Research Papers: Design Automation

Optimization on Metamodeling-Supported Iterative Decomposition

[+] Author and Article Information
Kambiz Haji Hajikolaei

Product Design and
Optimization Laboratory (PDOL),
School of Mechatronic Systems Engineering,
Simon Fraser University,
BC, Canada
e-mail: khajihaj@sfu.ca

George H. Cheng

Product Design and
Optimization Laboratory (PDOL),
School of Mechatronic Systems Engineering,
Simon Fraser University,
BC, Canada
e-mail: ghc2@sfu.ca

G. Gary Wang

Product Design and
Optimization Laboratory (PDOL),
School of Mechatronic Systems Engineering,
Simon Fraser University,
BC, Canada
e-mail: gary_wang@sfu.ca

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received December 2, 2014; final manuscript received October 19, 2015; published online December 8, 2015. Assoc. Editor: Christopher Mattson.

J. Mech. Des 138(2), 021401 (Dec 08, 2015) (11 pages) Paper No: MD-14-1767; doi: 10.1115/1.4031982 History: Received December 02, 2014; Revised October 19, 2015

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.

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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. [CrossRef]
Shan, S. , and Wang, G. G. , 2010, “ Metamodeling for High Dimensional Simulation-Based Design Problems,” ASME J. Mech. Des., 132(5), p. 051009. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Freund, J. B. , 2010, “ Adjoint-Based Optimization for Understanding and Suppressing Jet Noise,” Procedia Eng., 6, pp. 54–63. [CrossRef]
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. [CrossRef]
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.
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.
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.
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.
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.
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Bloebaum, C. , Hajela, P. , and Sobieszczanski-Sobieski, J. , 1992, “ Non-Hierarchic System Decomposition in Structural Optimization,” Eng. Optim., 19(3), pp. 171–186. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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.
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.
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.
Yang, Z. , Tang, K. , and Yao, X. , 2008, “ Large Scale Evolutionary Optimization Using Cooperative Coevolution,” Inf. Sci., 178(15), pp. 2985–2999. [CrossRef]
Li, X. , and Yao, X. , 2012, “ Cooperatively Coevolving Particle Swarms for Large Scale Optimization,” IEEE Trans. Evol. Comput., 16(2), pp. 210–224. [CrossRef]
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.
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.
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.
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. [CrossRef]
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.
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.
Pirmoradi, Z. , Haji Hajikolaei, K. , and Wang, G. G. , 2012, “ Design Optimization on ‘White-Box' Uncovered by Metamodeling,” AIAA Paper No. 2012-1927.
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. [CrossRef]
Sobol, I. , 1993, “ Sensitivity Estimates for Nonlinear Mathematical Models,” Math. Model. Comput. Exp., 1(4), pp. 407–414.
Rabitz, H. , and Alis, O. F. , 1999, “ General Foundations of High-Dimensional Model Representations,” J. Math. Chem., 25, pp. 197–233. [CrossRef]
Li, G. , Rosenthal, C. , and Rabitz, H. , 2001, “ High Dimensional Model Representations,” J. Phys. Chem., 105(33), pp. 7765–7777. [CrossRef]
Alı, Ö. F. , and Rabitz, H. , 2001, “ Efficient Implementation of High Dimensional Model Representations,” J. Math. Chem., 29(2), pp. 127–142. [CrossRef]
Hajikolaei, K. H. , and Wang, G. G. , 2013, “ High Dimensional Model Representation With Principal Component Analysis,” ASME J. Mech. Des., 136(1), p. 011003. [CrossRef]
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. [CrossRef]
Hock, W. , and Schittkowski, K. , 1980, “ Test Examples for Nonlinear Programming Codes,” J. Optim. Theory Appl., 30(1), pp. 127–129. [CrossRef]
Schittkowski, K. , 1987, More Test Examples for Nonlinear Programming Codes, Springer-Verlag, New York.
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. [CrossRef]
“Dimensional Control Systems, Inc.,” Last accessed 15 Sept., 2013, http://www.3dcs.com/


Grahic Jump Location
Fig. 2

Decomposed subproblems during OMID procedure for problem #1

Grahic Jump Location
Fig. 3

Decomposed subproblems during OMID procedure for problem #10




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