Abstract
This paper presents a novel surrogate-assisted evolutionary algorithm, CSMOEA, for multi-objective optimization problems (MOPs) with computationally expensive objectives. Considering most surrogate-assisted evolutionary algorithms (SAEAs) do not make full use of population information and only use population information in either the objective space or the design space independently, to address this limitation, we propose a new strategy for comprehensive utilization of population information of objective and design space. The proposed CSMOEA adopts an adaptive clustering strategy to divide the current population into good and bad groups, and the clustering centers in the design space are obtained, respectively. Then, a bi-level sampling strategy is proposed to select the best samples in both the design and objective space, using distance to the clustering centers and approximated objective values of radial basis functions. The effectiveness of CSMOEA is compared with five state-of-the-art algorithms on 21 widely used benchmark problems, and the results show high efficiency and a good balance between convergence and diversity. Additionally, CSMOEA is applied to the shape optimization of blend-wing-body underwater gliders with 14 decision variables and two objectives, demonstrating its effectiveness in solving real-world engineering problems.
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References
Bachmayer R, Leonard N E, Graver J, Fiorelli E, Bhatta P, Paley D (2004) Underwater gliders: Recent developments and future applications. In: Proceedings of the 2004 international symposium on underwater technology (IEEE Cat. No. 04EX869). IEEE. pp 195–200, https://doi.org/10.1109/UT.2004.1405540
Bosman PAN, Thierens D (2003) The balance between proximity and diversity in multiobjective evolutionary algorithms. IEEE Trans Evolut Comput 7(2):174–188. https://doi.org/10.1109/TEVC.2003.810761
Cai L, Qu S, Yuan Y, Yao X (2015) A clustering-ranking method for many-objective optimization. Appl Soft Comput 35:681–694. https://doi.org/10.1016/j.asoc.2015.06.020
Cai X, Gao L, Li X (2019) Efficient generalized surrogate-assisted evolutionary algorithm for high-dimensional expensive problems. IEEE Trans Evolut Comput 24(2):365–379. https://doi.org/10.1109/TEVC.2019.2919762
Cheng R, Rodemann T, Fischer M, Olhofer M, Jin Y (2017) Evolutionary many-objective optimization of hybrid electric vehicle control: from general optimization to preference articulation. IEEE Trans Emerg Topics Comput Intell 1(2):97–111. https://doi.org/10.1109/TETCI.2017.2669104
Chugh T, Jin Y, Miettinen K, Hakanen J, Sindhya K (2018) A surrogate-assisted reference vector guided evolutionary algorithm for computationally expensive many-objective optimization. IEEE Trans Evolut Comput 22(1):129–142. https://doi.org/10.1109/TEVC.2016.2622301
Deb K (2005) Scalable test problems for evolutionary multiobejctive optimization. Evolut Multiobjective Optim Theor Adv Appl. https://doi.org/10.1007/1-84628-137-7_6
Deb K, Pratap A, Agarwal S, Meyarivan TAMT (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evolut Comput 6(2):182–197. https://doi.org/10.1109/4235.996017
Denysiuk R, Costa L, Santo IE (2014) Clustering-based selection for evolutionary many-objective optimization. Int Conf Parallel Probl Solving Nat. https://doi.org/10.1007/978-3-319-10762-2_53
Dong H, Dong Z (2020) Surrogate-assisted Grey wolf optimization for high-dimensional, computationally expensive black-box problems. Swarm Evolut Comput 57:100713. https://doi.org/10.1016/j.swevo.2020.100713
Dong H, Wang P, Yu X, Song B (2020) Surrogate-assisted teaching-learning-based optimization for high-dimensional and computationally expensive problems. Appl Soft Comput 99(2):106934. https://doi.org/10.1016/j.asoc.2020.106934
Dyn N, Levin D, Rippa S (1986) Numerical procedures for surface fitting of scattered data by radial functions. SIAM J Sci Stat Comput 7(2):639–659. https://doi.org/10.1137/0907043
Geman S, Bienenstock E, Doursat R (1992) Neural networks and the bias/variance dilemma. Neural Comput 4(1):1–58. https://doi.org/10.1162/neco.1992.4.1.1
Gunn SR (1998) Support vector machines for classification and regression. ISIS Tech Rep 14(1):5–16. https://doi.org/10.1039/B918972F
Gunst RF, Myers RH, Montgomery DC (1996) Response surface methodology: process and product optimization using designed experiments | Clc. Technometrics 38(3):285. https://doi.org/10.1080/00401706.1996.10484509
Guo D, Jin Y, Ding J, Chai T (2018) Heterogeneous ensemble-based infill criterion for evolutionary multiobjective optimization of expensive problems. IEEE Trans Cyber 49(3):1012–1025. https://doi.org/10.1109/TCYB.2018.2794503
Guo D, Wang X, Gao K, Jin Y, Ding J, Chai T (2021) Evolutionary Optimization of High-Dimensional Multiobjective and Many-Objective Expensive Problems Assisted by a Dropout Neural Network. IEEE Trans Syst, Man, Cyber Syst. https://doi.org/10.1109/TSMC.2020.3044418
Hicks RM, Henne PA (1978) Wing design by numerical optimization. J Aircr 15(7):407–412. https://doi.org/10.2514/3.58379
Hua Y, Jin Y, Hao K (2018) A clustering-based adaptive evolutionary algorithm for multiobjective optimization with irregular Pareto fronts. IEEE Trans Cyber 49(7):2758–2770. https://doi.org/10.1109/TCYB.2018.2834466
Huband S, Hingston P, Barone L, While L (2006) A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans Evolut Comput 10(5):477–506. https://doi.org/10.1109/TEVC.2005.861417
Jin Y (2005) A comprehensive survey of fitness approximation in evolutionary computation. Soft Comput 9(1):3–12. https://doi.org/10.1007/s00500-003-0328-5
Jin Y, Sendhoff B (2009) A systems approach to evolutionary multiobjective structural optimization and beyond. IEEE Comput Intell Mag 4(3):62–76. https://doi.org/10.1109/MCI.2009.933094
Jin R, Simpson TW (2001) Comparative studies of metamodeling techniques under multiple modeling criteria. Struct Multidiscip Optim 23(1):1–13. https://doi.org/10.2514/6.2000-4801
Krkkinen I, Frnti P (2000) Minimization of the value of Davies-Bouldin index
Li F, Cheng R, Liu J, Jin Y (2018) A two-stage R2 indicator based evolutionary algorithm for many-objective optimization. Appl Soft Comput 67:245–260. https://doi.org/10.1016/j.asoc.2018.02.048
Li J, Wang P, Dong H, Shen J (2022) A two-stage surrogate-assisted evolutionary algorithm (TS-SAEA) for expensive multi/many-objective optimization. Swarm Evolut Comput 73:101107. https://doi.org/10.1016/j.swevo.2022.101107
Li J, Wang P, Dong H, Shen J, Chen C (2022) A classification surrogate-assisted multi-objective evolutionary algorithm for expensive optimization. Knowled-Based Syst 242:108416. https://doi.org/10.1016/j.knosys.2022.108416
Liu J, Gong M, Miao Q, Wang X, Li H (2017) Structure learning for deep neural networks based on multiobjective optimization. IEEE Trans Neural Netw Learn Syst 29(6):2450–2463. https://doi.org/10.1109/TNNLS.2017.2695223
Liu Y, Liu J, Jin Y (2021) Surrogate-assisted multipopulation particle swarm optimizer for high-dimensional expensive optimization. IEEE Trans Syst, Man, Cyber Syst 52(7):4671–4684. https://doi.org/10.1109/TSMC.2021.3102298
Liu Q, Cheng R, Jin Y, Heiderich M, Rodemann T (2022a) Reference vector-assisted adaptive model management for surrogate-assisted many-objective optimization. IEEE Trans Syst Man, Cyber Syst. https://doi.org/10.1109/TSMC.2022.3163129
Liu Y, Liu J, Tan S, Yang Y, Li F (2022b) A bagging-based surrogate-assisted evolutionary algorithm for expensive multi-objective optimization. Neural Comput Appl 34(14):12097–12118. https://doi.org/10.1007/s00521-022-07097-5
Liu Y, Liu J, Tan S (2023) Decision space partition based surrogate-assisted evolutionary algorithm for expensive optimization. Expert Syst Appl 214:119075. https://doi.org/10.1016/j.eswa.2022.119075
Liu Y, Liu J, Jin Y, Li F, Zheng T (2023b) A surrogate-assisted two-stage differential evolution for expensive constrained optimization. IEEE Trans Emerg Topics Comput Intell. https://doi.org/10.1109/TETCI.2023.3240221
Macqueen J (1965) some methods for classification and analysis of multivariate observations. Proc of Berkeley symposium on mathematical statistics & probability
Martin JD, Simpson TW (2004) Use of kriging models to approximate deterministic computer models. AIAA J 43(4):853–863. https://doi.org/10.2514/1.8650
Pan L, He C, Tian Y, Wang H, Zhang X, Jin Y (2018) A classification-based surrogate-assisted evolutionary algorithm for expensive many-objective optimization. IEEE Trans Evolut Comput 23(1):74–88. https://doi.org/10.1109/TEVC.2018.2802784
Song Z, Wang H, He C, Jin Y (2021) A Kriging-assisted two-archive evolutionary algorithm for expensive many-objective optimization. IEEE Trans Evolut Comput 25(6):1013–1027. https://doi.org/10.1109/TEVC.2021.3073648
Sun C, Song B, Peng W (2015) Parametric geometric model and shape optimization of an underwater glider with blended-wing-body. Int J Naval Archit Ocean Eng 7(6):995–1006. https://doi.org/10.1515/ijnaoe-2015-0069
Tanino T, Kuk H (2003) Nonlinear multiobjective programming. Multiple criteria optimization state of the art annotated bibliographic: surveys. Springer, Boston, pp 71–128
Tian Y, Cheng R, Zhang X, Jin Y (2017a) PlatEMO: a MATLAB platform for evolutionary multi-objective optimization [Educational Forum]. IEEE Comput Intell Mag. https://doi.org/10.1109/MCI.2017.2742868
Tian Y, Cheng R, Zhang X, Cheng F, Jin Y (2017b) An indicator-based multiobjective evolutionary algorithm with reference point adaptation for better versatility. IEEE Trans Evolut Comput 22(4):609–622. https://doi.org/10.1109/TEVC.2017.2749619
Wang ZY, Yu JC, Zhang AQ, Wang YX, Zhao WT (2017) Parametric geometric model and hydrodynamic shape optimization of a flying-wing structure underwater glider. China Ocean Eng 31:709–715. https://doi.org/10.1007/s13344-017-0081-7
Wang X, Wang GG, Song B, Wang P, Wang Y (2019) A novel evolutionary sampling assisted optimization method for high dimensional expensive problems. IEEE Trans Evolut Comput 23(99):815–827. https://doi.org/10.1109/TEVC.2019.2890818
Wang W, Wang X, Dong H, Wang P, Shen J (2023) A model-based shape conceptual design framework of blend-wing-body underwater gliders with curved wings. Ships Offshore Struct. https://doi.org/10.1080/17445302.2023.2181494
While L, Hingston P, Barone L, Huband S (2006) A faster algorithm for calculating hypervolume. IEEE Trans Evolut Comput 10(1):29–38. https://doi.org/10.1109/TEVC.2005.851275
Xu R, Wunsch D (2005) Survey of clustering algorithms. IEEE Trans Neural Netw 16(3):645–678. https://doi.org/10.1109/TNN.2005.845141
Zhang Q, Li H (2007) MOEA/D: A multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evolut Comput 11(6):712–731. https://doi.org/10.1109/TEVC.2007.892759
Zhang Q, Liu W, Tsang E, Virginas B (2010) Expensive multiobjective optimization by MOEA/D with gaussian process model. IEEE Trans Evolut Comput 14(3):456–474. https://doi.org/10.1109/TEVC.2009.2033671
Zhang H, Song S, Zhou A, Gao XZ (2014) A clustering based multiobjective evolutionary algorithm. IEEE World Congress Comput Intell. https://doi.org/10.1109/CEC.2014.6900519
Zhang J, Zhou A, Zhang G (2015) A classification and Pareto domination based multiobjective evolutionary algorithm. In: 2015 IEEE congress on evolutionary computation (CEC). IEEE, pp 2883–2890. https://doi.org/10.1109/CEC.2015.7257247
Zhou A, Qu BY, Li H, Zhao SZ, Suganthan PN, Zhang Q (2011) Multiobjective evolutionary algorithms: a survey of the state of the art. Swarm Evolut Comput 1(1):32–49. https://doi.org/10.1016/j.swevo.2011.03.001
Zitzler E, Deb K, Thiele L (2000) Comparison of multiobjective evolutionary algorithms: empirical results. Evolut Comput 8(2):173–195. https://doi.org/10.1162/106365600568202
Acknowledgements
This project is supported by the National Natural Science Foundation of China (Grant No. 52175251, 52205268) and the National Basic Scientific Research Program under Grant of JCKY2021206B005. Besides, the research work is also supported by the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University.
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Wenxin Wang contribution lies in the design, study conception and writing of the first draft. Huachao Dong and Peng Wang contribution lies in data collection and analysis, Xinjing Wang provided many constructive comments during the revision process and made significant contributions to improving the quality of this article. Jiangtao Shen provided assistance in completing the numerical experiment. The authors read and approved the final manuscript.
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Wang, W., Dong, H., Wang, P. et al. A clustering-based surrogate-assisted evolutionary algorithm (CSMOEA) for expensive multi-objective optimization. Soft Comput 27, 10665–10686 (2023). https://doi.org/10.1007/s00500-023-08227-4
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DOI: https://doi.org/10.1007/s00500-023-08227-4