Applied Computational Intelligence and Soft Computing

Volume 2010 (2010), Article ID 185063, 11 pages

http://dx.doi.org/10.1155/2010/185063

Review Article

## A Review of Constraint-Handling Techniques for Evolution Strategies

International Computer Science Institute, Berkeley, CA 94704, USA

Received 24 September 2009; Accepted 6 January 2010

Academic Editor: Chuan-Kang Ting

Copyright © 2010 Oliver Kramer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- C. Floudas and P. Pardalos,
*A Collection of Test Problems for Constrained Global Optimization Algorithms*, Springer, Berlin, Germany, 1990. - O. Kramer, “Premature convergence in constrained continuous search spaces,” in
*Proceedings of the 10th International Conference on Parallel Problem Solving from Nature (PPSN '08)*, pp. 62–71, Springer, Dortmund, Germany, 2008. - O. Kramer, A. Barthelmes, and G. Rudolph, “Surrogate constraint functions for CMA evolution strategies,” in
*Proceedings of the 32nd German Annual Conference on Artificial Intelligence (KI '09)*, pp. 169–176, Paderborn, Germany, September 2009. - O. Kramer, S. Brugger, and D. Lazovic, “Sex and death: towards biologically inspired heuristics for constraint handling,” in
*Proceedings of the 9th Conference on Genetic and Evolutionary Computation (GECCO '07)*, pp. 666–673, ACM Press, London, UK, July 2007. - O. Kramer and H.-P. Schwefel, “On three new approaches to handle constraints within evolution strategies,”
*Natural Computing*, vol. 5, no. 4, pp. 363–385, 2006. View at Google Scholar - O. Kramer, C.-K. Ting, and H. Kleine Büning, “A mutation operator for evolution strategies to handle constrained problems,” in
*Proceedings of the 7th Conference on Genetic and Evolutionary Computation (GECCO '05)*, pp. 917–918, Washington, DC, USA, June 2005. View at Publisher · View at Google Scholar - O. Kramer, C.-K. Ting, and H. Kleine Büning, “A new mutation operator for evolution strategies for constrained problems,” in
*Proceedings of the IEEE Congress on Evolutionary Computation (CEC '05)*, pp. 2600–2606, Edinburgh, UK, September 2005. - I. Rechenberg,
*Evolutionsstrategie: Optimierung Technischer Systeme nach Prinzipien der Biologischen Evolution*, Frommann-Holzboog, Stuttgart, Germany, 1973. - H.-P. Schwefel,
*Numerische Optimierung von Computer-Modellen Mittel der Evolutionsstrategie*, Birkhäuser, Basel, Switzerland, 1977. - H.-G. Beyer and H.-P. Schwefel, “Evolution strategies—a comprehensive introduction,”
*Natural Computing*, vol. 1, pp. 3–52, 2002. View at Google Scholar - A. E. Eiben and J. E. Smith,
*Introduction to Evolutionary Computing*, Springer, Berlin, Germany, 2003. - J. C. Bean and A. B. Hadj-Alouane, “A dual genetic algorithmfor bounded integer programs,” Tech. Rep., University of Michigan, Kalamazoo, Mich, USA, 1992. View at Google Scholar
- A. Fiacco and G. McCormick, “The sequential unconstrained minimization technique for nonlinear programming—a primal-dual method,”
*Management Science*, vol. 10, pp. 360–366, 1964. View at Google Scholar - A. Homaifar, S. H. Y. Lai, and X. Qi, “Constrained optimization via genetic algorithms,”
*Simulation*, vol. 62, no. 4, pp. 242–254, 1994. View at Google Scholar - J. Joines and C. Houck, “On the use of non-stationary penalty functions to solve nonlinear constrained optimization problems with GAs,” in
*Proceedings of the 1st IEEE Conference on Evolutionary Computation*, D. B. Fogel, Ed., pp. 579–584, IEEE Press, Orlando, Fla, USA, June 1994. - A. Kuri-Morales and C. V. Quezada, “A universal eclectic genetic algorithm for constrained optimization,” in
*Proceedings 6th European Congress on Intelligent Techniques & Soft Computing (EUFIT '98)*, pp. 518–522, Mainz, Aachen, Germany, September 1998. - R. G. L. Riche, C. Knopf-Lenoir, and R. T. Haftka, “A segregated genetic algorithm for constrained structural optimization,” in
*Proceedings of the 6th International Conference on Genetic Algorithms (ICGA '95)*, L. J. Eshelman, Ed., pp. 558–565, University of Pittsburgh, Morgan Kaufmann, San Francisco, Calif, USA, July 1995. - C. A. Coello Coello, “Use of a self-adaptive penalty approach for engineering optimization problems,”
*Computers in Industry*, vol. 41, no. 2, pp. 113–127, 2000. View at Publisher · View at Google Scholar - K. Deb, “An efficient constraint handling method for genetic algorithms,”
*Computer Methods in Applied Mechanics and Engineering*, pp. 971–978, 2001. View at Google Scholar - D. Powell and M. M. Skolnick, “Using genetic algorithms in engineering design optimization with non-linear constraints,” in
*Proceedings of the 5th International Conference on Genetic Algorithms (ICGA '93)*, S. Forrest, Ed., pp. 424–431, University of Illinois at Urbana-Champaign, Morgan Kaufmann, San Francisco, Calif, USA, July 1993. - F. Hoffmeister and J. Sprave, “Problem-independent handling of constraints by use of metric penalty functions,” in
*Proceedings of the 5th Conference on Evolutionary Programming (EP '96)*, L. J. Fogel, P. J. Angeline, and T. Bäck, Eds., pp. 289–294, MIT Press, Cambridge, UK, February 1996. - A. I. Oyman, K. Deb, and H.-G. Beyer, “An alternative constraint handling method for evolution strategies,” in
*Proceedings of the Congress on Evolutionary Computation (CEC '99)*, vol. 1, pp. 612–619, IEEE Service Center, Piscataway, NJ, USA, July 1999. - T. P. Runarsson, “Approximate evolution strategy using stochastic ranking,” in
*Proceedings of the IEEE Congress on Evolutionary Computation (CEC '06)*, pp. 2760–2767, IEEE, Vancouver, Canada, July 2006. - S. V. Belur, “CORE: constrained optimization by randomevolution,” in
*Proceedings of the Late Breaking Papers at the Genetic Programming Conference*, J. R. Koza, Ed., pp. 280–286, Stanford University, Stanford, Calif, USA, July 1997. - C. A. Coello Coello, “Theoretical and numerical constraint handling techniques used with evolutionary algorithms: a survey of the state of the art,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 191, no. 11-12, pp. 1245–1287, 2002. View at Google Scholar - S. Koziel and Z. Michalewicz, “Evolutionary algorithms, homomorphous mappings, and constrained parameter optimization,”
*Evolutionary Computation*, vol. 7, no. 1, pp. 19–44, 1999. View at Google Scholar - Z. Michalewicz and D. B. Fogel,
*How to Solve It: Modern Heuristics*, Springer, Berlin, Germany, 2000. - J. Paredis, “Co-evolutionary constraint satisfaction,” in
*Proceedings of the 3rd Conference on Parallel Problem Solving from Nature (PPSN '94)*, pp. 46–55, Springer, Jerusalem, Israel, October 1994. - M. Schoenauer and Z. Michalewicz, “Evolutionary computation at the edge of feasibility,” in
*Proceedings of the 4th Conference on Parallel Problem Solving from Nature (PPSN '96)*, H.-M. Voigt, W. Ebeling, I. Rechenberg, and H.-P. Schwefel, Eds., pp. 245–254, Berlin, Germany, September 1996. - C. A. Coello Coello, “Constraint handling through a multiobjective optimization technique,” in
*Proceedings of the Genetic and Evolutionary Computation Conference*, A. S. Wu, Ed., pp. 117–118, Orlando, Fla, USA, July 1999. - C. A. Coello Coello, “Treating constraints as objectives for single-objective evolutionary optimization,”
*Engineering Optimization*, vol. 32, no. 3, pp. 275–308, 2000. View at Google Scholar - F. Jimenez and J. L. Verdegay, “Evolutionary techniques for constrained optimization problem,” in
*Proceedings of the 7th European Congress on Intelligent Techniques and Soft Computing (EUFIT '99)*, H.-J. Zimmermann, Ed., Mainz, Aachen, Germany, September 1999. - E. Mezura-Montes and C. A. Coello Coello, “Constrained optimization via multiobjective evolutionary algorithms,”
*Multi-Objective Problem Solving from Nature: From Concepts to Applications*, pp. 53–75, 2008. View at Google Scholar - I. C. Parmee and G. Purchase, “The development of a directed genetic search technique for heavily constrained design spaces,” in
*Proceedings of the Conference on Adaptive Computing in Engineering Design and Control (PEDC '94)*, I. C. Parmee, Ed., pp. 97–102, University of Plymouth, Plymouth, UK, September 1994. - M. Schoenauer and S. Xanthakis, “Constrained GA optimization,” in
*Proceedings of the 5th International Conference on Genetic Algorithms (ICGA '93)*, S. Forrest, Ed., pp. 573–580, Morgan Kaufman, San Francisco, Calif, USA, July 1993. - P. D. Surry, N. J. Radcliffe, and I. D. Boyd, “Amulti-objective approach to constrained optimisation of gas supply networks: the COMOGA Method,” in
*Proceedings of the Evolutionary Computing, AISB Workshop*, T. C. Fogarty, Ed., Lecture Notes in Computer Science, pp. 166–180, Springer, Sheffield, UK, April 1995. - E. M. Montes and C. A. Coello Coello, “A simple multi-membered evolution strategy to solve constrained optimization problems,”
*IEEE Transactions on Evolutionary Computation*, vol. 9, no. 1, pp. 1–17, 2005. View at Google Scholar - T. Takahama and S. Sakai, “Constrained optimization by the e constrained differential evolution with gradient-based mutation and feasible elites,” in
*Proceedings of the IEEE Congress on Evolutionary Computation (CEC '06)*, G. G. Yen, S. M. Lucas, G. Fogel et al., Eds., pp. 1–8, IEEE Press, Vancouver, Canada, July 2006. - J. Liang and P. Suganthan, “Dynamic multi-swarm particle swarm optimizer with a novel constraint-handling mechanism,” in
*Proceedings of the IEEE Congress on Evolutionary Computation*, G. G. Yen, S. M. Lucas, G. Fogel et al., Eds., pp. 9–16, IEEE Press, Vancouver, Canada, July 2006. - E. Mezura-Montes, J. Velazquez-Reyes, and C. A. Coello Coello, “Modified differential evolution for constrained optimization,” in
*Proceedings of the IEEE Congress on Evolutionary Computation*, G. G. Yen, S. M. Lucas, G. Fogel et al., Eds., pp. 25–32, Vancouver, Canada, July 2006. - E. Mezura-Montes, Ed.,
*Constraint-Handling in Evolutionary Computation*, vol. 198 of*Studies in Computational Intelligence*, Springer, Berlin, Germany, 2009. - H.-P. Schwefel,
*Evolutionsstrategie und numerische optimierung*, Ph.D. thesis, TU Berlin, Berlin, Germany, 1975. - H.-P. Schwefel,
*Evolution and Optimum Seeking. Sixth-Generation Computer Technology*, Wiley Interscience, New York, NY, USA, 1995. - D. V. Arnold and D. Brauer, “On the behaviour of the (1+1)-es for a simple constrained problem,” in
*Proceedings of the 10th International Conference on Parallel Problem Solving From Nature (PPSN '08)*, pp. 1–10, Dortmund, Germany, September 2008. - H.-P. Schwefel, “Adaptive mechanismen in der biologischen evolution und ihr einfluss auf die evolutionsgeschwindigkeit,” in
*Interner Bericht der Arbeitsgruppe Bionik und Evolutionstechnik am Institut für Mess- und Regelungstechnik*, TU Berlin, Berlin, Germany, July 1974. View at Google Scholar - N. Hansen, “The cma evolution strategy: a tutorial,” Tech. Rep., TU Berlin, ETH Zürich, Germany, 2005. View at Google Scholar
- A. Ostermeier, A. Gawelczyk, and N. Hansen, “A derandomized approach to self adaptation of evolution strategies,”
*Evolutionary Computation*, vol. 2, no. 4, pp. 369–380, 1994. View at Google Scholar - M. Emmerich, A. Giotis, M. Özdemir, T. Bäck, and K. Giannakoglou, “Metamodel-assisted evolution strategies,” in
*Proceedings of the 7th International Conference on Parallel Problem Solving from Nature (PPSN '02)*, pp. 361–370, Granada, Spain, September 2002. - S. Kern, N. Hansen, and P. Koumoutsakos, “Local meta-models for optimization using evolution strategies,” in
*Proceedings of the 9th International Conference on Parallel Problem Solving from Nature (PPSN '06)*, pp. 939–948, Reykjavik, Iceland, 2006. - H. Ulmer, F. Streichert, and A. Zell,
*Optimization by Gaussian Processes Assisted Evolution Strategies*, Springer, Heidelberg, Germany, 2003. - G. Marsaglia, “Choosing a point from the surface of a sphere,”
*The Annals of Mathematical Statistics*, vol. 43, pp. 645–646, 1972. View at Google Scholar - T. Hastie, R. Tibshirani, and J. Friedman,
*The Elements of Statistical Learning*, Springer, Berlin, Germany, 2009.