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Global Optimization Method Using SLE and Adaptive RBF Based on
ZHU Huaguang, LIU Li*, LONG Teng, and ZHAO Junfeng
School of Aerospace Engineering, Beijing Institute of
Technology, Beijing 100081, China
Received July 14, 2010; revised March 1, 2012; accepted March
Abstract: High fidelity analysis models, which are beneficial to
improving the design quality, have been more and more widely
utilized in the modern engineering design optimization problems.
However, the high fidelity analysis models are so
computationally expensive that the time required in design
optimization is usually unacceptable. In order to improve the
efficiency of optimization involving high fidelity analysis
models, the optimization efficiency can be upgraded through
applying surrogates to approximate the computationally expensive
models, which can greately reduce the computation time. An
efficient heuristic global optimization method using adaptive
radial basis function (RBF) based on fuzzy clustering (ARFC) is
proposed. In this method, a novel algorithm of maximin Latin
hypercube design using successive local enumeration (SLE) is
employed to obtain sample points with good performance in both
space-filling and projective uniformity properties, which does a
great deal of good to metamodels accuracy. RBF method is adopted
for constructing the metamodels, and with the increasing the
number of sample points the approximation accuracy of RBF is
gradually enhanced. The fuzzy c-means clustering method is
applied to identify the reduced attractive regions in the
original design space. The numerical benchmark examples are used
for validating the performance of ARFC. The results demonstrates
that for most application examples the global optima are
effectively obtained and comparison with adaptive response
surface method (ARSM) proves that the proposed method can
intuitively capture promising design regions and can efficiently
identify the global or near-global design optimum. This method
improves the efficiency and global convergence of the
optimization problems, and gives a new optimization strategy for
engineering design optimization problems involving
computationally expensive models.
Key words: global optimization, Latin hypercube design, radial
basis function, fuzzy clustering, adaptive response surface
ZHU Huaguang, born in 1981, is currently a PhD candidate at School of
Aerospace Engineering, Beijing Institute of Technology, China. His
research interests include multidisciplinary design optimization, flight
vehicle conceptual design, and flight vehicle engineering structural
Tel: +86-10-68913290; E-mail: email@example.com
LIU Li, born in 1964, is currently a professor at School of Aerospace
Engineering, Beijing Institute of Technology, China. Her research
interests include flight vehicle conceptual design, flight vehicle
engineering structural optimization design, multidisciplinary design
optimization, flight vehicle guidance and control.
Tel: +86-10-68914534; E-mail: firstname.lastname@example.org
LONG Teng, born in 1982, is currently a lecturer at School of Aerospace
Engineering, Beijing Institute of Technology, China. His research
interests include flight vehicle conceptual design, theory and
applications of multidisciplinary design optimization.
Tel: +86-10-68911926; E-mail: email@example.com
ZHAO Junfeng, born in 1984, is currently a PhD candidate at School of
Aerospace Engineering, Beijing Institute of Technology, China. His
research interests include flight vehicle engineering structural
optimization design, multi-body dynamics, and flight vehicle conceptual
Tel: +86-10-68913290; E-mail: firstname.lastname@example.org
 SHAN S Q, WANG G G. Survey of modeling and optimization strategies
for high-dimensional design problems[C]//12th AIAA/ISSMO
Multidisciplinary Analysis and Optimization Conference, Victoria,
Canada, September 10–12, 2008: 1–24.
 JONES D, SCHONLAU M, WELCH W. Efficient global optimization of
expensive black-box functions[J]. Journal of Global Optimization, 1998,
 ADEL Y, DONG Z M. Trends, features, and tests of common and recently
introduced global optimization methods[J]. Engineering Optimization,
2010, 42(8): 691–718.
 SHAN S Q, WANG G G. Metamodeling for high dimensional
simulation-based design problems[J]. Journal of Mechanical Design, 2010,
 SACKS J, SCHILLER S B, WELCH W J. Designs and analysis of computer
experiments[J]. Statistical Science, 1989, 4(4): 409–435.
 FANG Kaitai, MA Changxing, WINKER P. Centered L2-discrepancy of
random sampling and Latin hypercube design and construction of uniform
designs[J]. Mathematics of Computation, 2002, 71(237): 275–296.
 MORRIS M D, MITCHELL T J. Exploratory designs for computer
experiments[J]. Journal of Statistical Planning and Inference, 1995,
 EDWIN R D, BART H. Maximin Latin hypercube designs in two
dimensions[J]. Operations Research, 2005, 55(1): 158–169.
 XIONG Fenfen, XIONG Y, CHEN W, et al. Optimizing Latin hypercube
design for sequential sampling of computer experiments[J]. Engineering
Optimization, 2009, 41(8): 793–810.
 MYERS R H. Response surface methodology-Current status and future
directions[J]. Journal of Quality Technology, 1999, 31(1): 30–44.
 MACKMAN T J, ALLEN C B. Multidimensional adaptive sampling for
global metamodeling[C]//48th AIAA Aerospace Sciences Meeting Including
the New Horizons Forum and Aerospace Exposition, Orlando, USA, January
4–7, 2010: 1–12.
 HAFTKA R T. Combining global and local approximations[J]. AIAA
Journal, 1991, 29(9): 1 523–1 525.
 HUTCHISON M G, UNGER E, MASON W H, et al. Variable-complexity
aerodynamic optimization of a high speed civil transport wing[J].
Journal of Aircraft, 1994, 31(1): 110–116.
 WANG G G, DONG Z M, PETER A. Adaptive response surface method—A
global optimization scheme for approximation-based design problems[J].
Engineering Optimization, 2001, 33(6): 707–733.
 WANG G G. Adaptive response surface method using inherited Latin
hypercube design points[J]. Journal of Mechanical Design, 2003, 125(6):
 WANG G G, SIMPSON T W. Fuzzy clustering based hierarchical
metamodeling for space reduction and design optimization[J]. Engineering
Optimization, 2004, 36(3): 313–335.
 DAM E R, HUSSLAGE B, HERTOG D, et al. Maximin Latin hypercube
design in two dimensions[J]. Operations Research, 2007, 55(1): 158–169.
 BUHMANN M D. Radial basis functions: Theory and implementations[M].
Cambridge: United Kingdom at the University Press, 2003.
 KRISHNAMURTHY T. Comparison of response surface construction
methods for derivative estimation using moving least squares, Kriging
and radial basis functions[C]//46th AIAA/ASME/ ASCE/AHS/ASC Structures,
Structural Dynamics & Materials Conference, Austin, USA, April 18-21,
 BEZDEK J C. Patten recognition with fuzzy objective function
algorithms[M]. New York: Plenum Press, 1981.