Current metamodel-based design optimization methods rarely deal with problems of not only expensive objective functions but also expensive constraints. In this work, we propose a novel metamodel-based optimization method, which aims directly at reducing the number of evaluations for both objective function and constraints. The proposed method builds on existing mode pursuing sampling method and incorporates two intriguing strategies: (1) generating more sample points in the neighborhood of the promising regions, and (2) biasing the generation of sample points toward feasible regions determined by the constraints. The former is attained by a discriminative sampling strategy, which systematically generates more sample points in the neighborhood of the promising regions while statistically covering the entire space, and the latter is fulfilled by utilizing the information adaptively obtained about the constraints. As verified through a number of test benchmarks and design problems, the above two coupled strategies result in significantly low number of objective function evaluations and constraint checks and demonstrate superior performance compared with similar methods in the literature. To the best of our knowledge, this is the first metamodel-based global optimization method, which directly aims at reducing the number of evaluations for both objective function and constraints.

1.
Simpson
,
T.
,
Peplinski
,
J.
,
Koch
,
P.
, and
Allen
,
J.
, 2001, “
Metamodels for Computer-Based Engineering Design: Survey and Recommendations
,”
Eng. Comput.
0177-0667,
17
(
2
), pp.
129
150
.
2.
Wang
,
G.
, and
Shan
,
S.
, 2007, “
Review of Metamodeling Techniques in Support of Engineering Design Optimization
,”
ASME J. Mech. Des.
0161-8458,
129
(
4
), pp.
370
380
.
3.
Schonlau
,
M.
,
Welch
,
W. J.
, and
Jones
,
D. R.
, 1998, “
Global Versus Local Search in Constrained Optimization of Computer Models
,”
Lecture Notes—Monograph Series
,
34
, pp.
11
25
.
4.
Regis
,
R. G.
, and
Shoemaker
,
C. A.
, 2005, “
Constrained Global Optimization of Expensive Black Box Functions Using Radial Basis Functions
,”
J. Global Optim.
0925-5001,
31
(
1
), pp.
153
171
.
5.
Sasena
,
M. J.
,
Papalambros
,
P.
, and
Goovaerts
,
P.
, 2002, “
Exploration of Metamodeling Sampling Criteria for Constrained Global Optimization
,”
Eng. Optimiz.
0305-215X,
34
, pp.
263
278
.
6.
Yannou
,
B.
,
Simpson
,
T. W.
, and
Barton
,
R.
, 2005, “
Towards a Conceptual Design Explorer Using Metamodeling Approaches and Constraint Programming
,”
ASME
Paper No. DETC2003/DAC-48766.
7.
Yannou
,
B.
,
Moreno
,
F.
,
Thevenot
,
H.
, and
Simpson
,
T.
, 2005, “
Faster Generation of Feasible Design Points
,”
ASME
Paper No. DETC2005/DAC-85449.
8.
Moghaddam
,
R.
,
Wang
,
G.
,
Yannou
,
B.
, and
Wu
,
C.
, 2006, “
Applying Constraint Programming for Design Space Reduction in Metamodeling Based Optimization
,”
16th International Institution for Production Engineering Research (CIRP) International Design Seminar
, Paper No. 10081.
9.
Yannou
,
B.
, and
Harmel
,
G.
, 2006, “
Use of Constraint Programming for Design
,”
Advances in Design
,
Springer
,
London, UK
, pp.
145
157
.
10.
Wang
,
G.
, 2003, “
Adaptive Response Surface Method Using Inherited Latin Hypercube Design Points
,”
ASME J. Mech. Des.
0161-8458,
125
(
2
), pp.
210
220
.
11.
Arora
,
J.
, 2004,
Introduction to Optimum Design
,
Elsevier Academic
,
New York
.
12.
Coello
,
C.
, 2002, “
Theoretical and Numerical Constraint-Handling Techniques Used With Evolutionary Algorithms: A Survey of the State of the Art
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
191
(
11–12
), pp.
1245
1287
.
13.
2009,
Constraint-Handling in Evolutionary Optimization
,
E.
Mezura-Montes
, ed.,
Springer
,
New York
.
14.
Wang
,
L.
,
Shan
,
S.
, and
Wang
,
G.
, 2004, “
Mode-Pursuing Sampling Method for Global Optimization of Expensive Black-Box Functions
,”
Eng. Optimiz.
0305-215X,
36
(
4
), pp.
419
438
.
15.
Sharif
,
B.
,
Wang
,
G.
, and
ElMekkawy
,
T.
, 2008, “
Mode Pursuing Sampling Method for Discrete Variable Optimization on Expensive Black-Box Functions
,”
ASME J. Mech. Des.
0161-8458,
130
(
2
), p.
021402
.
16.
Duan
,
X.
,
Wang
,
G.
,
Kang
,
X.
,
Niu
,
Q.
,
Naterer
,
G.
, and
Peng
,
Q.
, 2009, “
Performance Study of Mode-Pursuing Sampling Method
,”
Eng. Optimiz.
0305-215X,
41
(
1
), pp.
1
21
.
17.
Kazemi
,
M.
,
Wang
,
G. G.
,
Rahnamayan
,
S.
, and
Gupta
,
K.
, 2010, “
Constraint Importance Mode Pursuing Sampling for Continuous Global Optimization
,”
ASME
Paper No. DETC2010-28355.
18.
Fu
,
J.
, and
Wang
,
L.
, 2002, “
A Random-Discretization Based Monte Carlo Sampling Method and Its Applications
,”
Methodol. Comput. Appl. Probab.
1387-5841,
4
(
1
), pp.
5
25
.
19.
Runarsson
,
T. P.
, and
Yao
,
X.
, 2000, “
Stochastic Ranking for Constrained Evolutionary Optimization
,”
IEEE Trans. Evol. Comput.
1089-778X,
4
(
3
), pp.
284
294
.
20.
Floudas
,
C. A.
, and
Pardalos
,
P. M.
, 1990,
A Collection of Test Problems for Constrained Global Optimization Algorithms
,
Springer-Verlag
,
New York
.
21.
Himmelblau
,
D. M.
, 1972,
Applied Nonlinear Programming
,
McGraw-Hill
,
New York
.
22.
Hock
,
W.
, and
Schittkowski
,
K.
, 1981,
Test Examples for Nonlinear Programming Codes
,
Springer-Verlag
,
Secaucus, NJ
.
23.
Koziel
,
S.
, and
Michalewicz
,
Z.
, 1999, “
Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization
,”
Evol. Comput.
1063-6560,
7
(
1
), pp.
19
44
.
24.
Belegundu
,
A. D.
, and
Arora
,
J. S.
, 1985, “
A Study of Mathematical Programming Methods for Structural Optimization. Part II: Numerical Results
,”
Int. J. Numer. Methods Eng.
0029-5981,
21
(
9
), pp.
1601
1623
.
25.
Mahdavi
,
M.
,
Fesanghary
,
M.
, and
Damangir
,
E.
, 2007, “
An Improved Harmony Search Algorithm for Solving Optimization Problems
,”
Appl. Math. Comput.
0096-3003,
188
(
2
), pp.
1567
1579
.
26.
Coello
,
C. A. C.
, and
Mezura-Montes
,
E.
, 2002, “
Constraint-Handling in Genetic Algorithms Through the Use of Dominance-Based Tournament Selection
,”
Adv. Eng. Inf.
1474-0346,
16
(
3
), pp.
193
203
.
27.
Geem
,
Z.
,
Kim
,
J.
, and
Loganathan
,
G.
, 2001, “
A New Heuristic Optimization Algorithm: Harmony Search
,”
Simulation
0037-5497,
76
(
2
), pp.
60
68
.
28.
Powell
,
M.
, 1978, “
Algorithms for Nonlinear Constraints That Use Lagrangian Functions
,”
Math. Program.
0025-5610,
14
(
1
), pp.
224
248
.
29.
Wu
,
S.
, and
Chow
,
P.
, 1995, “
Genetic Algorithms for Nonlinear Mixed Discrete-Integer Optimization Problems via Meta-Genetic Parameter Optimization
,”
Eng. Optimiz.
0305-215X,
24
(
2
), pp.
137
159
.
30.
Lee
,
K.
, and
Geem
,
Z.
, 2005, “
A New Meta-Heuristic Algorithm for Continues Engineering Optimization: Harmony Search Theory and Practice
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
194
(
36–38
), pp.
3902
3933
.
31.
Sandgren
,
E.
, 1990, “
Nonlinear Integer and Discrete Programming in Mechanical Design Optimization
,”
ASME J. Mech. Des.
0161-8458,
112
(
2
), pp.
223
229
.
You do not currently have access to this content.