# A multi-objective optimization methodology based on multi-mid-range meta-models for multimodal deterministic/robust problems

- 99 Downloads

## Abstract

The success of meta-model-based optimization primarily relies on how accurately the black-box functions are being represented. However, sometimes a global meta-model fails to achieve sufficient accuracy. This can be the case in multi-objective deterministic problems involving multimodal functions with competitive local optima or robust problems which require an accurate local description. This paper proposes a new methodology that deals with this type of situations and that provides the required accuracy both locally and globally. We use a set of mid-range meta-models which, in contrast to other works, are not used to construct a global meta-model but are managed both to compete and collaborate to solve the problem. They are defined across overlapping regions of interest generated by a process which resizes and moves adaptively these regions until tracking the Pareto front. The accuracy of these mid-range meta-models is also improved by a new design-of-experiment (DoE) adaptive technique allowing the suppression of some inefficient DoE points. The proposed method is implemented using standard techniques, such as non-dominated sorting genetic algorithm-II (NSGA-II), whereas the optimal shape factor of radial basis functions (RBF) is calculated by combining NSGA-II and particle swarm optimization (PSO). We also use Hager’s method to detect ill-conditioned systems and avoid propagating their outcome, which significantly improves the performance. This method is tested against difficult deterministic and robust multi-objective multimodal benchmarks and is applied to the robust optimization of an aerodynamic design case*.*

## Keywords

Meta-model-based optimization Robust design Multimodal functions CFD## Notes

### Acknowledgments

The authors would like to thank anonymous reviewers for having provided valuable criticisms and recommendations which have greatly helped improve the quality of the paper.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## Supplementary material

## References

- Baudoui V, Klotz P, Hiriart-Urruty JB, Jan S, Morel F (2012) Local uncertainty processing (LOUP) method for multidisciplinary robust design optimization. Struct Multidiscip Optim 46(5):711–726CrossRefGoogle Scholar
- Deb K, Gupta H (2006) Introducing robustness in multi-objective optimization. Evol Comput 14(4):463–494CrossRefGoogle Scholar
- Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197CrossRefGoogle Scholar
- Deb K, Hussein R, Roy P, and Toscano G, (2017) Classifying metamodeling methods for evolutionary multi-objective optimization: first results. EMO 2017 9th International Conference on Evolutionary Multi-Criterion Optimization, 10173: 160–175Google Scholar
- Diaz-Manriquez A, Toscano-Pulido G, and Gomez-Flores W. (2011) On the selection of surrogate models in evolutionary optimization algorithms. proceeding of: Evolutionary Computation (CEC), IEEE Congress on, 2155–2162Google Scholar
- Dow EA and Wang Q, (2014) The implications of tolerance optimization on compressor blade design. ASME J Turbomach 137(10):101008_1–101008_7Google Scholar
- Emmerich MTM, Giannakoglou KC, Naujoks B (2006) Single- and multiobjective evolutionary optimization assisted by Gaussian random field metamodels. IEEE Trans Evol Comput 10(4):421–439CrossRefGoogle Scholar
- Fonseca CM, Fleming PJ (1998) Multiobjective optimization and multiple constraint handling with evolutionary algorithms—part II: application example. IEEE Trans Syst Man Cybern Syst Hum 28(1):38–47CrossRefGoogle Scholar
- Forrester AIJ, Keane AJ (2009) Recent advances in surrogate-based optimization. Prog Aerosp Sci 45(1–3):50–79CrossRefGoogle Scholar
- Giunta AA, and Eldred MS, (2000) Implementation of a trust region model management strategy in the Dakota optimization toolkit. Proceedings of 8th AIAA/USAF/ NASA/ISSMO Symposium on Multi- disciplinary Analysis and Optimization, Long Beach, CA, U.S.A.Google Scholar
- Gray WANGG, Shan S (2007) Review of metamodeling techniques in support of engineering design optimization. ASME J Mechan Design 129(4):370–380CrossRefGoogle Scholar
- Gu X, Renaud JE, Batill SM, Brach RM, Budhiraja AS (2000) Worst case propagated uncertainty of multidisciplinary systems in robust design optimization. Struct Multidiscip Optim 20(3):190–213CrossRefGoogle Scholar
- Gumbert CR, Newman PA, and Hou GJ-W, (2002) Effect of random geometric uncertainty on the computational design of a 3-D Flexible Wing. 20th AIAA Applied Aerodynamics Conference, AIAA 2002–2806Google Scholar
- Haftka RT, Nachlas JA, Watson LT, Rizzo T, Desai R (1987) Two-point constraint approximation in structural optimization. Comput Methods Appl Mech Eng 60(3):289–301CrossRefGoogle Scholar
- Hager WW (1984) Condition estimates. SIAM J Sci Stat Comput 5(2):311–316MathSciNetCrossRefGoogle Scholar
- Havinga J, van den Boogaard AH, Klaseboer G (2017) Sequential improvement for robust optimization using an uncertainty measure for radial basis functions. Struct Multidiscip Optim 55(4):1345–1363MathSciNetCrossRefGoogle Scholar
- Janusevskis J, Riche RL (2013) Simultaneous kriging-based estimation and optimization of mean response. J Glob Optim 55(2):313–336MathSciNetCrossRefGoogle Scholar
- Jin R, Chen W, Simpson TW (2001) Comparative studies of metamodeling techniques under multiple modeling criteria. Struct Multidiscip Optim 23(1):1–13CrossRefGoogle Scholar
- Jin R, Du X, Chen W (2003) The use of metamodeling techniques for optimization under uncertainty. Struct Multidiscip Optim 25(2):99–116CrossRefGoogle Scholar
- Jin R, Chen W, and Sudjianto A, (2004) Analytical metamodel-based global sensitivity analysis and uncertainty propagation for robust design. SAE Technical Paper 2004-01-0429Google Scholar
- Jin R, Chen W, Sudjianto A (2005) An efficient algorithm for constructing optimal design of computer experiments. J Stat Plann Infer 134(1):268–287MathSciNetCrossRefGoogle Scholar
- Jones DR, Schonlau M, Welch WJ (1998) Efficient global optimization of expensive black-box functions. J Glob Optim 13(4):455–492MathSciNetCrossRefGoogle Scholar
- Jurecka F, Ganser M, Bletzinger KU (2007) Update scheme for sequential spatial correlation approximations in robust design optimization. Comput Struct 85(10):606–614CrossRefGoogle Scholar
- Kennedy J and Eberhart R, (1995) Particle swarm optimization. Proceedings of the IEEE International Conference on Neural Networks, 4: 1942–1948Google Scholar
- Khalfallah S, Ghenaiet A, Benini E, Bedon G (2015) Surrogate-based shape optimization of stall margin and efficiency of a centrifugal compressor. AIAA J Propuls Power 31(6):1607–1162CrossRefGoogle Scholar
- Koch PN, Simpson TW, Allen JK, Mistree F (1999) Statistical approximations for multidisciplinary optimization: the problem of size. AIAA Journal of Aircraft 36(1):275–286CrossRefGoogle Scholar
- Korolev YM and Toropov VV, (2017) Design optimization under uncertainty using the multipoint approximation method, 58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials ConferenceGoogle Scholar
- Kumar A, Nair PB, Keane AJ, Shahpar S (2008) Robust design using Bayesian Monte Carlo. Int J Numer Methods Eng 73(11):1497–1517CrossRefGoogle Scholar
- Ladson CL, (1988) Effects of independent variation of Mach and Reynolds numbers on the low-speed aerodynamic characteristics of the NACA 0012 airfoil section. Tech. Rep., NASA Langley Research CenterGoogle Scholar
- Leary SJ, Bhaskar A, Keane AJ (2003) A knowledge-based approach to response surface modelling in multifidelity optimization. J Glob Optim 26(3):297–319MathSciNetCrossRefGoogle Scholar
- Lee KH, Park GJ (2006) A global robust optimization using kriging based approximation model. Jsme Int J Series C-Mechan Syst Mach Elements Manuf 49(3):779–788CrossRefGoogle Scholar
- Lehman JS, Santner TJ, Notz WI (2004) Designing computer experiments to determine robust control variables. Stat Sin 14(2):571–590MathSciNetzbMATHGoogle Scholar
- Lelièvre N, Beaurepaire P, Mattrand C, Gayton N, Otsmane A (2016) On the consideration of uncertainty in design: optimization - reliability – robustness. Struct Multidiscip Optim 54(6):1423–1437MathSciNetCrossRefGoogle Scholar
- Liu H, Chen W, Kokkolaras M, Papalambros PY, Kim HM (2006) Probabilistic analytical target cascading: a moment matching formulation for multilevel optimization under uncertainty. ASME J Mechan Design 128(4):991–1000CrossRefGoogle Scholar
- Lock RC, (1970) Test cases for numerical methods in two-dimensional transonic flows. Report of Advisory Group for Aerospace Research and Development (AGARD), No. 575Google Scholar
- Ong YS, Nair PB, Keane AJ (2003) Evolutionary optimization of computationally expensive problems via surrogate modeling. AIAA J 41(4):687–696CrossRefGoogle Scholar
- Padulo M, Maginot J, Guenov M, and Holden C, (2009) Airfoil design under uncertainty with robust geometric parameterization. 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Structures, Structural Dynamics, and Materials and Co-located Conferences, AIAA 2009–2270Google Scholar
- Pedrielli G and Hui Ng S, (2016) G-STAR: a new kriging-based trust region method for global optimization. IEEE Proceedings of the 2016 Winter Simulation Conference, 803–814Google Scholar
- Polynkin A and Toropov VV, (2009) Multiple mid-range and global metamodel building based on linear regression. Collection of Technical Papers - AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials ConferenceGoogle Scholar
- Polynkin A, Toropov VV (2012) Mid-range metamodel assembly building based on linear regression for large scale optimization problems. Struct Multidiscip Optim 45(4):515–527CrossRefGoogle Scholar
- Qin N, Carnie G, Wang Y, Shahpar S (2014) Design optimization of casing grooves using zipper layer meshing. ASME J Turbomach 136(3):031002–1-031002-12CrossRefGoogle Scholar
- Rippa S (1999) An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv Comput Math 11(2):193–210MathSciNetCrossRefGoogle Scholar
- Rumpfkeil MP (2013) Optimizations under uncertainty using gradients, hessians, and surrogate models. AIAA J 51(2):444–451CrossRefGoogle Scholar
- Samareh JA, (2000) Multidisciplinary aerodynamic-structural shape optimization using deformation (MASSOUD). 8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization. AIAA-2000-4911, Long Beach, CAGoogle Scholar
- Simpson TW, Mauery TM, Korte JJ (2001) Kriging models for global approximation in simulation-based multidisciplinary design optimization. AIAA J 39(12):2233–2241CrossRefGoogle Scholar
- Simpson TW, Toropov V, Balabanov V, and Viana F, (2008) 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, CanadaGoogle Scholar
- Toropov VV, Filatov AA, Polynkin AA (1993) Multiparameter structural optimization using FEM and multipoint explicit approximations. Struct Optimization 6(1):7–14CrossRefGoogle Scholar
- Venter G, Haftka RT (1999) Using response surface approximations in fuzzy set based design optimization. Struct Multidiscip Optim 18(4):218–227CrossRefGoogle Scholar
- Wang XD, Hirsch C, Kang SH, Lacor C (2011) Multi-objective optimization of turbomachinery using improved NSGA-II and approximation model. Comput Methods Appl Mech Eng 200(9–12):883–895MathSciNetCrossRefGoogle Scholar
- Wilson B, Cappelleri D, Simpson TW, Frecker M (2001) Efficient Pareto frontier exploration using surrogate approximations. Optim Eng 2(1):31–50MathSciNetCrossRefGoogle Scholar
- Yahyaie F, Filizadeh S (2011) A surrogate-model based multi-modal optimization algorithm. Eng Optim 43(7):779–799CrossRefGoogle Scholar
- Zhang J, Chowdhury S, Messac A (2012) An adaptive hybrid surrogate model. Struct Multidiscip Optim 46(2):223–238CrossRefGoogle Scholar
- Zhao D, Xue D (2011) A multi-surrogate approximation method for metamodeling. Eng Comput 27(2):139–153CrossRefGoogle Scholar
- Zhou Z, Ong YS, Nair PB, Keane AJ, Kai-Yew L (2007) Combining global and local surrogate models to accelerate evolutionary optimization. IEEE Trans Syst Man and Cybernet (SMC), Part C 37:66–76CrossRefGoogle Scholar