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Elite-driven surrogate-assisted CMA-ES algorithm by improved lower confidence bound method

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Abstract

To relieve the computational burden and improve the global optimizing ability of Covariance Matrix Adaptation Evolution Strategy (CMA-ES) for real-world expensive problems, an elite-driven surrogate-assisted CMA-ES (ES-CMA-ES) algorithm by the improved Lower Confidence Bound (ILCB) method is proposed in this paper. Firstly, the ILCB method is established by introducing the step size, which captures the trend of exploration and exploitation in CMA-ES, to control the uncertainty term of the ILCB formula adaptively. Next, based on the ILCB method, a novel model management consisting of the efficient pre-screening strategy and the competitive chaotic operator is developed. In each generation of ES-CMA-ES, a large number of candidate points are sampled first, and then a few of them with better ILCB predicted values are screened out by the efficient pre-screening strategy, aiming to enhance the sampling quality and accelerate the optimization convergence. Moreover, the local search is performed on the best-performing screened sample points utilizing the competitive chaotic operator, with the purpose of increasing the diversity of populations in ES-CMA-ES and avoiding being trapped in the local optima. By means of the above procedures of the model management, the elite sample points are finally obtained which will be evaluated by true fitness function in each generation of ES-CMA-ES. To verify the effectiveness of ES-CMA-ES, five known black-box optimization algorithms are employed to make a comparison. Firstly, seven typical numerical examples of 10-dimensional and 20-dimensional benchmark functions are carried out, respectively. Furthermore, a 20-dimensional engineering example of the aerospace variable-stiffness composite shell under combined loadings is studied. Results indicate the outstanding efficiency, global optimizing ability and applicability of the proposed ES-CMA-ES compared to its counterpart algorithms.

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References

  1. Clarke A, Miles JC (2012) Strategic fire and rescue service decision making using evolutionary algorithms. Adv Eng Softw 50:29–36

    Google Scholar 

  2. Arya Azar N, Kardan N, Ghordoyee Milan S (2021) Developing the artificial neural network–evolutionary algorithms hybrid models (ANN–EA) to predict the daily evaporation from dam reservoirs. Eng Comput 37:1–19

    Google Scholar 

  3. Wang Y, Ni C, Fan X, et al (2021) Cellular differential evolutionary algorithm with double-stage external population-leading and its application. Eng Comput 1–20. https://doi.org/10.1007/s00366-021-01311-z

  4. Hansen N, Ostermeier A (1996) Adapting arbitrary normal mutation distributions in evolution strategies: the covariance matrix adaptation. Proceedings of IEEE international conference on evolutionary computation. IEEE 312–317

  5. Pitra Z, Bajer L, Repický J, et al (2017) Overview of surrogate-model versions of covariance matrix adaptation evolution strategy. Proceedings of the Genetic and Evolutionary Computation Conference Companion 1622–1629

  6. Bajer L, Pitra Z, Repický J et al (2019) Gaussian process surrogate models for the CMA evolution strategy. Evol Comput 27(4):665–697

    Google Scholar 

  7. Li W, Lei Z, Yuan J et al (2021) Enhancing the competitive swarm optimizer with covariance matrix adaptation for large scale optimization. Appl Intell 51:4984–5006

    Google Scholar 

  8. Islam J, Vasant PM, Negash BM et al (2020) A holistic review on artificial intelligence techniques for well placement optimization problem. Adv Eng Softw 141:102767

    Google Scholar 

  9. Fujii G, Takahashi M, Akimoto Y (2018) CMA-ES-based structural topology optimization using a level set boundary expression—application to optical and carpet cloaks. Comput Methods Appl Mech Eng 332:624–643

    MathSciNet  MATH  Google Scholar 

  10. Reddy SS, Panigrahi BK, Kundu R et al (2013) Energy and spinning reserve scheduling for a wind-thermal power system using CMA-ES with mean learning technique. Int J Electr Power Energy Syst 53:113–122

    Google Scholar 

  11. Loshchilov I (2014) A computationally efficient limited memory CMA-ES for large scale optimization. Proceedings of the 2014 Annual Conference on Genetic and Evolutionary Computation 397–404

  12. Molina D, Lozano M, García-Martínez C et al (2010) Memetic algorithms for continuous optimization based on local search chains. Evol Comput 18(1):27–63

    Google Scholar 

  13. Auger A, Hansen N (2005) A restart CMA evolution strategy with increasing population size. 2005 IEEE congress on evolutionary computation. IEEE 2:1769–1776

    Google Scholar 

  14. Liao T, de Oca MAM, Stützle T (2013) Computational results for an automatically tuned CMA-ES with increasing population size on the CEC’05 benchmark set. Soft Comput 17(6):1031–1046

    Google Scholar 

  15. Bouzarkouna Z, Auger A, Ding DY (2010) Investigating the local-meta-model CMA-ES for large population sizes. European Conference on the Applications of Evolutionary Computation. Springer, Berlin, Heidelberg, 6024:402–411.

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

    Google Scholar 

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

    Google Scholar 

  18. Zhou Q, Wu J, Xue T et al (2021) A two-stage adaptive multi-fidelity surrogate model-assisted multi-objective genetic algorithm for computationally expensive problems. Eng Comput 37(1):623–639

    Google Scholar 

  19. Li E, Wang H (2016) An alternative adaptive differential evolutionary algorithm assisted by expected improvement criterion and cut-HDMR expansion and its application in time-based sheet forming design. Adv Eng Softw 97:96–107

    Google Scholar 

  20. Jin Y, Wang H, Chugh T et al (2018) Data-driven evolutionary optimization: an overview and case studies. IEEE Trans Evol Comput 23(3):442–458

    Google Scholar 

  21. Wang H, Jin Y, Jansen JO (2016) Data-driven surrogate-assisted multi-objective evolutionary optimization of a trauma system. IEEE Trans Evol Comput 20(6):939–952

    Google Scholar 

  22. Hong L, Li H, Peng K (2021) A combined radial basis function and adaptive sequential sampling method for structural reliability analysis. Appl Math Model 90:375–393

    MathSciNet  MATH  Google Scholar 

  23. Meng Z, Zhang Z, Li G et al (2019) An active weight learning method for efficient reliability assessment with small failure probability. Struct Multidiscip Optim 61:1–14

    MathSciNet  Google Scholar 

  24. Song LK, Fei CW, Wen J et al (2017) Multi-objective reliability-based design optimization approach of complex structure with multi-failure modes. Aerosp Sci Technol 64:52–62

    Google Scholar 

  25. Song LK, Bai GC, Li XQ (2021) A novel metamodeling approach for probabilistic LCF estimation of turbine disk. Eng Fail Anal 120:105074

    Google Scholar 

  26. Guo H, Nguyen H, Bui XN et al (2021) A new technique to predict fly-rock in bench blasting based on an ensemble of support vector regression and GLMNET. Eng Comput 37(1):421–435

    Google Scholar 

  27. Bajer L, Pitra Z, Holeňa M (2015) Benchmarking Gaussian processes and random forests surrogate models on the BBOB noiseless testbed. Proceedings of the Companion Publication of the 2015 Annual Conference on Genetic and Evolutionary Computation 1143–1150

  28. Pitra Z, Bajer L, Holeňa M (2016) Doubly trained evolution control for the surrogate CMA-ES. International Conference on Parallel Problem Solving from Nature. Springer, Cham 9921:59–68

  29. Huang C, Radi B, El Hami A et al (2018) CMA evolution strategy assisted by kriging model and approximate ranking. Appl Intell 48(11):4288–4304

    Google Scholar 

  30. Yi J, Gao L, Li X et al (2019) An on-line variable-fidelity surrogate-assisted harmony search algorithm with multi-level screening strategy for expensive engineering design optimization. Knowl-Based Syst 170:1–19

    Google Scholar 

  31. Kolahchi R, Tian K, Keshtegar B, et al (2020) AK-GWO: a novel hybrid optimization method for accurate optimum hierarchical stiffened shells. Eng Comput 1–13

  32. Tian K, Wang B, Zhang K et al (2018) Tailoring the optimal load-carrying efficiency of hierarchical stiffened shells by competitive sampling. Thin-Walled Struct 133:216–225

    Google Scholar 

  33. Li F, Shen W, Cai X et al (2020) A fast surrogate-assisted particle swarm optimization algorithm for computationally expensive problems. Appl Soft Comput 92:106303

    Google Scholar 

  34. Gräning L, Jin Y, Sendhoff B (2005) Efficient evolutionary optimization using individual-based evolution control and neural networks: a comparative study. ESANN 273–278.

  35. Gräning L, Jin Y, Sendhoff B (2007) Individual-based management of meta-models for evolutionary optimization with application to three-dimensional blade optimization. Evolutionary computation in dynamic and uncertain environments. Springer, Berlin, pp 225–250

    Google Scholar 

  36. Yu H, Tan Y, Sun C et al (2019) A generation-based optimal restart strategy for surrogate-assisted social learning particle swarm optimization. Knowl-Based Syst 163:14–25

    Google Scholar 

  37. Baykasoğlu A, Ozsoydan FB (2017) Evolutionary and population-based methods versus constructive search strategies in dynamic combinatorial optimization. Inf Sci 420:159–183

    MATH  Google Scholar 

  38. Regis RG (2014) Particle swarm with radial basis function surrogates for expensive black-box optimization. J Comput Sci 5(1):12–23

    MathSciNet  Google Scholar 

  39. Jin Y, Olhofer M, Sendhoff B (2002) A framework for evolutionary optimization with approximate fitness functions. IEEE Trans Evol Comput 6(5):481–494

    Google Scholar 

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

    Google Scholar 

  41. Tian K, Li ZC, Huang L et al (2020) Enhanced variable-fidelity surrogate-based optimization framework by Gaussian process regression and fuzzy clustering. Comput Methods Appl Mech Eng 366:113045

    MathSciNet  MATH  Google Scholar 

  42. Tian J, Tan Y, Zeng J et al (2018) Multiobjective infill criterion driven Gaussian process-assisted particle swarm optimization of high-dimensional expensive problems. IEEE Trans Evol Comput 23(3):459–472

    Google Scholar 

  43. Yu H, Tan Y, Zeng J et al (2018) Surrogate-assisted hierarchical particle swarm optimization. Inf Sci 454:59–72

    MathSciNet  Google Scholar 

  44. Guo D, Jin Y, Ding J et al (2018) Heterogeneous ensemble-based infill criterion for evolutionary multiobjective optimization of expensive problems. IEEE Transact Cybern 49(3):1012–1025

    Google Scholar 

  45. Hansen N (2016) The CMA evolution strategy: a tutorial. arXiv preprint arXiv 1604.00772.

  46. Hansen N (2006) The CMA evolution strategy: a comparing review. Towards a new evolutionary computation, vol 192. Springer, Berlin, pp 75–102

    Google Scholar 

  47. Lin Q, Hu D, Hu J et al (2021) A screening-based gradient-enhanced Gaussian process regression model for multi-fidelity data fusion. Adv Eng Inform 50:101437

    Google Scholar 

  48. Gao Y, Jiao Y, Liu Y (2021) Efficient high-dimensional material reliability analysis with explicit voxel-level stochastic microstructure representation. Appl Math Model 91:1117–1140

    MathSciNet  MATH  Google Scholar 

  49. Kaintura A, Spina D, Couckuyt I et al (2017) A Kriging and Stochastic Collocation ensemble for uncertainty quantification in engineering applications. Eng Comput 33(4):935–949

    Google Scholar 

  50. Horn D, Wagner T, Biermann D, et al (2015) Model-based multi-objective optimization: taxonomy, multi-point proposal, toolbox and benchmark. International Conference on Evolutionary Multi-Criterion Optimization. Springer, Cham, p 64–78

  51. Wang X, Jin Y, Schmitt S et al (2020) An adaptive Bayesian approach to surrogate-assisted evolutionary multi-objective optimization. Inf Sci 519:317–331

    MathSciNet  MATH  Google Scholar 

  52. Meng Z, Zhang D, Li G et al (2019) An importance learning method for non-probabilistic reliability analysis and optimization. Struct Multidiscip Optim 59(4):1255–1271

    MathSciNet  Google Scholar 

  53. Meng Z, Zhang Z, Zhang D et al (2019) An active learning method combining Kriging and accelerated chaotic single loop approach (AK-ACSLA) for reliability-based design optimization. Comput Methods App Mech Eng 357:112570

    MathSciNet  MATH  Google Scholar 

  54. Liu Q, Jin Y, Heiderich M et al (2022) Surrogate-assisted evolutionary optimization of expensive many-objective irregular problems. Knowledge-Based Syst 240:108197

    Google Scholar 

  55. Cox DD, John S (1992) A statistical method for global optimization. 1992 IEEE International Conference on Systems, Man, and Cybernetics. IEEE 1241–1246.

  56. Zheng J, Li Z, Gao L et al (2016) A parameterized lower confidence bounding scheme for adaptive metamodel-based design optimization. Eng Comput 33:2165–2184

    Google Scholar 

  57. Jiang P, Cheng J, Zhou Q et al (2019) Variable-fidelity lower confidence bounding approach for engineering optimization problems with expensive simulations. AIAA J 57(12):5416–5430

    Google Scholar 

  58. Qian J, Yi J, Zhang J et al (2020) An entropy weight-based lower confidence bounding optimization approach for engineering product design. Appl Sci 10(10):3554

    Google Scholar 

  59. Tian D (2017) Particle swarm optimization with chaos-based initialization for numerical optimization. Intell Automat Soft Comput 1–12

  60. Alatas B, Akin E, Ozer AB (2009) Chaos embedded particle swarm optimization algorithms. Chaos Solitons Fractals 40(4):1715–1734

    MathSciNet  MATH  Google Scholar 

  61. Assarzadeh Z, Naghsh-Nilchi AR (2015) Chaotic particle swarm optimization with mutation for classification. J Med Signals Sens 5(1):12

    Google Scholar 

  62. Wang Y, Liu JH (2010) Chaotic particle swarm optimization for assembly sequence planning. Robot Comput-Integrated Manuf 26(2):212–222

    Google Scholar 

  63. Shan L, Qiang H, Li J et al (2005) Chaotic optimization algorithm based on Tent map. Control Decision 20(2):179–182

    MATH  Google Scholar 

  64. Fuerle F, Sienz J (2011) Formulation of the Audze-Eglais uniform Latin hypercube design of experiments for constrained design spaces. Adv Eng Softw 42(9):680–689

    MATH  Google Scholar 

  65. Kang F, Xu Q, Li J (2016) Slope reliability analysis using surrogate models via new support vector machines with swarm intelligence. Appl Math Model 40(11–12):6105–6120

    MathSciNet  MATH  Google Scholar 

  66. Dong H, Dong Z (2020) Surrogate-assisted Grey wolf optimization for high-dimensional, computationally expensive black-box problems. Swarm Evol Comput 57:100713

    Google Scholar 

  67. Chen G, Li Y, Zhang K et al (2021) Efficient hierarchical surrogate-assisted differential evolution for high-dimensional expensive optimization. Inf Sci 542:228–246

    MathSciNet  MATH  Google Scholar 

  68. Chu SC, Du ZG, Peng YJ et al (2021) Fuzzy hierarchical surrogate assists probabilistic particle swarm optimization for expensive high dimensional problem. Knowledge-Based Syst 220:106939

    Google Scholar 

  69. Suganthan PN, Hansen N, Liang JJ, et al (2005) Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. Nat Comput 2005005:341–357

    Google Scholar 

  70. Whitley D (1994) A genetic algorithm tutorial. Stat Comput 4(2):65–85

    Google Scholar 

  71. Carrasco J, García S, Rueda MM et al (2020) Recent trends in the use of statistical tests for comparing swarm and evolutionary computing algorithms: practical guidelines and a critical review. Swarm Evol Comput 54:100665

    Google Scholar 

  72. Schmidt C, Schultz C, Weber P et al (2014) Evaluation of eddy current testing for quality assurance and process monitoring of automated fiber placement. Compos B Eng 56:109–116

    Google Scholar 

  73. Guo Q, Hang J, Wang S et al (2020) Buckling optimization of variable stiffness composite cylinders by using multi-fidelity surrogate models. Thin-Walled Struct 156:107014

    Google Scholar 

  74. Hao P, Yuan X, Liu C et al (2018) An integrated framework of exact modeling, isogeometric analysis and optimization for variable-stiffness composite panels. Comput Methods Appl Mech Eng 339:205–238

    MathSciNet  MATH  Google Scholar 

  75. Hyer MW, Charette RF (1991) Use of curvilinear fiber format in composite structure design. AIAA J 29(6):1011–1015

    Google Scholar 

  76. Yoo K, Bacarreza O, Aliabadi MHF (2020) A novel multi-fidelity modelling-based framework for reliability-based design optimisation of composite structures. Eng Comput 38:595–608

    Google Scholar 

  77. Rouhi M, Ghayoor H, Hoa SV et al (2014) Effect of structural parameters on design of variable-stiffness composite cylinders made by fiber steering. Compos Struct 118:472–481

    Google Scholar 

  78. Pan Z, Zhang LW, Liew KM (2021) Adaptive surrogate-based harmony search algorithm for design optimization of variable stiffness composite materials. Comput Methods Appl Mech Eng 379:113754

    MathSciNet  MATH  Google Scholar 

  79. Jing Z, Fan X, Sun Q (2015) Stacking sequence optimization of composite laminates for maximum buckling load using permutation search algorithm. Compos Struct 121:225–236

    Google Scholar 

  80. White SC, Weaver PM, Wu KC (2015) Post-buckling analyses of variable-stiffness composite cylinders in axial compression. Compos Struct 123:190–203

    Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China [No. 11902065, No. 11825202], and Fundamental Research Funds for the Central Universities [No. DUT21RC(3)013, No. DUT20ZD104, No. DUT2019TD37].

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  1. Department of Engineering Mechanics, Key Laboratory of Digital Twin for Industrial Equipment, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, China

    Zengcong Li, Tianhe Gao, Kuo Tian & Bo Wang

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  1. Zengcong Li
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  2. Tianhe Gao
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  3. Kuo Tian
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Correspondence to Kuo Tian.

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Li, Z., Gao, T., Tian, K. et al. Elite-driven surrogate-assisted CMA-ES algorithm by improved lower confidence bound method. Engineering with Computers 39, 2543–2563 (2023). https://doi.org/10.1007/s00366-022-01642-5

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