Abstract
To solve engineering problems with evolutionary algorithms, many expensive function evaluations (FEs) are required. To alleviate this difficulty, surrogate-assisted evolutionary algorithms (SAEAs) have attracted increasingly more attention in both academia and industry. Most existing SAEAs either waste computational resources due to the lack of accuracy of the surrogate model or easily fall into the local optimum as the dimension increases. To address these problems, this paper proposes an adaptive surrogate-assisted particle swarm optimization algorithm. In the proposed algorithm, a surrogate model is adaptively selected from a single model and an ensemble model by comparing the best existing solution and the latest obtained solution. Additionally, a model output criterion based on the standard deviation is suggested to improve the stability and generalization ability of the ensemble model. To verify the performance of the proposed algorithm, 10 benchmark functions with different modalities from 10 to 50 dimensions are tested, and the results are compared with those of five state-of-the-art SAEAs. The experimental results indicate that the proposed algorithm performs well for most benchmark functions within a limited number of FEs. Moreover, the performance of the proposed algorithm in solving engineering problems is verified by applying the algorithm to the PX oxidation process.
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References
Alizadeh R, Allen JK, Mistree F (2020) Managing computational complexity using surrogate models: a critical review. Res Eng Des 31(3):275–298. https://doi.org/10.1007/s00163-020-00336-7
Awad NH, Ali MZ, Mallipeddi R, Suganthan PN (2018) An improved differential evolution algorithm using efficient adapted surrogate model for numerical optimization. Inf Sci 451:326–347. https://doi.org/10.1016/j.ins.2018.04.024
Díaz-Manríquez A, Toscano G, Coello Coello CA (2017) Comparison of metamodeling techniques in evolutionary algorithms. Soft Comput 21(19):5647–5663. https://doi.org/10.1007/s00500-016-2140-z
Emmerich MTM, Giannakoglou KC, Naujoks B (2006) Single- and multi-objective evolutionary optimization assisted by gaussian random field metamodels. IEEE Trans Evol Comput 10(4):421–439. https://doi.org/10.1109/TEVC.2005.859463
Fan CD, Hou B, Xiao LY, Yi LZ (2020) A surrogate-assisted particle swarm optimization using ensemble learning for expensive problems with small sample datasets. Appl Soft Comput. https://doi.org/10.1016/j.asoc.2020.106242
Fan QQ, Yan XF (2015) Differential evolution algorithm with self-adaptive strategy and control parameters for p-xylene oxidation process optimization. Soft Comput 19(5):1363–1391. https://doi.org/10.1007/s00500-014-1349-y
Fu CB, Wang P, Zhao L, Wang XJ (2020) A distance correlation-based Kriging modeling method for high-dimensional problems. Knowl-Based Syst. https://doi.org/10.1016/j.knosys.2020.106356
Gao KF, Mei G, Cuomo S, Piccialli F, Xu NX (2020) ARBF: adaptive radial basis function interpolation algorithm for irregularly scattered point sets. Soft Comput. https://doi.org/10.1007/s00500-020-05211-0
Goel T, Haftka RT, Shyy W, Queipo NV (2007) Ensemble of surrogates. Struct Multidiscip Optim 33(3):199–216. https://doi.org/10.1007/s00158-006-0051-9
Huang CW, Radi B, Hami AE, Bai H (2018) CMA evolution strategy assisted by Kriging model and approximate ranking. Appl Intell 48(11):4288–4304. https://doi.org/10.1007/s10489-018-1193-3
Hüllen G, Zhai JY, Kim SH et al (2020) Managing uncertainty in data-driven simulation-based optimization. Comput Chem Eng. https://doi.org/10.1016/j.compchemeng.2019.106519
Jia LY, Alizadeh R, Jia H et al (2020) A rule-based method for automated surrogate model selection. Adv Eng Inf. https://doi.org/10.1016/j.aei.2020.101123
Jin YC (2011) Surrogate-assisted evolutionary computation: recent advances and future challenges. Swarm Evol Comput 1(2):61–70. https://doi.org/10.1016/j.swevo.2011.05.001
Liang S, Rasheed K (2008) ASAGA: an adaptive surrogate-assisted genetic algorithm. In: Proceeding of the 2008 conference on genetic and evolutionary computation conference(GECCO). https://doi.org/10.1145/1389095.1389289
Li F, Cai XW, Gao L (2019) Ensemble of surrogates assisted particle swarm optimization of medium scale expensive problems. Appl Soft Comput 74:291–305. https://doi.org/10.1016/j.asoc.2018.10.037
Li F, Shen WM, Cai XW (2020) A fast surrogate-assisted particle swarm optimization algorithm for computationally expensive problems. Appl Soft Comput 92:106303. https://doi.org/10.1016/j.asoc.2020.106303
Li YH (2020) A Kriging-based multi-point sequential sampling optimization method for complex black-box problem. Evol Intell. https://doi.org/10.1007/s12065-020-00352-5
Liu B, Grout V, Nikolaeva A (2018) Efficient global optimization of actuator based on a surrogate model assisted hybrid algorithm. IEEE Trans Ind Electron 65(7):5712–5721. https://doi.org/10.1109/TIE.2017.2782203
Liu B, Zhang QF, Gielen GGE (2014) A Gaussian process surrogate model assisted evolutionary algorithm for medium scale expensive optimization problems. IEEE Trans Evol Comput 18(2):180–192. https://doi.org/10.1109/TEVC.2013.2248012
Liu HT, Meng JG, Xu SL, Yang SH, Wang XF (2016) Optimal weighted pointwise ensemble of radial basis functions with different basis functions. AIAA J 54(10):3117–3133. https://doi.org/10.2514/1.J054664
Liu HT, Cai JF, Ong Y (2017) An adaptive sampling approach for Kriging metamodeling by maximizing expected prediction error. Comput Chem Eng 106:171–182. https://doi.org/10.1016/j.compchemeng.2017.05.025
Liu RR, Li ZM (2014) Soft sensor for COx content in tail gas of PX oxidation side reactions based on particle filters and EM algorithm. In: 2013 International Conference on Future Software Engineering and Multimedia Engineering 6: 63–71. https://doi.org/10.1016/j.ieri.2014.03.011
Loshchilov I, Schoenauer M, Sebag M (2012) Self-adaptive surrogate-assisted covariance matrix adaptation evolution strategy. In: Proceedings of the 2012 international conference on genetic and evolutionary computation. https://doi.org/10.1145/2330163.2330210
Masato K, Hidetoshi M (2020) The limiting distribution of combining the t and Wilcoxon rank sum tests. Stat 54(4):871–884. https://doi.org/10.1080/02331888.2020.1809662
Mohamed WA (2017) A novel differential evolution algorithm for solving constrained engineering optimization problems. J Intell Manuf 29(3):659–692. https://doi.org/10.1007/s10845-017-1294-6
Nguyen DD, Long N (2021) An adaptive control for surrogate assisted multi-objective evolutionary algorithms. pp: 123–132. https://doi.org/10.1007/978-981-15-8289-9_12
Pan JS, Liu NX, Chu SC, Lai TT (2021) An efficient surrogate-assisted hybrid optimization algorithm for expensive optimization problems. Inf Sci 561:304–325. https://doi.org/10.1016/j.ins.2020.11.056
Pan LQ, He C, Tian Y, Wang HD, Zhang XY, Jin YC (2019) A classification-based surrogate-assisted evolutionary algorithm for expensive many-objective optimization. IEEE Trans Evol Comput 23(1):74–88. https://doi.org/10.1109/TEVC.2018.2802784
Regis RG (2014) Particle swarm with radial basis function surrogates for expensive black-box optimization. J Comput Sci 5(1):12–23. https://doi.org/10.1016/j.jocs.2013.07.004
Si T, Jana ND, Sil J (2011) Constrained function optimization using PSO with polynomial mutation. In: International conference on swarm 7076: 209-216
Sun CL, Jin YC, Cheng R, Ding JL, Zeng JC (2017) Surrogate-assisted cooperative swarm optimization of high-dimensional expensive problems. IEEE Trans Evol Comput 21(4):644–660. https://doi.org/10.1109/TEVC.2017.2675628
Sun CL, Jin YC, Zeng JC, Yu Y (2015) A two-layer surrogate-assisted particle swarm optimization algorithm. Soft Comput 19(6):1461–1475. https://doi.org/10.1007/s00500-014-1283-z
Urquhart M, Ljungskog E, Sebben S (2020) Surrogate-based optimization using adaptively scaled radial basis functions. Appl Soft Comput 88:106050. https://doi.org/10.1016/j.asoc.2019.106050
Wang HD, Jin YC, Sun CL, Doherty J (2019) Offline data-driven evolutionary optimization using selective surrogate ensembles. IEEE Trans Evol Comput 23(2):203–216. https://doi.org/10.1109/TEVC.2018.2834881
Wang HD, Jin YC, Doherty J (2017) Committee-based active learning for surrogate-assisted particle swarm optimization of expensive problems. IEEE Trans Cybern 47(9):2664–2677. https://doi.org/10.1109/TCYB.2017.2710978
Wu JL, Luo Z, Zhang N, Gao W (2018) A new sequential sampling method for constructing the high-order polynomial surrogate models. Eng Comput 35(2):529–564. https://doi.org/10.1108/EC-05-2016-0160
Ye PC, Pan G, Dong ZM (2018) Ensemble of surrogate based global optimization methods using hierarchical design space reduction. Struct Multidiscip Optim 58(2):537–554. https://doi.org/10.1007/s00158-018-1906-6
Yondo R, Andrés E, Valero E (2018) A review on design of experiments and surrogate models in aircraft real-time and many-query aerodynamic analyses. Prog Aeosp Sci 96:23–61. https://doi.org/10.1016/j.paerosci.2017.11.003
Zhai JY, Boukouvala F (2019) Nonlinear variable selection algorithms for surrogate modeling. AIChE J 65(8):e16601. https://doi.org/10.1002/aic.16601
Zhang XY, Tian Y, Cheng R, Jin YC (2015) An efficient approach to non-dominated sorting for evolutionary multiobjective optimization. IEEE Trans Evol Comput 19(2):201–213. https://doi.org/10.1109/TEVC.2014.2308305
Zhu H, Hu YM, Zhu WD (2019) A dynamic adaptive particle swarm optimization and genetic algorithm for different constrained engineering design optimization problems. Adv Mech Eng 11(3):1–27. https://doi.org/10.1177/1687814018824930
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The authors of this paper appreciate the support from the National Natural Science Foundation of China (Project No. 21676086).
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Xuemei Li contributed to conceptualization, methodology, software, writing-reviewing and editing. Shaojun Li contributed to conceptualization, writing-reviewing, resources, supervision, project administration and funding acquisition.
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Li, X., Li, S. An adaptive surrogate-assisted particle swarm optimization for expensive problems. Soft Comput 25, 15051–15065 (2021). https://doi.org/10.1007/s00500-021-06348-2
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DOI: https://doi.org/10.1007/s00500-021-06348-2