Many types of optimization problems involving expensive functions have been handled successfully by meta-model-based optimization (MBO) methods. However, some particular difficult problems, such as multimodal functions, which require an accurate approximation even around local optima, still prove challenging for MBO. Generally, this type of problems is solved by combining or assembling meta-models. This paper proposes a new strategy of combining a global meta-model with many mid-range ones. The latter are constructed on sub-regions where the global meta-model is inaccurate. These mid-range meta-models are constructed in two phases. The first phase aims to explore accurately all the design space; whereas the second phase aims to provide more accurate solutions than those of the global meta-model on regions of interest (exploitation). The set of sub-regions can be updated adaptively until reaching the meta-model target accuracy. This two phases’ adaptive process prevents the over sampling of non-interesting regions, which allows gaining significantly in the cost. The proposed strategy is tested using the following techniques: Radial basis function (RBF) and Non-dominated Sorting Genetic Algorithm (NSGA-II). This strategy is tested both on difficult mathematical benchmarks and on an engineering application giving satisfactory results. The considered engineering application aims to optimize the operation of a vertical axe wind turbine.
WangXDHirschCKangSH,et al.Multi-Objective optimization of turbomachinery using improved NSGA-II and approximation model. Comput Methods Appl Mech Eng2011; 200: 883–895.
2.
KhalfallahSGhenaietA. Radial basis function-based shape optimization of centrifugal impeller using sequential sampling. Proc IMechE Part G: J Aerospace Engineering2014; 229: 648–665.
3.
QinNCarnieGWangY,et al.Design optimization of casing grooves using zipper layer meshing. ASME Journal of Turbomachinery2014; 136: 031002–1–031002-12.
4.
SimpsonTWMaueryTMKorteJJ. Kriging models for global approximation in simulation-based multidisciplinary design optimization. AIAA J. 2001;39: 2233–2241.
5.
GuoZZhangYDingW. Optimization of the aerodynamic drag reduction of a passenger hatchback car. Proc IMechE Part G: J Aerospace Engineering2018; 233: 2819–2836.
6.
KhalfallahSLehtihetHE. A multi-objective optimization methodology based on multi-mid-range meta-models for multimodal deterministic/robust problems. Struct Multidiscipl Optim2019; 60: 2373–2389.
7.
HavingaJvan den BoogaardAHKlaseboerG. Sequential improvement for robust optimization using an uncertainty measure for radial basis functions, struct. Multidisc. Optim.2017; 55: 1345–1363.
8.
Gray WangGShanS. Review of metamodeling techniques in support of engineering design optimization. ASME Journal of Mechanical Design2007; 129: 370–380.
JonesDRSchonlauMWelchWJ. Efficient global optimization of expensive black-box functions. Journal of Global Optimization1998; 13: 455–492.
11.
WangYYinDYangS,et al.Global and local surrogate-assisted differential evolution for expensive constrained optimization problems with inequality constraints. IEEE Transactions on Cybernetics2019; 49: 1642–1656.
12.
AkhtarTShoemakerC. Combining local surrogates and adaptive restarts for global optimization of moderately expensive functions. AIP Conference Proceedings.2019; 2070: 020048.
13.
RanRLiWLiY. A two-level global optimization method based on hybrid metamodel for expensive problems. Advances in Mechanical Engineering2018; 10: 168781401876954.
14.
ZhangXLiHXiangG,et al.The ensemble of surrogate model based on local and global errors. IOP Conf Ser: Mater Sci Eng2021; 1043: 052049.
15.
ZhangJYueXQiuJ, et al.A unified ensemble of surrogates with global and local measures for global metamodelling. Eng Optim2021; 53: 474–495.
16.
ChenLMQiuHBJiangC, et al.Ensemble of surrogates with hybrid method using global and local measures for engineering design. Struct Multidiscipl Optim2018; 57: 1711–1729.
17.
ChristelisaVKopsiaftisGMantoglouA. Performance comparison of multiple and single surrogate models for pumping optimization of coastal aquifers. Hydrol Sci J2019; 64: 336–349.
18.
KochPNSimpsonTWAllenJK,et al.Statistical approximations for multidisciplinary optimization: the problem of size. J Aircr1999; 36: 275–286.
19.
BaudouiVKlotzPHiriart-UrrutyJB, et al.Local uncertainty processing (LOUP) method for multidisciplinary robust design optimization. Struct. Multidisc. Optim.2012; 46: 711–726.
20.
PolynkinAToropovVV. Mid-range metamodel assembly building based on linear regression for large scale optimization problems. Struct. Multidisc. Optim2012; 45: 515–527.
21.
BettebghorDBartoliNGrihonS, et al.Surrogate modeling approximation using a mixture of experts based on EM joint estimation. Struct Multidisc Optim2011; 43: 243–259.
22.
VianaFACHaftkaRTSteffenV. Multiple surrogates: how cross-validation errors can help us to obtain the best predictor. Struct Multidisc Optim2009; 39: 439–457.
23.
ZerpaLEQueipoNVPintosS,et al.An optimization methodology of alkaline-surfactant polymer flooding processes using field scale numerical simulation and multiple surrogates. J Pet Sci Eng2005; 47: 197–208.
24.
SimpsonTWToropovVBalabanovV,et al.Design and Analysis of Computer Experiments in Multidisciplinary Design Optimization: A Review of How Far We Have Come - Or Not, 12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Multidisciplinary Analysis Optimization Conferences, Victoria, British Columbia, 2008, Canada, doi: 10.2514/6.2008-5802.
25.
HaftkaRT. Combining global and local approximations. AIAA J1991; 29: 1523–1525.
26.
ZhouZOngYSNairPB, et al.Combining global and local surrogate models to accelerate evolutionary optimization. IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews)2007; 37: 66–76.
27.
YangZQiuHCaiXL, et al.A surrogate-assisted particle swarm optimization algorithm based on efficient global optimization for expensive black-box problems. Eng Optim2019; 51: 549–566.
28.
KhatouriHBenamaraTBreitkopfP, et al.Metamodeling techniques for CPU-intensive simulation-based design optimization: a survey. Adv. Model. and Simul. in Eng. Sci.2022; 9: 1.
29.
LiXYangOWangY, et al.Development of surrogate models in reliability-based design optimization: a review. Math Biosci Eng2021; 18: 6386–6409.
30.
ChenGZhangKZhangL, et al.Global and local surrogate-model-assisted differential evolution for waterflooding production optimization. SPE J2019; 25: 105–118.
31.
ZhaoDXueD. A multi-surrogate approximation method for metamodeling. Eng. Comput2011; 27: 139–153.
32.
LuoLLiuDZhuM, et al.Maximum energy conversion from human motion using piezoelectric flex transducer: a multi-level surrogate modeling strategy. J Intell Mater Syst Struct 2018; 29: 3097–3107.
33.
DongGLiuCLiuD, et al.Adaptive multi-level search for global optimization: an integrated swarm intelligence-metamodelling technique. Appl. Sci2021; 11: 2277.
34.
Diaz-ManriquezAToscano-PulidoGGomez-FloresW. On the selection of surrogate models in evolutionary optimization algorithms. In: Evolutionary computation (CEC), inst. Of electrical and electronics engineers congress. New Orleans, LA: Inst. of Electrical and Electronics Engineers, 2011, pp.2155–2162.
35.
PeterJMarceletM. Comparison of surrogate models for turbomachinery design. WSEAS Trans Fluid Mech2008; 3: 10–17.
36.
JinRChenWSimpsonTW. Comparative studies of metamodeling techniques under multiple modeling criteria. Journal of Structural and Multidisciplinary Optimization2001; 23: 1–13.
37.
LiangJKhailahLZYE. Research on the modified surrogate model based on local RBF for concrete dam static and dynamic response analysis. Front. Earth Sci2023; 11: 1125691.
38.
DerbalSKhalfallahSCerdounM, et al.Robust optimization of the aerodynamic design of a helicopter rotor blade based on multi-fidelity meta-models. Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering2022; 237: 095440892211115.
39.
DebKGuptaH. Introducing robustness in multi-objective optimization. Evol Comput2006; 14: 463–494.
40.
RippaS. An algorithm for selecting a good value for the parameter C in radial basis function interpolation. Adv Comput Math1999; 11: 193–210.
41.
BenedictMLakshminarayanVPinoJ,et al.Aerodynamics of a small-scale vertical-axis wind turbine with dynamic blade pitching. AIAA J2016; 54: 924–935.
KursaweF. A variant of evolution strategies for vector optimization. In: SchwefelH-PMännerR (eds) Parallel problem solving from nature. Berlin, Germany: Springer-Verlag, 1990, pp.193–197.
