
Hansen, N., Ostermeier, A.: Completely derandomized self-adaptation in evolution strategies. Evolutionary Computation 9(2), 159–195 (2001)
Kern, S., Müller, S., Hansen, N., Büche, D., Ocenasek, J., Koumoutsakos, P.: Learning probability distributions in continuous evolutionary algorithms – A comparative review. Natural Computing 3, 77–112 (2004)
Igel, C., Hansen, N., Roth, S.: Covariance matrix adaptation for multi-objective optimization. Evolutionary Computation (Accepted, 2006)
Deb, K., Agrawal, S., Pratap, A., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6(2), 182–197 (2002)
Emmerich, M., Beume, N., Naujoks, B.: An EMO algorithm using the hypervolume measure as selection criterion. In: Coello Coello, C.A., Hernández Aguirre, A., Zitzler, E. (eds.) EMO 2005. LNCS, vol. 3410, pp. 62–76. Springer, Heidelberg (2005)
Beume, N., Naujoks, B., Emmerich, M.: SMS-EMOA: Multiobjective selection based on dominated hypervolume. European Journal of Operational Research (In press, 2007)
Igel, C., Suttorp, T., Hansen, N.: A computational efficient covariance matrix update and a (1+1)-CMA for evolution strategies. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2006), pp. 453–460. ACM Press, New York (2006)
Hansen, N., Müller, S.D., Koumoutsakos, P.: Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES). Evolutionary Computation 11(1), 1–18 (2003)
Grewal, M.S., Andrews, A.P.: Kalman Filtering: Theory and Practice. Prentice-Hall, Englewood Cliffs (1993)
Deb, K.: Multi-Objective Optimization Using Evolutionary Algorithms. Wiley, Chichester (2001)
Zitzler, E., Thiele, L.: Multiobjective optimization using evolutionary algorithms – a comparative case study. In: Eiben, A.E., Bäck, T., Schoenauer, M., Schwefel, H.-P. (eds.) Parallel Problem Solving from Nature - PPSN V. LNCS, vol. 1498, pp. 292–301. Springer, Heidelberg (1998)
Whitley, L.D.: The GENITOR algorithm and selection pressure: Why rank–based allocation of reproductive trials is best. In: Schaffer, J.D. (ed.) Proceedings of the Third International Conference on Genetic Algorithms (ICGA’89), pp. 116–121. Morgan Kaufmann, San Francisco (1989)
Syswerda, G.: Uniform crossover in genetic algorithms. In: Schaffer, J.D. (ed.) Proceedings of the Third International Conference on Genetic Algorithms (ICGA’89), pp. 2–9. Morgan Kaufmann, San Francisco (1989)
Iorio, A., Li, X.: Solving rotated multi-objective optimization problems using differential evolution. In: Webb, G.I., Yu, X. (eds.) AI 2004. LNCS (LNAI), vol. 3339, pp. 861–872. Springer, Heidelberg (2004)
Knowles, J., Thiele, L., Zitzler, E.: A tutorial on the performance assessment of stochastic multiobjective optimizers. TIK-Report 214, Computer Engineering and Networks Laboratory (TIK), Swiss Federal Institute of Technology (ETH) Zurich (2005)
Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., Grunert da Fonseca, V.: Performance assesment of multiobjective optimizers: An analysis and review. IEEE Transactions on Evolutionary Computation 7(2), 117–132 (2003)
Bleuler, S., Laumanns, M., Thiele, L., Zitzler, E.: PISA – A platform and programming language independent interface for search algorithms. In: Fonseca, C.M., Fleming, P.J., Zitzler, E., Deb, K., Thiele, L. (eds.) EMO 2003. LNCS, vol. 2632, pp. 494–508. Springer, Heidelberg (2003)
Fonseca, C.M., Fleming, P.J.: Multiobjective optimization and multiple constraint handling with evolutionary algorithms—Part II: Application example. IEEE Transactions on Systems, Man, and Cybernetics, Part A: Systems and Humans 28(1), 38–47 (1998)
Zitzler, E., Deb, K., Thiele, L.: Comparison of multiobjective evolutionary algorithms: Empirical results. Evolutionary Computation 8(2), 173–195 (2000)
Deb, K., Sinha, A., Kukkonen, S.: Multi-objective test problems, linkages, and evolutionary methodologies. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2006), pp. 1141–1148. ACM Press, New York (2006)
Iorio, A.W., Li, X.: Rotated test problems for assessing the performance of multi-objective optimization algorithms. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2006), pp. 683–690. ACM Press, New York (2006)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Igel, C., Suttorp, T., Hansen, N. (2007). Steady-State Selection and Efficient Covariance Matrix Update in the Multi-objective CMA-ES. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T. (eds) Evolutionary Multi-Criterion Optimization. EMO 2007. Lecture Notes in Computer Science, vol 4403. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70928-2_16
Download citation
DOI: https://doi.org/10.1007/978-3-540-70928-2_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70927-5
Online ISBN: 978-3-540-70928-2
eBook Packages: Computer ScienceComputer Science (R0)