Soft Computing

, Volume 16, Issue 2, pp 331–351 | Cite as

Modeling dynamics of a real-coded CHC algorithm in terms of dynamical probability distributions

  • Jesús MarínEmail author
  • Daniel Molina
  • Francisco Herrera
Original Paper


Some theoretical models have been proposed in the literature to predict dynamics of real-coded evolutionary algorithms. These models are often applied to study very simplified algorithms, simple real-coded functions or sometimes these make difficult to obtain quantitative measures related to algorithm performance. This paper, trying to reduce these simplifications to obtain a more useful model, proposes a model that describes the behavior of a slightly simplified version of the popular real-coded CHC in multi-peaked landscape functions. Our approach is based on predicting the shape of the search pattern by modeling the dynamics of clusters, which are formed by individuals of the population. This is performed in terms of dynamical probability distributions as a basis to estimate its averaged behavior. Within reasonable time, numerical experiments show that is possible to achieve accurate quantitative predictions in functions of up to 5D about performance measures such as average fitness, the best fitness reached or number of fitness function evaluations.


CHC algorithm Probability distribution Real-coded optimization Algorithm dynamics 



This work is supported by the Ministerio de Ciencia e Innovación (Spain) under grant TEC2008-02754/TEC and TIN2008-05854.


  1. Barnett L (1998) Collapsing the state space by applying markov analysis to evolutionary systems. In: Workshop presentation at the sixth international conference on artificial life, UCLAGoogle Scholar
  2. Beyer H, Schwefel H (2002) Evolution strategies—a comprehensive introduction. Nat Comput 1(1):3–52CrossRefzbMATHMathSciNetGoogle Scholar
  3. Beyer HG, Schwefel HP, Wegener I (2002) How to analyse evolutionary algorithms. Theor Comput Sci 287(1):101–130CrossRefzbMATHMathSciNetGoogle Scholar
  4. Bornholdt S (1998) Genetic algorithm dynamics on a rugged landscape. Phys Rev E 57(4):3853–3860CrossRefGoogle Scholar
  5. Cano JR, Herrera F, Lozano M (2003) Using evolutionary algorithms as instance selection for data reduction in kdd: an experimental study. IEEE Trans Evol Comput 7(6):561–575CrossRefGoogle Scholar
  6. Cano JR, Herrera F, Lozano M (2005) Strategies for scaling up evolutionary instance reduction algorithms for data mining. In: Ghosh A, Jain L (eds) Evolutionary computation in data mining. Studies in fuzziness and soft computing, vol 163. Springer, Berlin, pp 21–39Google Scholar
  7. Cervone G, Kaufman KK, Michalski RS (2000) Experimental validations of the learnable evolution model. In: Proceedings of the 2000 congress on evolutionary computation, pp 1064–1071Google Scholar
  8. Chellapilla K, Fogel DB (1999) Fitness distributions in evolutionary computation: motivation and examples in the continuous domain. Biosystems 54(1-2):15–29CrossRefGoogle Scholar
  9. Cordón O, Damas S, Santamaría J (2006) Feature-based image registration by means of the chc evolutionary algorithm. Image Vis Comput 24(5):525–533CrossRefGoogle Scholar
  10. Delgado M, Pegalajar M, Pegalajar M (2006) Evolutionary training for dynamical recurrent neural networks: an application in finantial time series prediction. Mathware Soft Comput 13:89–110zbMATHMathSciNetGoogle Scholar
  11. Eiben AE, Rudolph G (1999) Theory of evolutionary algorithms: a bird’s eye view. Theor Comput Sci 229(1):3–9CrossRefzbMATHMathSciNetGoogle Scholar
  12. Eshelman L (1990) The chc adaptive search algorithm. In: Rawlins G (ed) Foundations of genetic algorithms. Morgan Kaufmann, San Francisco, pp 265–283Google Scholar
  13. Eshelman L, Caruana A, Schaffer J (1993) Real-coded genetic algorithms and interval-schemata. Found Genetic Algorithms 2:187–202Google Scholar
  14. Eshelman LJ, Schaffer JD (1992) Real-coded genetic algorithms and interval-schemata. In: Foundation of genetic algorithms, pp 187–202Google Scholar
  15. Eshelman LJ, Schaffer JD (1993) Real-coded genetic algorithms in genetic algorithms by preventing incest. Foundation of genetic algorithms, vol 2, pp 187–202Google Scholar
  16. Herrera F, Lozano M, Molina D (2010) Components and parameters of DE, real-coded CHC, and G-CMA-ES. Tech. rep., SCI2S, University of Granada, Spain.
  17. Holland J (1992) Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence. 1st edn. MIT Press, CambridgeGoogle Scholar
  18. Huang W, Wang X (2007) An improved CHC algorithm for damage diagnosis of offshore platforms. J Ocean Univ China 6:85–89 (english edition)CrossRefGoogle Scholar
  19. Kesur KB (2009) Advances in genetic algorithm optimization of traffic signals. J Transport Eng 135(4):160–173CrossRefGoogle Scholar
  20. Lozano M, Herrera F, Krasnogor N, Molina D (2004) Real-coded memetic algorithms with crossover hill-climbing. Evol Comput 12(3):273–302CrossRefGoogle Scholar
  21. Luzón M, Barreiro E, Yeguas E, Joan-Arinyo R (2004) GA and CHC two evolutionary algorithms to solve the root identification problem in geometric constraint solving. In: Bubak M, van Albada GD, Sloot PMA, Dongarra JJ (eds) Computational science—ICCS 2004. Lecture notes in computer science, vol 3039. Springer, Berlin, pp 139–146Google Scholar
  22. Marín J, Solé RV (1999) Macroevolutionary algorithms: a new optimization method on fitness landscapes. IEEE Trans Evol Comput 3(4):272–286CrossRefGoogle Scholar
  23. Nebro AJ, Alba E, Molina G, Chicano F, Luna F, Durillo JJ (2007) Optimal antenna placement using a new multi-objective chc algorithm. In: GECCO ’07: proceedings of the 9th annual conference on genetic and evolutionary computation. ACM, New York, NY, USA, pp 876–883Google Scholar
  24. Nomura T (1997) An analysis on crossovers for real number chromosomes in an infinite population size. In: IJCAI’97: proceedings of the fifteenth international joint conference on artificial intelligence. Morgan Kaufmann, San Francisco, CA, USA, pp 936–941Google Scholar
  25. Poli R, Langdon W, Clerc M, Stephens C (2007) Continuous optimisation theory made easy? Finite-element models of evolutionary strategies, genetic algorithms and particle swarm optimizers. In: Stephens C, Toussaint M, Whitley D, Stadler P (eds) Foundations of genetic algorithms. Lecture notes in computer science, vol 4436. Springer, Berlin, pp 165–193Google Scholar
  26. Prügel-Bennett A, Rogers A (2001) Modelling GA dynamics. Nat Comput. SpringerGoogle Scholar
  27. Qi X, Palmieri F (1994) Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space. Part I: basic properties of selection and mutation. IEEE Trans Neural Netw 5(1):102–119CrossRefGoogle Scholar
  28. Qi X, Palmieri F (1994) Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space. Part II: analysis of the diversification role of crossover. IEEE Trans Neural Netw 5(1):120–129CrossRefGoogle Scholar
  29. Sánchez AM, Lozano M, García-Martínez C, Molina D, Herrera F (2008) Real-parameter crossover operators with multiple descendents: an experimental study. Int J Intell Syst 23:246–268CrossRefzbMATHGoogle Scholar
  30. Santamaría J, Cordón O, Damas S, García-Torres JM, Quirin A (2009) Performance evaluation of memetic approaches in 3d reconstruction of forensic objects. Soft Comput 13:883–904. Google Scholar
  31. Schmitt LM (2004) Theory of genetic algorithms II: models for genetic operators over the string-tensor representation of populations and convergence to global optima for arbitrary fitness function under scaling. Theor Comput Sci 310(1–3):181–231CrossRefzbMATHGoogle Scholar
  32. Stephens C, Waelbroeck H (1998) Schemata evolution and building blocks. Evol Comput 7:109–124CrossRefGoogle Scholar
  33. van Nimwegen E, Crutchfield JP, Mitchell M (1999) Statistical dynamics of the Royal Road genetic algorithm. Theor Comput Sci 229(1–2):41–102CrossRefzbMATHGoogle Scholar
  34. Vose MD (1998) The simple genetic algorithm: foundations and theory. MIT Press, CambridgeGoogle Scholar
  35. Whitley D, Lunacek M, Sokolov A (2006) Comparing the niches of CMA-ES, CHC and pattern search using diverse benchmarks. In: Runarsson T, Beyer HG, Burke E, Merelo-Guervs J, Whitley L, Yao X (eds) Parallel problem solving from nature—PPSN IX. Lecture notes in computer science, vol 4193. Springer, Berlin, pp 988–997Google Scholar
  36. Witt C (2008) Population size versus runtime of a simple evolutionary algorithm. Theor Comput Sci 403(1):104–120CrossRefzbMATHMathSciNetGoogle Scholar
  37. Wright AH, Rowe JE, Neil JR (2002) Analysis of the simple genetic algorithm on the single-peak and double-peak landscapes. In: Proceedings of the congress on evolutionary computation (CEC) 2002. IEEE Press, pp 214–219Google Scholar
  38. Zhao X, Gao XS, Hu ZC (2007) Evolutionary programming based on non-uniform mutation. Appl Math Comput 192(1):1–11CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Jesús Marín
    • 1
    Email author
  • Daniel Molina
    • 2
  • Francisco Herrera
    • 3
  1. 1.Department of Automatic Control (ESAII)Universitat Politècnica de Catalunya, EUETIBBarcelonaSpain
  2. 2.Department of Computer Science and EngineeringUniversity of CádizCádizSpain
  3. 3.Department of Computer Science and Artificial IntelligenceUniversity of GranadaGranadaSpain

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