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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

Researcher
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

Professor
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.

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Figures

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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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