The recently developed metamodel-based decomposition strategy relies on quantifying the variable correlations of black-box functions so that high-dimensional problems are decomposed to smaller subproblems, before performing optimization. Such a two-step method may miss the global optimum due to its rigidity or requires extra expensive sample points for ensuring adequate decomposition. This work develops a strategy to iteratively decompose high-dimensional problems within the optimization process. The sample points used during the optimization are reused to build a metamodel called principal component analysis-high dimensional model representation (PCA-HDMR) for quantifying the intensities of variable correlations by sensitivity analysis. At every iteration, the predicted intensities of the correlations are updated based on all the evaluated points, and a new decomposition scheme is suggested by omitting the weak correlations. Optimization is performed on the iteratively updated subproblems from decomposition. The proposed strategy is applied for optimization of different benchmarks and engineering problems, and results are compared to direct optimization of the undecomposed problems using trust region mode pursuing sampling method (TRMPS), genetic algorithm (GA), cooperative coevolutionary algorithm with correlation-based adaptive variable partitioning (CCEA-AVP), and divide rectangles (DIRECT). The results show that except for the category of undecomposable problems with all or many strong (i.e., important) correlations, the proposed strategy effectively improves the accuracy of the optimization results. The advantages of the new strategy in comparison with the previous methods are also discussed.

References

References
1.
Shan
,
S.
, and
Wang
,
G. G.
,
2010
, “
Survey of Modeling and Optimization Strategies to Solve High-Dimensional Design Problems With Computationally-Expensive Black-Box Functions
,”
Struct. Multidiscip. Optim.
,
41
(
2
), pp.
219
241
.
2.
Shan
,
S.
, and
Wang
,
G. G.
,
2010
, “
Metamodeling for High Dimensional Simulation-Based Design Problems
,”
ASME J. Mech. Des.
,
132
(5), p.
051009
.
3.
Haftka
,
R. T.
,
Scott
,
E. P.
, and
Cruz
,
J. R.
,
1998
, “
Optimization and Experiments: A Survey
,”
ASME Appl. Mech. Rev.
,
51
(
7
), pp.
435
448
.
4.
Martins
,
J. R. R. A.
,
Sturdza
,
P.
, and
Alonso
,
J. J.
,
2003
, “
The Complex-Step Derivative Approximation
,”
ACM Trans. Math. Software
,
29
(
3
), pp.
245
262
.
5.
Freund
,
J. B.
,
2010
, “
Adjoint-Based Optimization for Understanding and Suppressing Jet Noise
,”
Procedia Eng.
,
6
, pp.
54
63
.
6.
Mader
,
C. A.
,
Martins
,
J. R. R. A.
,
Alonso
,
J. J.
, and
Van Der Weide
,
E.
,
2008
, “
ADjoint: An Approach for the Rapid Development of Discrete Adjoint Solvers
,”
AIAA J.
,
46
(
4
), pp.
863
873
.
7.
Bates
,
R. A.
,
Buck
,
R. J.
,
Riccomagno
,
E.
, and
Wynn
,
H. P.
,
1996
, “
Experimental Design and Observation for Large Systems
,”
J. R. Stat. Soc. Ser. B
,
58
(
1
), pp.
77
94
.
8.
Srivastava
,
A.
,
Hacker
,
K.
,
Lewis
,
K.
, and
Simpson
,
T. W.
,
2004
, “
A Method for Using Legacy Data for Metamodel-Based Design of Large-Scale Systems
,”
Struct. Multidiscip. Optim.
,
28
(
2–3
), pp.
146
155
.
9.
Koch
,
P. N.
,
Simpson
,
T. W.
,
Allen
,
J. K.
, and
Mistree
,
F.
,
1999
, “
Statistical Approximations for Multidisciplinary Design Optimization: The Problem of Size
,”
Struct. Multidiscip. Optim.
,
36
(1), pp.
275
286
.
10.
Myers
,
R. H.
,
Montgomery
,
D. C.
, and
Anderson-Cook
,
C. M.
,
2009
,
Response Surface Methodology: Process and Product Optimization Using Designed Experiments
,
Wiley
,
New York
.
11.
Ding
,
C.
,
He
,
X.
,
Zha
,
H.
, and
Simon
,
H. D.
,
2002
, “
Adaptive Dimension Reduction for Clustering High Dimensional Data
,” International
IEEE
Conference on Data Mining (ICDM)
,
Maebashi City, Japan
, pp.
147
154
.
12.
Kaya
,
H.
,
Kaplan
,
M.
, and
Saygın
,
H.
,
2004
, “
A Recursive Algorithm for Finding HDMR Terms for Sensitivity Analysis
,”
Comput. Phys. Commun.
,
158
(
2
), pp.
106
112
.
13.
Wang
,
G. G.
,
Dong
,
Z.
, and
Aitchison
,
P.
,
2001
, “
Adaptive Response Surface Method—A Global Optimization Scheme for Approximation-Based Design Problems
,”
Eng. Optim.
,
33
(
6
), pp.
707
734
.
14.
Wang
,
G. G.
, and
Simpson
,
T.
,
2004
, “
Fuzzy Clustering Based Hierarchical Metamodeling for Design Space Reduction and Optimization
,”
Eng. Optim.
,
36
(
3
), pp.
313
335
.
15.
Winer
,
E. H.
, and
Bloebaum
,
C. L.
,
2002
, “
Development of Visual Design Steering as an Aid in Large-Scale Multidisciplinary Design Optimization. Part I: Method Development
,”
Struct. Multidiscip. Optim.
,
23
(
6
), pp.
412
424
.
16.
Winer
,
E. H.
, and
Bloebaum
,
C. L.
,
2002
, “
Development of Visual Design Steering as an Aid in Large-Scale Multidisciplinary Design Optimization. Part II: Method Validation
,”
Struct. Multidiscip. Optim.
,
23
(
6
), pp.
425
435
.
17.
Bloebaum
,
C.
,
Hajela
,
P.
, and
Sobieszczanski-Sobieski
,
J.
,
1992
, “
Non-Hierarchic System Decomposition in Structural Optimization
,”
Eng. Optim.
,
19
(
3
), pp.
171
186
.
18.
Kim
,
H. M.
,
Michelena
,
N. F.
,
Papalambros
,
P. Y.
, and
Jiang
,
T.
,
2003
, “
Target Cascading in Optimal System Design
,”
ASME J. Mech. Des.
,
125
(
3
), pp.
474
480
.
19.
Michelena
,
N.
,
Papalambros
,
P.
,
Park
,
H.
, and
Kulkarni
,
D.
,
1999
, “
Hierarchical Overlapping Coordination for Large-Scale Optimization by Decomposition
,”
AIAA J.
,
37
(
7
), pp.
890
896
.
20.
Allison
,
J. T.
,
Kokkolaras
,
M.
, and
Papalambros
,
P. Y.
,
2009
, “
Optimal Partitioning and Coordination Decisions in Decomposition-Based Design Optimization
,”
ASME J. Mech. Des.
,
131
(
8
), p.
081008
.
21.
Alexander
,
M. J.
,
Allison
,
J. T.
, and
Papalambros
,
P. Y.
,
2011
, “
Reduced Representations of Vector-Valued Coupling Variables in Decomposition-Based Design Optimization
,”
Struct. Multidiscip. Optim.
,
44
(
3
), pp.
379
391
.
22.
Liu
,
Y.
,
Yao
,
X.
,
Zhao
,
Q.
, and
Higuchi
,
T.
,
2001
, “
Scaling Up Fast Evolutionary Programming With Cooperative Coevolution
,” 2001
Congress on Evolutionary Computation
,
Seoul, Korea
, May 27–May30, pp.
1101
1108
.
23.
Potter
,
M.
, and
De Jong
,
K.
,
1994
, “
A Cooperative Coevolutionary Approach to Function Optimization
,”
Proceedings of the International Conference on Evolutionary Computation. The Third Conference on Parallel Problem Solving from Nature: Parallel Problem Solving from Nature
(
PPSN III
),
Y
.
Davidor
,
H.-P
.
Schwefel
, and
R
.
Mãnner
, eds.,
Springer-Verlag
,
London
, pp. 249–257.
24.
Shi
,
Y.
,
Teng
,
H.
, and
Li
,
Z.
,
2005
, “
Cooperative Co-Evolutionary Differential Evolution for Function Optimization
,” Advances in Natural Computation
SE-147
,
L
.
Wang
,
K
.
Chen
, and
Y
.
Ong
, eds., Springer, Berlin, Heidelberg, pp.
1080
1088
.
25.
Yang
,
Z.
,
Tang
,
K.
, and
Yao
,
X.
,
2008
, “
Large Scale Evolutionary Optimization Using Cooperative Coevolution
,”
Inf. Sci.
,
178
(
15
), pp.
2985
2999
.
26.
Li
,
X.
, and
Yao
,
X.
,
2012
, “
Cooperatively Coevolving Particle Swarms for Large Scale Optimization
,”
IEEE Trans. Evol. Comput.
,
16
(
2
), pp.
210
224
.
27.
Omidvar
,
M. N.
,
Li
,
X.
, and
Yao
,
X.
,
2011
, “
Smart Use of Computational Resources Based on Contribution for Cooperative Co-Evolutionary Algorithms
,”
13th Annual Conference on Genetic and Evolutionary Computation
,
Dublin, Ireland
, pp.
1115
1122
.
28.
Yang
,
Z.
,
Tang
,
K.
, and
Yao
,
X.
,
2008
, “
Multilevel Cooperative Coevolution for Large Scale Optimization
,”
IEEE
Congress on Evolutionary Computation
,
Hong Kong
, June 1–6, pp.
1305
1312
.
29.
Omidvar
,
M. N.
,
Mei
,
Y.
, and
Li
,
X.
,
2014
, “
Effective Decomposition of Large-Scale Separable Continuous Functions for Cooperative Co-Evolutionary Algorithms
,”
2014 IEEE Congress on Evolutionary Computation
(
CEC
),
Beijing
, July 6–11, pp.
1305
1312
.
30.
Omidvar
,
M. N.
,
Li
,
X.
, and
Mei
,
Y.
,
2014
, “
Cooperative Co-Evolution With Differential Grouping for Large Scale Optimization
,”
IEEE Trans. Evol. Comput.
,
18
(
3
), pp.
378
393
.
31.
Singh
,
H.
, and
Ray
,
T.
,
2010
, “
Divide and Conquer in Coevolution: A Difficult Balancing Act
,”
Agent-Based Evolutionary Search SE-6
,
R.
Sarker
and
T.
Ray
, eds.,
Springer
,
Berlin
, pp.
117
138
.
32.
Mahdavi
,
S.
,
Shiri
,
M. E.
, and
Rahnamayan
,
S.
,
2014
, “
Cooperative Co-Evolution With a New Decomposition Method for Large-Scale
,”
IEEE
World Congress on Computational Intelligence
,
Beijing
, July 6–11, pp. 1285–1292.
33.
Pirmoradi
,
Z.
,
Haji Hajikolaei
,
K.
, and
Wang
,
G. G.
,
2012
, “
Design Optimization on ‘White-Box' Uncovered by Metamodeling
,”
AIAA
Paper No. 2012-1927.
34.
Hajikolaei
,
K. H.
,
Pirmoradi
,
Z.
,
Cheng
,
G. H.
, and
Wang
,
G. G.
,
2014
, “
Decomposition for Large Scale Global Optimization Based on Quantified Variable Correlations Uncovered by Metamodeling
,”
Eng. Optim.
,
47
(
4
), pp.
429
452
.
35.
Sobol
,
I.
,
1993
, “
Sensitivity Estimates for Nonlinear Mathematical Models
,”
Math. Model. Comput. Exp.
,
1
(
4
), pp.
407
414
.
36.
Rabitz
,
H.
, and
Alis
,
O. F.
,
1999
, “
General Foundations of High-Dimensional Model Representations
,”
J. Math. Chem.
,
25
, pp.
197
233
.
37.
Li
,
G.
,
Rosenthal
,
C.
, and
Rabitz
,
H.
,
2001
, “
High Dimensional Model Representations
,”
J. Phys. Chem.
,
105
(
33
), pp.
7765
7777
.
38.
Alı
,
Ö. F.
, and
Rabitz
,
H.
,
2001
, “
Efficient Implementation of High Dimensional Model Representations
,”
J. Math. Chem.
,
29
(
2
), pp.
127
142
.
39.
Hajikolaei
,
K. H.
, and
Wang
,
G. G.
,
2013
, “
High Dimensional Model Representation With Principal Component Analysis
,”
ASME J. Mech. Des.
,
136
(
1
), p.
011003
.
40.
Fu
,
J.
, and
Wang
,
L.
,
2002
, “
A Random-Discretization Based Monte Carlo Sampling Method and Its Applications
,”
Methodol. Comput. Appl. Probab.
,
4
(
1
), pp.
5
25
.
41.
Hock
,
W.
, and
Schittkowski
,
K.
,
1980
, “
Test Examples for Nonlinear Programming Codes
,”
J. Optim. Theory Appl.
,
30
(
1
), pp.
127
129
.
42.
Schittkowski
,
K.
,
1987
,
More Test Examples for Nonlinear Programming Codes
,
Springer-Verlag
,
New York
.
43.
Cheng
,
G. H.
,
Younis
,
A.
,
Hajikolaei
,
K. H.
, and
Wang
,
G. G.
,
2015
, “
Trust Region Based MPS Method for Global Optimization of High Dimensional Design Problems
,”
ASME J. Mech. Des.
,
137
(
2
), p.
021407
.
44.
Dimensional Control Systems, Inc.
,” Last accessed 15 Sept.,
2013
, http://www.3dcs.com/
You do not currently have access to this content.