Research Papers

Updating Kriging Surrogate Models Based on the Hypervolume Indicator in Multi-Objective Optimization

[+] Author and Article Information
Koji Shimoyama

Assistant Professor
Institute of Fluid Science,
Tohoku University,
Sendai 980-8577, Japan
e-mail: shimoyama@edge.ifs.tohoku.ac.jp

Koma Sato

Hitachi Research Laboratory,
Hitachi, Ltd.,
Hitachinaka, Ibaraki 312-0034, Japan
e-mail: koma.sato.ky@hitachi.com

Shinkyu Jeong

Associate Professor
Department of Mechanical Engineering,
Kyunghee University,
Yongin 446-701, Korea
e-mail: icarus@khu.ac.kr

Shigeru Obayashi

Institute of Fluid Science,
Tohoku University,
Sendai 980-8577, Japan
e-mail: obayashi@ifs.tohoku.ac.jp

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the Journal of Mechanical Design. Manuscript received September 24, 2012; final manuscript received May 15, 2013; published online July 2, 2013. Assoc. Editor: Bernard Yannou.

J. Mech. Des 135(9), 094503 (Jul 02, 2013) (7 pages) Paper No: MD-12-1469; doi: 10.1115/1.4024849 History: Received September 24, 2012; Revised May 15, 2013; Accepted June 04, 2013

This paper presents a comparison of the criteria for updating the Kriging surrogate models in multi-objective optimization: expected improvement (EI), expected hypervolume improvement (EHVI), estimation (EST), and those in combination (EHVI + EST). EI has been conventionally used as the criterion considering the stochastic improvement of each objective function value individually, while EHVI has recently been proposed as the criterion considering the stochastic improvement of the front of nondominated solutions in multi-objective optimization. EST is the value of each objective function estimated nonstochastically by the Kriging model without considering its uncertainties. Numerical experiments were implemented in the welded beam design problem, and empirically showed that, in an unconstrained case, EHVI maintains a balance between accuracy, spread, and uniformity in nondominated solutions for Kriging-model-based multiobjective optimization. In addition, the present experiments suggested future investigation into techniques for handling constraints with uncertainties to enhance the capability of EHVI in constrained cases.

Copyright © 2013 by ASME
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Myers, R. H., and Montgomery, D. C., 1995, Response Surface Methodology: Process and Product Optimization Using Designed Experiments, Wiley, New York.
Wang, G. G., 2003, “Adaptive Response Surface Method Using Inherited Latin Hypercube Design Points,” ASME J. Mech. Des., 125(2), pp. 210–220. [CrossRef]
Bishop, C. M., 1995, Neural Networks for Pattern Recognition, Oxford University, Oxford.
Sacks, J., Welch, W. J., Mitchell, T. J., and Wynn, H. P., 1989, “Design and Analysis of Computer Experiments,” Statist. Sci., 4(4), pp. 409–435. [CrossRef]
Jones, D. R., Schonlau, M., and Welch, W. J., 1998, “Efficient Global Optimization of Expensive Black-Box Function,” J. Global Optim., 13, pp. 455–492. [CrossRef]
Jeong, S., Minemura, Y., and Obayashi, S., 2006, “Optimization of Combustion Chamber for Diesel Engine Using Kriging Model,” J. Fluid Sci. Technol., 1(2), pp. 138–146. [CrossRef]
Li, M., Li, G., and Azarm, S., 2008, “A Kriging Metamodel Assisted Multi-Objective Genetic Algorithm for Design Optimization,” ASME J. Mech. Des., 130(3), p. 031401. [CrossRef]
Li, M., 2011, “An Improved Kriging-Assisted Multi-Objective Genetic Algorithm,” ASME J. Mech. Des., 133(7), p. 071008. [CrossRef]
Martin, J. D., 2009, “Computational Improvements to Estimating Kriging Metamodel Parameters,” ASME J. Mech. Des., 131(8), p. 084501. [CrossRef]
Shimoyama, K., Lim, J. N., Jeong, S., Obayashi, S., and Koishi, M., 2009, “Practical Implementation of Robust Design Assisted by Response Surface Approximation and Visual Data-Mining,” ASME J. Mech. Des., 131(6), p. 0610071. [CrossRef]
Hamel, J. M., and Azarm, S., 2011, “Reducible Uncertain Interval Design by Kriging Metamodel Assisted Multi-Objective Optimization,” ASME J. Mech. Des., 133(1), p. 011002. [CrossRef]
Wang, G. G., and Shan, S., 2007, “Review of Metamodeling Techniques in Support of Engineering Design Optimization,” ASME J. Mech. Des., 129(4), pp. 370–380. [CrossRef]
Łaniewski-Wołłk, Ł., Obayashi, S., and Jeong, S., 2010, “Development of Expected Improvement for Multi-Objective Problem,” Proceedings of 42nd Fluid Dynamics Conference/Aerospace Numerical Simulation Symposium 2010.
Emmerich, M. T. M., Deutz, A. H., and Klinkenberg, J. W., 2011, “Hypervolume-Based Expected Improvement: Monotonicity Properties and Exact Computation,” Proceedings of the 2011 IEEE Congress on Evolutionary Computation, IEEE Press, pp. 2147–2154.
Zitzler, E., and Thiele, L., 1998, “Multiobjective Optimization Using Evolutionary Algorithms—A Comparative Case Study,” Proceedings of the 5th International Conference on Parallel Problem Solving From Nature, Springer-Verlag, pp. 292–301.
Beume, N., Fonseca, C. M., López-Ibáñez, M., Paquete, L., and Vahrenhold, J., 2009, “On the Complexity of Computing the Hypervolume Indicator,” IEEE Trans. Evol. Comput., 13(5), pp. 1075–1082. [CrossRef]
McKay, M. D., Beckman, R. J., and Conover, W. J., 1979, “A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output From a Computer Code,” Technometrics, 21(2), pp. 239–245.
Sato, K., Kumano, T., Yonezawa, M., Yamashita, H., Jeong, S., and Obayashi, S., 2008, “Low-Boom and Low-Drag Optimization of the Twin Engine Version of Silent Supersonic Business Jet,” J. Fluid Sci. Technol., 3(4), pp. 576–585. [CrossRef]
Shimoyama, K., Yoshimizu, S., Jeong, S., Obayashi, S., and Yokono, Y., 2011, “Multi-Objective Design Optimization for a Steam Turbine Stator Blade Using LES and GA,” J. Comput. Sci. Technol., 5(3), pp. 134–147. [CrossRef]
Van Veldhuizen, D. A., and Lamont, G. B., 1998, “Multiobjective Evolutionary Algorithm Research: A History and Analysis, Air Force Institute of Technology, Dayton, OH, Technical Report No. TR–98–03.
Schott, J. R., 1998, “Fault Tolerant Design Using Single and Multicriteria Genetic Algorithm Optimization,” Master's thesis, Massachusetts Institute of Technology, Cambridge, MA.
Zitzler, E., 1999, “Evolutionary Algorithms for Multiobjective Optimization: Methods and Applications,” Ph.D. thesis, Swiss Federal Institute of Technology Zürich, Zürich, Switzerland.
Van Veldhuizen, D. A., 1999, “Multiobjective Evolutionary Algorithms: Classifications, Analyses, and New Innovations,” Ph.D. thesis, Air Force Institute of Technology, Dayton, OH.
While, L., and Bradstreet, L., 2012, “Applying the WFG Algorithm to Calculate Incremental Hypervolumes,” Proceedings of the 2012 IEEE Congress on Evolutionary Computation, IEEE Press, pp. 489–496.


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Fig. 2

Criteria for updating the Kriging model: (a) EI and (b) EHVI

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Fig. 3

Welded beam design problem

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Fig. 4

Flowchart of updating the Kriging models

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Fig. 5

Scatter plots of the sample points in objective function space obtained on the first trial in problem 1: (a) EST, (b) EI, (c) EHVI, and (d) EHVI + EST

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Fig. 6

Scatter plots of the sample points in objective function space obtained on the first trial in problem 2: (a) EST, (b) EI, (c) EHVI, and (d) EHVI + EST




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