Skip to main content
SpringerLink
Log in

CMA evolution strategy assisted by kriging model and approximate ranking

  • Published:
Applied Intelligence Aims and scope Submit manuscript

Abstract

The covariance matrix adaptation evolution strategy (CMA-ES) is a competitive evolutionary algorithm (EA) for difficult continuous optimization problems. However, expensive function evaluation of many real-world optimization problems poses a serious challenge to the application of CMA-ES (and other EAs) to these problems. To address this challenge, surrogate-assisted EAs has attracted increasing attention and become popular. In this paper, a new surrogate-assisted CMA-ES algorithm in which Kriging model is used to enhance CMA-ES via approximate ranking procedure is proposed. In the proposed algorithm, the approximate ranking procedure which estimates the rank of current population by using Kriging model and the exact fitness function together is adopted. In addition, the confidence interval method of training set selection is introduced for surrogate model construction. An initial sampling is performed before entering the evolution loop. In each iteration (generation), after the population sampling, the approximate ranking procedure is called instead of the original fitness evaluation, then, parameters of the sampling distribution are updated. This iterative search process continues until the target fitness is reached or the computational budget is exhausted. The proposed algorithm and confidence interval method of training set selection are analyzed through experimental study. The results demonstrate that the confidence interval method works well in Kriging-assisted CMA-ES, and that the proposed algorithm significantly reduces the number of function evaluations of CMA-ES and outperforms the Kriging-assisted CMA-ES using pre-selection and generation-based control on the tested problems.

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

Access this article

Log in via an institution

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
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Amali SMJ, Baskar S (2015) Surrogate assisted-hybrid differential evolution algorithm using diversity control. Expert Syst J Knowl Eng 32(4):531–545

    Article  Google Scholar 

  2. Back T (1996) Evolutionary algorithms in theory and practice: evolution strategies, evolutionary programming, genetic algorithms. Oxford University Press, Oxford

    MATH  Google Scholar 

  3. Bäck T, Foussette c, Krause P (2013) Contemporary evolution strategies. Springer, Berlin Heidelberg

    Book  Google Scholar 

  4. Beyer HG, Schwefel HP (2002) Evolution strategies–a comprehensive introduction. Natur Comput 1(1):3–52

    Article  MathSciNet  Google Scholar 

  5. Bouzarkouna Z, Auger A, Ding DY (2010) Investigating the local-meta-model cma-es for large population sizes. In: European conference on the applications of evolutionary computation. Springer, pp 402–411

  6. Branke J, Schmidt C (2005) Faster convergence by means of fitness estimation. Soft Comput 9(1):13–20

    Article  Google Scholar 

  7. Emmerich M, Giotis A, Özdemir M, Bäck T, Giannakoglou K (2002a) Metamodel—assisted evolution strategies. In: International conference on parallel problem solving from nature. Springer, pp 361–370

  8. Emmerich M, Giotis A, Özdemir M, Bäck T, Giannakoglou K (2002b) Metamodel—assisted evolution strategies. In: International conference on parallel problem solving from nature. Springer, pp 361–370

  9. Hansen N (2006) The cma evolution strategy: a comparing review. In: Towards a new evolutionary computation. Springer, pp 75–102

  10. Hansen N (2016) The cma evolution strategy: a tutorial. arXiv:160400772

  11. Hansen N, Ostermeier A (2001) Completely derandomized self-adaptation in evolution strategies. Evol Comput 9(2):159–195

    Article  Google Scholar 

  12. Hong YS, Lee H, Tahk MJ (2003) Acceleration of the convergence speed of evolutionary algorithms using multi-layer neural networks. Eng Optim 35(1):91–102

    Article  MathSciNet  Google Scholar 

  13. Huang C, Radi B, El Hami A (2016) Uncertainty analysis of deep drawing using surrogate model based probabilistic method. Int J Adv Manuf Technol 86(9-12):3229–3240

    Article  Google Scholar 

  14. Huang C, El Hami A, Radi B (2017) Metamodel-based inverse method for parameter identification: elastic–plastic damage model. Eng Optim 49(4):633–653

    Article  MathSciNet  Google Scholar 

  15. Jamil M, Yang XS (2013) A literature survey of benchmark functions for global optimisation problems. Int J Math Modell Numer Optim 4(2):150–194

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Jin Y (2011) Surrogate-assisted evolutionary computation: Recent advances and future challenges. Swarm Evol Comput 1(2):61–70

    Article  Google Scholar 

  18. Jin Y, Sendhoff B (2002) Fitness approximation in evolutionary computation-a survey. In: Proceedings of the genetic and evolutionary computation conference. Morgan Kaufmann Publishers Inc., pp 1105–1112

  19. Jin Y, Olhofer M, Sendhoff B (2000) On evolutionary optimization with approximate fitness functions. In: Proceedings of the 2nd annual conference on genetic and evolutionary computation. Morgan Kaufmann Publishers Inc., pp 786–793

  20. Jin Y, Hüsken M, Sendhoff B (2003) Quality measures for approximate models in evolutionary computation. In: GECCO, pp 170–173

  21. Jones DR, Schonlau M, Welch WJ (1998) Efficient global optimization of expensive black-box functions. J Glob Optim 13(4):455–492

    Article  MathSciNet  Google Scholar 

  22. Kaymaz I (2005) Application of kriging method to structural reliability problems. Struct Saf 27(2):133–151

    Article  Google Scholar 

  23. Kern S, Hansen N, Koumoutsakos P (2004) Fast quadratic local meta-models for evolutionary optimization of anguilliform swimmers. In: Neittaanmaki et al. (eds) EUROGEN 2007, Helsinki. Finland, https://hal.inria.fr/inria-00173469

  24. Kern S, Hansen N, Koumoutsakos P (2006) Local meta-models for optimization using evolution strategies. In: Parallel problem solving from nature-PPSN IX. Springer, pp 939–948

  25. Kim HS, Cho SB (2001) An efficient genetic algorithm with less fitness evaluation by clustering. In: 2001. Proceedings of the 2001 congress on evolutionary computation, vol 2. IEEE, pp 887–894

  26. Kramer O (2014) A brief introduction to continuous evolutionary optimization. Springer International Publishing

  27. Loeppky JL, Sacks J, Welch WJ (2009) Choosing the sample size of a computer experiment: a practical guide. Technometrics 51(4):366–376

    Article  MathSciNet  Google Scholar 

  28. Loshchilov I (2013) Surrogate-assisted evolutionary algorithms. Theses, Université Paris Sud - Paris XI; Institut national de recherche en informatique et en automatique - INRIA. https://tel.archives-ouvertes.fr/tel-00823882

  29. Loshchilov I (2013) Surrogate-assisted evolutionary algorithms. PhD thesis, Université Paris Sud-Paris XI; Institut national de recherche en informatique et en automatique-INRIA

  30. Mallipeddi R, Lee M (2015) An evolving surrogate model-based differential evolution algorithm. Appl Soft Comput 34:770–787

    Article  Google Scholar 

  31. Martin JD, Simpson TW (2003) A study on the use of kriging models to approximate deterministic computer models. In: ASME 2003 international design engineering technical conferences and computers and information in engineering conference. American Society of Mechanical Engineers, pp 567–576

  32. Pitra Z, Bajer L, Holeňa M (2016) Doubly trained evolution control for the surrogate cma-es. In: International conference on parallel problem solving from nature. Springer, pp 59–68

  33. Pitra Z, Bajer L, Repickỳ J, Holeňa M (2017) Overview of surrogate-model versions of covariance matrix adaptation evolution strategy. In: Proceedings of the genetic and evolutionary computation conference companion. ACM, pp 1622–1629

  34. Rasheed K, Hirsh H (2000) Informed operators: speeding up genetic-algorithm-based design optimization using reduced models. In: Proceedings of the 2nd annual conference on genetic and evolutionary computation. Morgan Kaufmann Publishers Inc., pp 628–635

  35. Rasmussen CE (2004) Gaussian processes in machine learning. In: Advanced lectures on machine learning. Springer, pp 63–71

  36. Ratle A (1998) Accelerating the convergence of evolutionary algorithms by fitness landscape approximation. In: International conference on parallel problem solving from nature. Springer, pp 87–96

  37. Rechenberg I (1973) Evolutionsstrategie–optimierung technisher systeme nach prinzipien der biologischen evolution

  38. Rencher AC (2003) Methods of multivariate analysis, vol 492. Wiley, New York

    Google Scholar 

  39. Runarsson TP (2004) Constrained evolutionary optimization by approximate ranking and surrogate models. In: International conference on parallel problem solving from nature. Springer, pp 401–410

  40. Sacks J, Welch WJ, Mitchell TJ, Wynn HP (1989) Design and analysis of computer experiments. Stat Sci 4:409–423

    Article  MathSciNet  Google Scholar 

  41. Shi L, Rasheed K (2010) A survey of fitness approximation methods applied in evolutionary algorithms. In: Computational intelligence in expensive optimization problems. Springer, pp 3–28

  42. Smith RE, Dike BA, Stegmann S (1995) Fitness inheritance in genetic algorithms. In: Proceedings of the 1995 ACM symposium on applied computing. ACM, pp 345–350

  43. Suganthan PN, Hansen N, Liang JJ, Deb K, Chen Y, Auger A, Tiwari S (2005) Problem definitions and evaluation criteria for the cec 2005 special session on realparameter optimization. Technical Report, Nanyang Technological University, Singapore, May 2005 AND KanGAL Report 2005005, IIT Kanpur, India

  44. Sun C, Jin Y, Zeng J, Yu Y (2015) A two-layer surrogate-assisted particle swarm optimization algorithm. Soft Comput 19(6):1461–1475

    Article  Google Scholar 

  45. Ulmer H, Streichert F, Zell A (2003) Evolution strategies assisted by gaussian processes with improved preselection criterion. In: 2003. CEC’03. The 2003 congress on evolutionary computation, vol 1. IEEE, pp 692–699

  46. Ulmer H, Streichert F, Zell A (2004) Optimization by gaussian processes assisted evolution strategies. In: Operations research proceedings 2003. Springer, pp 435–442

  47. Ulmer H, Streichert F, Zell A (2005) Model assisted evolution strategies. In: Knowledge incorporation in evolutionary computation. Springer, pp 333–355

  48. Venturelli G, Benini E, ukasz aniewski W (2017) A kriging-assisted multiobjective evolutionary algorithm. Appl Soft Comput 58:155–175. https://doi.org/10.1016/j.asoc.2017.04.017, http://www.sciencedirect.com/science/article/pii/S1568494617301898

    Article  Google Scholar 

  49. Wang H, Jin Y, Doherty J (2017) Committee-based active learning for surrogate-assisted particle swarm optimization of expensive problems. IEEE Trans Cybern 47(9):2664–2677

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported by China Scholarship Council (Award Number: 201304490045) and the Science and Technology Innovation Committee Foundation of Shenzhen (Grant No. ZDSYS201703031748284).

Author information

Authors and Affiliations

  1. Shenzhen Key Laboratory of Computational Intelligence, Department of Computer Science and Engineering, Southern University of Science and Technology, Shenzhen, 518055, China

    Changwu Huang

  2. LIMII, FST Settat, BP: 577, Route de Casa, Settat, Morocco

    Bouchaïb Radi

  3. INSA Rouen, LMN, Normandie University, 76000, Rouen, France

    Abdelkhalak El Hami & Hao Bai

Authors
  1. Changwu Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bouchaïb Radi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abdelkhalak El Hami
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hao Bai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changwu Huang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, C., Radi, B., El Hami, A. et al. CMA evolution strategy assisted by kriging model and approximate ranking. Appl Intell 48, 4288–4304 (2018). https://doi.org/10.1007/s10489-018-1193-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10489-018-1193-3

Keywords

Search

Navigation