Skip to main content
Log in

A comparative study on surrogate models for SAEAs

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

Surrogate model assisted evolutionary algorithms (SAEAs) are metamodel-based strategies usually employed on the optimization of problems that demand a high computational cost to be evaluated. SAEAs employ metamodels, like Kriging and radial basis function (RBF), to speed up convergence towards good quality solutions and to reduce the number of function evaluations. However, investigations concerning the influence of metamodels in SAEAs performance have not been developed yet. In this context, this paper performs an investigative study on commonly adopted metamodels to compare the ordinary Kriging (OK), first-order universal Kriging (UK1), second-order universal Kriging (UK2), blind Kriging (BK) and RBF metamodels performance when embedded into a single-objective SAEA Framework (SAEA/F). The results obtained suggest that the OK metamodel presents a slightly better improvement than the others, although it does not present statistically significant difference in relation to UK1, UK2, and BK. The RBF showed the lowest computational cost, but the worst performance. However, this worse performance is around 2% in relation to the other metamodels. Furthermore, the results show that BK presents the highest computational cost without any significant improvement in solution quality when compared to OK, UK1, and UK2.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. https://github.com/monicavaladao/PhD/blob/master/supplementary-material.pdf.

References

  1. Coello, C.C., Lamont, G.B., van Veldhuizen, D.A.: Evolutionary Algorithms for Solving Multi-objective Problems. Genetic and Evolutionary Computation, 2nd edn. Springer, Berlin (2007)

    MATH  Google Scholar 

  2. Collette, Y., Siarry, P.: Multiobjective Optimization: Principles and Case Studies. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  3. Deb, K.: Multi-objective Optimization using Evolutionary Algorithms, 1st edn. Wiley, Hoboken (2001)

    MATH  Google Scholar 

  4. Jin, Y.: A comprehensive survey of fitness approximation in evolutionary computation. Soft. Comput. 9(1), 3–12 (2005)

    Article  Google Scholar 

  5. Emmerich, M.T.M., Giannakoglou, K.C., Naujoks, B.: Single- and multiobjective evolutionary optimization assisted by Gaussian random field metamodels. IEEE Trans. Evol. Comput. 10(4), 421–439 (2006)

    Article  Google Scholar 

  6. Liu, B., Zhang, Q., Gielen, G.G.E.: A Gaussian process surrogate model assisted evolutionary algorithm for medium scale expensive optimization problems. IEEE Trans. Evol. Comput. 18(2), 180–192 (2014)

    Article  Google Scholar 

  7. Sun, C., Jin, Y., Cheng, R., Ding, J., Zeng, J.: Surrogate-assisted cooperative swarm optimization of high-dimensional expensive problems. IEEE Trans. Evol. Comput. 21(4), 644–660 (2017)

    Article  Google Scholar 

  8. Büche, D., Scharaudolph, N.N., Koumountsakos, P.: Accelerating evolutionary algorithms with Gaussian process fitness function models. IEEE Trans. Syst. Man Cybern. C (Appl. Rev) 35(2), 183–194 (2005)

    Article  Google Scholar 

  9. Zhou, Z., Ong, Y.S., Nair, P.B., Keane, A.J., Lum, K.Y.: Combining global and local surrogate models to accelerate evolutionary optimization. IEEE Trans. Syst. Man Cybern. C (Appl. Rev) 37(1), 66–76 (2007)

    Article  Google Scholar 

  10. Lim, D., Jin, Y., Ong, Y.S., Sendhoff, B.: Generalizing surrogate-assisted evolutionary computation. IEEE Trans. Evol. Comput. 14, 329–355 (2010)

    Article  Google Scholar 

  11. Hao, W., Shaoping, W., Tomovic, M.M.: Modified sequential kriging optimization for multidisciplinary complex product simulation. Chin. J. Aeronaut. 23(5), 616–622 (2010)

    Article  Google Scholar 

  12. Schonlau, M.: Computer experiments and global optimization. Ph.D. thesis, University of Waterloo (1997)

  13. Forrester, A.I.J., Sóbester, A., Keane, A.J.: Engineering Design via Surrogate Modelling: A Practical Guide. Wiley, Hoboken (2008)

    Book  Google Scholar 

  14. Jin, Y., Olhofer, M., Sendhoff, B.: On evolutioary optimization with approximate fitness function. In: Genetic and Evolutionary Computation Conference, pp. 786–793 (2000)

  15. Zhao, L., Choi, K.K., Lee, I.: Metamodeling method using dynamic kriging for design optimization. AIAA J. 49(9), 2034–2046 (2011)

    Article  Google Scholar 

  16. Xia, B., Baatar, N., Ren, Z., Koh, C.S.: A numerically efficient muli-objective optimization algorithm: combination of dynamic Taylor Kriging and differential evolution. IEEE Trans. Magn. 51(3), 1–4 (2015)

    Google Scholar 

  17. Xia, B., Ren, Z., seop Koh, C.: Comparative study on Kriging surrogate models for metaheuristic optimization of multidimensional eletromagnetic problems. IEEE Trans. Magn. 51(3), 1–4 (2015)

    Google Scholar 

  18. Palar, P.S., Shimoyama, K.: On efficient global optimization via universal Kriging surrogate models. Struct. Multidiscip. Optim. 57(6), 2377–2397 (2017)

    Article  MathSciNet  Google Scholar 

  19. Jones, D.R., Schonlau, M., Welch, W.J.: Efficient global optimization of expensive black-box functions. J. Global Optim. 13(4), 455–492 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Giunta, A.A., Watson, L.T.: A comparison of approximation modeling techniques-polynomial versus interpolating models. In: 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, p. 4758 (1998)

  21. Jin, R., Chen, W., Simpson, T.W.: Comparative studies of metamodelling techniques under multiple modelling criteria. Struct. Multidiscip. Optim. 23(1), 1–13 (2001)

    Article  Google Scholar 

  22. Daberkow, D.D., Mavris, D.N.: New approaches to conceptual and preliminary aircraft design: a comparative assessment of a neural network formulation and a response surface methodology. In: 1998 World Aviation Conference, 985509, pp. 1–13. American Institute of Aeronautics and Astronautics (AIAA), Anaheim, CA (1998)

  23. Trosset, M.W., Torczon, V.: Numerical optimization using computer experiments. Technical Report 97-38, Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton VA (1997)

  24. Torczon, V., Trosset, M.W.: Using approximation to accelerate engineering design optimization. Technical Report 98-33, Institute for Computer Applications in Science and Engineering, NASA Langley Research Center (1998)

  25. Ma, H., Fei, M., Simon, D., Mo, H.: Update-based evolution control: a new fitness approximation method for evolutionary algorithms. Eng. Optim. 47(9), 1177–1190 (2015)

    Article  MathSciNet  Google Scholar 

  26. Regis, R.G., Shoemaker, C.A.: Local function approximation in evolutionary algorithms for the optimization of costly functions. IEEE Trans. Evol. Comput. 8(5), 490–505 (2004)

    Article  Google Scholar 

  27. Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P.: Design and analysis of computer experiments. Stat. Sci. 4(4), 409–423 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  28. Joseph, V.R., Hung, Y., Sudjianto, A.: Blind Kriging: a new method for developing metamodels. J. Mech. Des. 130, 031102(1–7) (2008)

  29. Mackay, D.J.C.: Introduction to Gaussian Process. Cambridge University (1998). http://www.inference.org.uk/mackay/gpB.pdf

  30. Lophanev, S.N., Nielsen, H.B., Søndergaard, J.: DACE—a MATLAB kriging toolbox. Technical Report IMM-TR-2002-12, Technical University of Denmark (2002)

  31. Roustant, O., Ginsbourger, D., Deville, Y.: DiceKriging, DiceOptim: two R packages for the analysis of computer experiments by Kriging-based metamodeling and optimization. J. Stat. Softw. 51(1), 1–55 (2012)

    Article  Google Scholar 

  32. Ginsbourger, D., Riche, R.L., Carraro, L.: Kriging is well-suited to parallelize optimization. In: Tenne, Y., Goh, C.K. (eds.) Computational Intelligence in Expensive Optimization Problems, Adaptation, Learning and Optimization, vol. 2, pp. 131–162. Springer, Berlin (2010). Chap. 6

    Google Scholar 

  33. Couckuyt, I., Forrester, A., Gorissen, D., Turck, F.D., Dhaene, T.: Blind Kriging: Implementation and performance analysis. Adv. Eng. Softw. 49, 1–13 (2012)

    Article  Google Scholar 

  34. Martin, J.D., Simpson, T.W.: Use of Kriging models to approximate deterministic computer models. AIAA J. 43(4), 853–863 (2005)

    Article  Google Scholar 

  35. Forrester, A.I.J., Keane, A.J.: Recent advances in surrogate-based optimization. Prog. Aerosp. Sci. 45(1–3), 50–79 (2009)

    Article  Google Scholar 

  36. Jin, R., Chen, W., Sudjianto, A.: On sequential sampling for global metamodeling in engineering design. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, p. 10. American Society of Mechanical Engineers (ASME), Montreal, Canada (2002)

  37. Rocha, H.: On the selection of the most adequate radial basis function. Appl. Math. Model. 33(3), 1573–1583 (2009)

    Article  Google Scholar 

  38. Laguna, M., Martí, R.: Experimental testing of advanced scatter search designs for global optimization of multimodal functions. J. Global Optim. 33(2), 235–255 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  39. Surjanovic, S., Bingham, D.: Virtual library of simulation experiments: test functions and datasets. http://www.sfu.ca/~ssurjano (2018). Retrieved 27 April 2018

  40. Price, K.V., Storn, R.M.: Differential Evolution: A Practical Approach to Global Optimization. Springer, Berlin (2005)

    MATH  Google Scholar 

  41. Couckuyt, I., Dhaene, T., Demeester, P.: ooDACE toolbox a Matlab Kriging toolbox: getting started, 3rd June edn (2013). http://sumo.intec.ugent.be/ooDACE

  42. Viana, F.A.C.: SURROGATES toolbox user’s guide, version 2.1 edn (2010). http://sites.google.com/site/felipeacviana/surrogatestoolbox

  43. Jēkabsons, G.: RBF: radial basis function interpolation for MATLAB/OCTAVE, version 1.1 edn (2009). http://www.cs.rtu.lv/jekabsons/regression.html

  44. Dean, A., Voss, D.: Design and Analysis of Experiments. Springer Texts in Statistic. Springer, New York (1999)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

This work was partially supported by PRPq/UFMG and by the Brazilian agencies CNPq, FAPEMIG and CAPES. The authors would also like to thank the comments and invaluable suggestions to improve the quality of this manuscript offered by our colleagues André L. Maravilha and Fillipe Goulart (ORCS Lab. / UFMG).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mônica A. C. Valadão.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (tex 16 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Valadão, M.A.C., Batista, L.S. A comparative study on surrogate models for SAEAs. Optim Lett 14, 2595–2614 (2020). https://doi.org/10.1007/s11590-020-01575-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-020-01575-2

Keywords

Navigation