Soft Computing

, Volume 21, Issue 19, pp 5647–5663 | Cite as

Comparison of metamodeling techniques in evolutionary algorithms

  • Alan Díaz-Manríquez
  • Gregorio Toscano
  • Carlos A. Coello Coello
Methodologies and Application


Although researchers have successfully incorporated metamodels in evolutionary algorithms to solve computational-expensive optimization problems, they have scarcely performed comparisons among different metamodeling techniques. This paper presents an in-depth comparison study over four of the most popular metamodeling techniques: polynomial response surface, Kriging, radial basis function neural network (RBF), and support vector regression. We adopted six well-known scalable test functions and performed experiments to evaluate their suitability to be coupled with an evolutionary algorithm and the appropriateness to surrogate problems by regions (instead of surrogating the entire problem). Notwithstanding that most researchers have undertaken accuracy as the main measure to discern among metamodels, this paper shows that the precision, measured with the ranking preservation indicator, gives a more valuable information for selecting purposes. Additionally, nonetheless each model has its own peculiarities; our results concur that RBF fulfills most of our interests. Furthermore, the readers can also benefit from this study if their problem at hand has certain characteristics such as a low budget of computational time or a low-dimension problem since they can assess specific results of our experimentation.


Surrogate models Evolutionary algorithms Local models 



G. Toscano gratefully acknowledges support from CONACyT through Project No. 105060. C. A. Coello Coello gratefully acknowledges support from CONACyT Project No. 221551.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Bäck T (1996) Evolutionary algorithms in theory and practice. Oxford University Press, OxfordzbMATHGoogle Scholar
  2. Barton R (1992) Metamodels for simulation input-output relations. In: Proceedings of the 24th conference on winter simulation (WSC’92). ACM, New York, pp 289–299Google Scholar
  3. Carpenter W, Barthelemy J (1993) A comparison of polynomial approximations and artificial neural nets as response surfaces. Struct Optim 5(3):166–174Google Scholar
  4. Chow C, Yuen S (2011) An evolutionary algorithm that makes decision based on the entire previous search history. IEEE Trans Evol Comput 15(6):741–769CrossRefGoogle Scholar
  5. De Jong K (1975) An analysis of the behavior of a class of genetic adaptive systems. Ph.D. thesis, University of Michigan, Ann ArborGoogle Scholar
  6. Díaz-Manríquez A, Toscano-Pulido G, Gomez-Flores W (2011) On the selection of surrogate models in evolutionary optimization algorithms. In: IEEE congress on evolutionary computation, pp 2155–2162Google Scholar
  7. Díaz-Manríquez A, Toscano-Pulido G, Coello Coello CA, Landa-Becerra R (2013) A ranking method based on the \(r2\) indicator for many-objective optimization. In: 2013 IEEE congress on evolutionary computation (CEC’13). IEEE Press, Cancún, pp 1523–1530. ISBN 978-1-4799-0454-9Google Scholar
  8. Draper N, Smith H (1981) Applied regression analysis. In: Wiley series in probability and mathematical statistics, 2nd edn. Wiley, New YorkGoogle Scholar
  9. Forgy EW (1965) Cluster analysis of multivariate data: efficiency versus interpretability of classifications. Biometrics 21:768–769Google Scholar
  10. Gaspar-Cunha A, Vieira A (2005) A multi-objective evolutionary algorithm using neural networks to approximate fitness evaluations. Int J Comput Syst Signal 6:18–36Google Scholar
  11. Georgopoulou C, Giannakoglou K (2009) Multiobjective metamodel-assisted memetic algorithms. In: Multi-objective memetic algorithms, studies in computational intelligence, vol 171. Springer, Berlin, pp 153–181Google Scholar
  12. Giunta A, Watson L (1998) A comparison of approximation modeling techniques: polynomial versus interpolating models. Tech. rep., NASA Langley Technical Report ServerGoogle Scholar
  13. Hansen N, Ostermeier A (2001) Completely derandomized self-adaptation in evolution strategies. Evol Comput 9(2):159–195CrossRefGoogle Scholar
  14. Hardy R (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76:1905–1915CrossRefGoogle Scholar
  15. Isaacs A, Ray T, Smith W (2007) An evolutionary algorithm with spatially distributed surrogates for multiobjective optimization. In: Randall M, Abbass H, Wiles J (eds) Progress in artificial life, vol 4828., Lecture notes in computer scienceSpringer, Berlin, pp 257–268CrossRefGoogle Scholar
  16. Jin R, Chen W, Simpson T (2001) Comparative studies of metamodelling techniques under multiple modelling criteria. Struct Multidiscip Optim 23(1):1–13Google Scholar
  17. Matheron G (1963) Principles of geostatistics. Econ Geol 58(8):1246–1266CrossRefGoogle Scholar
  18. McKay M, Beckman R, Conover W (1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2):239–245MathSciNetzbMATHGoogle Scholar
  19. Myers R, Anderson-Cook C (2009) Response surface methodology: process and product optimization using designed experiments, vol 705. Wiley, New YorkGoogle Scholar
  20. Nain P, Deb K (2002) A computationally effective multi-objective search and optimization technique using coarse-to-fine grain modeling. In: 2002 PPSN workshop on evolutionary multiobjective optimization comprehensive survey of fitness approximation in evolutionary computationGoogle Scholar
  21. Pilat M, Neruda R (2013) Aggregate meta-models for evolutionary multiobjective and many-objective optimization. Neurocomputing 116:392–402CrossRefGoogle Scholar
  22. Press W, Teukolsky SA, Vetterling W, Flannery B (2007) Numerical recipes 3rd edition: the art of scientific computing, 3rd edn. Cambridge University Press, New YorkzbMATHGoogle Scholar
  23. Rasheed K, Ni X, Vattam S (2002) Comparison of methods for developing dynamic reduced models for design optimization. In: IEEE congress on evolutionary computation, pp 390–395Google Scholar
  24. Rastrigin L (1974) Extremal control systems. In: Theoretical foundations of engineering cybernetics series. Nauka, MoscowGoogle Scholar
  25. Sacks J, Welch W, Mitchell T, Wynn H (1989) Design and analysis of computer experiments. Stat Sci 4(4):409–423MathSciNetCrossRefzbMATHGoogle Scholar
  26. Schumaker L (2007) Spline functions: basic theory. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  27. Schwefel H (1981) Numerical optimization of computer models. Wiley, New YorkzbMATHGoogle Scholar
  28. Shyy W, Papila N, Vaidyanathan R, Tucker K (2001) Global design optimization for aerodynamics and rocket propulsion components. Prog Aerosp Sci 37(1):59–118CrossRefGoogle Scholar
  29. Silverman B, Jones M (1989) An important contribution to nonparametric discriminant analysis and density estimation: commentary on fix and hodges. In: International statistical review/revue internationale de statistique, pp 233–238Google Scholar
  30. Simpson T, Mauery T, Korte J, Mistree F (1998) Comparison of response surface and Kriging models for multidiscilinary design optimizationGoogle Scholar
  31. Storn R, Price K (1997) Differential evolution? A simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 11(4):341–359MathSciNetCrossRefzbMATHGoogle Scholar
  32. Vapnik V (1998) Statistical learning theory. Wiley-Interscience, New YorkzbMATHGoogle Scholar
  33. Voutchkov I, Keane A (2006) Multiobjective optimization using surrogates. In: International conference on adaptive computing in design and manufacture. The M.C.Escher Company, Holland, pp 167–175Google Scholar
  34. Willmes L, Baeck T, Jin Y, Sendhoff B (2003) Comparing neural networks and Kriging for fitness approximation in evolutionary optimization. In: IEEE congress on evolutionary computation, pp 663–670Google Scholar
  35. Yuen S, Chow C (2009) A genetic algorithm that adaptively mutates and never revisits. IEEE Trans Evol Comput 13(2):454–472CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Alan Díaz-Manríquez
    • 1
  • Gregorio Toscano
    • 2
  • Carlos A. Coello Coello
    • 3
  1. 1.Facultad de Ingeniería y Ciencias, Centro Universitario VictoriaUniversidad Autónoma de TamaulipasCd. VictoriaMexico
  2. 2.CINVESTAV-IPN, Unidad Tamaulipas, Parque Científico y Tecnológico TECNOTAMCd. VictoriaMexico
  3. 3.Departamento de ComputaciónCINVESTAV-IPNMexicoMexico

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