Research Papers: Design Automation

An Adaptive Aggregation-Based Approach for Expensively Constrained Black-Box Optimization Problems

[+] Author and Article Information
George H. Cheng

Product Design and Optimization Laboratory
Simon Fraser University,
Surrey, BC V3T 0A3, Canada
e-mail: ghc2@sfu.ca

Timothy Gjernes

Hevvy/Toyo Pumps North America Corporation,
Coquitlam, BC V3K 7C1, Canada

G. Gary Wang

Product Design and Optimization Laboratory
Simon Fraser University,
Surrey, BC V3T 0A3, Canada
e-mail: gwa5@sfu.ca

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received November 23, 2017; final manuscript received May 29, 2018; published online June 26, 2018. Editor: Wei Chen.

J. Mech. Des 140(9), 091402 (Jun 26, 2018) (14 pages) Paper No: MD-17-1787; doi: 10.1115/1.4040485 History: Received November 23, 2017; Revised May 29, 2018

Expensive constraints are commonly seen in real-world engineering design. However, metamodel based design optimization (MBDO) approaches often assume inexpensive constraints. In this work, the situational adaptive Kreisselmeier and Steinhauser (SAKS) method was employed in the development of a hybrid adaptive aggregation-based constraint handling strategy for expensive black-box constraint functions. The SAKS method is a novel approach that hybridizes the modeling and aggregation of expensive constraints and adds an adaptive strategy to control the level of hybridization. The SAKS strategy was integrated with a modified trust region-based mode pursuing sampling (TRMPS) algorithm to form the SAKS-trust region optimizer (SAKS-TRO) for single-objective design optimization problems with expensive black-box objective and constraint functions. SAKS-TRO was benchmarked against five popular constrained optimizers and demonstrated superior performance on average. SAKS-TRO was also applied to optimize the design of an industrial recessed impeller.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Gu, L. , 2001, “ A Comparison of Polynomial Based Regression Models in Vehicle Safety Analysis,” ASME Paper No. DAC-21063.
Duddeck, F. , 2008, “ Multidisciplinary Optimization of Car Bodies,” Struct. Multidiscip. Optim., 35(4), pp. 375–389. [CrossRef]
Mitchell, M. , 1996, An Introduction to Genetic Algorithms, MIT Press, Cambridge, MA.
Bertsimas, D. , and Tsitsiklis, J. , 1993, “ Simulated Annealing,” Stat. Sci., 8(1), pp. 10–15. [CrossRef]
Poli, R. , Kennedy, J. , and Blackwell, T. , 2007, “ Particle Swarm Optimization: An Overview,” Swarm Intell., 1(1), pp. 33–57. [CrossRef]
Jones, D. , Perttunen, C. , and Stuckman, B. , 1993, “ Lipschitzian Optimization Without the Lipschitz Constant,” J. Optim. Theory Appl., 79(1), pp. 157–181. [CrossRef]
Wang, L. , Shan, S. , and Wang, G. , 2004, “ Mode-Pursuing Sampling Method for Global Optimization on Expensive Black-Box Functions,” J. Eng. Optim., 36(4), pp. 419–438. [CrossRef]
Cheng, G. H. , Younis, A. , Haji Hajikolaei, K. , 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]
Jones, D. R. , Schonlau, M. , and Welch, W. J. , 1998, “ Efficient Global Optimization of Expensive Black-Box Functions,” J. Global Optim., 13(4), pp. 455–492. [CrossRef]
Wang, G. G. , and Shan, S. , 2006, “ Review of Metamodeling Techniques in Support of Engineering Design Optimization,” ASME J. Mech. Des., 129(4), pp. 370–380. [CrossRef]
Kazemi, M. , Wang, G. G. , Rahnamayan, S. , and Gupta, K. , 2011, “ Global Optimization for Problems With Expensive Objective and Constraint Functions,” ASME J. Mech. Des., 133(1) , p. 014505.
Homaifar, A. , Qi, C. , and Lai, S. , 1994, “ Constrained Optimization Via Genetic Algorithms,” Simulation, 62(4), pp. 242–254. [CrossRef]
Joines, J. , and Houck, C. , 1994, “ On the Use of Non-Stationary Penalty Functions to Solve Nonlinear Constrained Optimization Problems With GAs,” First IEEE Conference on Evolutionary Computation, Orlando, FL, June 27–29, pp. 579–584.
Michalewicz, Z. , and Attia, N. , 1994, “ Evolutionary Optimization of Constrained Problems,” Third Annual Conference on Evolutionary Programming, San Diego, CA, Feb. 24–26.
Hadj-Alouane, A. , and Bean, J. , 1997, “ A Genetic Algorithm for the Multiple-Choice Integer Program,” Oper. Res., 45(1), pp. 92–101. [CrossRef]
Tessema, B. , and Yen, G. , 2006, “ A Self Adaptive Penalty Function Based Algorithm for Constrained Optimization,” IEEE Congress on Evolutionary Computation, Vancouver, BC, Canada, July 16–21, pp. 246–253.
Coello, C. , 2000, “ Use of a Self-Adaptive Penalty Approach for Engineering Optimization Problems,” Comput. Ind., 41(2), pp. 113–127. [CrossRef]
Holmström, K. , Quttineh, N. , and Edvall, M. , 2008, “ An Adaptive Radial Basis Algorithm (ARBF) for Expensive Black-Box Mixed-Integer Constrained Global Optimization,” Optim. Eng., 9(4), pp. 311–339. [CrossRef]
Riche, R. , and Haftka, R. , 1995, “ Improved Genetic Algorithm for Minimum Thickness Composite Laminate Design,” Compos. Eng., 5(2), pp. 143–161. [CrossRef]
Ullah, A. , Sarker, R. , and Lokan, C. , 2012, “ Handling Equality Constraints in Evolutionary Optimization,” Eur. J. Oper. Res., 221(3), pp. 480–490. [CrossRef]
Paredis, J. , 1994, “ Co-Evolutionary Constraint Satisfaction,” Third Conference on Parallel Problem Solving From Nature, Jerusalem, Israel, Oct. 9–14.
Parmee, I. , and Purchase, G. , 1994, “ The Development of a Directed Genetic Search Technique for Heavily Constrained Design Spaces,” Adaptive Computing in Engineering Design and Control '94, Plymouth, UK, pp. 97–102.
Camponogara, E. , and Talukdar, S. , 1997, “ A Genetic Algorithm for Constrained and Multiobjective Optimization,” Third Nordic Workshop on Genetic Algorithms and Their Applications (3NWGA), Vaasa, Finland, pp. 49–62.
Parr, J. , Keane, A. , Forrester, A. , and Holden, C. , 2012, “ Infill Sampling Criteria for Surrogate-Based Optimization With Constraint Handling,” Eng. Optim., 44(10), pp. 1147–1166. [CrossRef]
Poon, N. , and Martins, J. , 2007, “ An Adaptive Approach to Constraint Aggregation Using Adjoint Sensitivity Analysis,” J. Struct. Multidiscip. Optim., 34(1), pp. 61–73. [CrossRef]
Wrenn, G. , 1989, “ An Indirect Method for Numerical Optimization Using the Kreisselmeier–Steinhauser Function,” NASA Langley Research Center, Hampton, VA, Contractor Report No. 4220. https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19890007408.pdf
Bloss, K. F. , Biegler, L. T. , and Schiesser, W. E. , 1999, “ Dynamic Process Optimization Through Adjoint Formulations and Constraint Aggregation,” Ind. Eng. Chem. Res., 38(2), pp. 421–432. [CrossRef]
Jiang, Z. , Cheng, G. H. , and Wang, G. G. , 2013, “ Mixed Discrete and Continuous Variable Optimization Based on Constraint Aggregation and Relative Sensitivity,” ASME Paper No. DETC2013-12668.
Kleijnen, J. , van Beers, W. , and van Nieuwenhuyse, I. , 2010, “ Constrained Optimization in Expensive Simulation: A Novel Approach,” Eur. J. Oper. Res., 202(1), pp. 164–174. [CrossRef]
Regis, R. , 2011, “ Stochastic Radial Basis Function Algorithms for Large Scale Optimization Involving Expensive Black-Box Objective and Constraint Functions,” J. Comput. Oper. Res., 38(5), pp. 837–853. [CrossRef]
Rashid, K. , Ambani, S. , and Cetinkaya, E. , 2013, “ An Adaptive Multiquadric Radial Basis Function Method for Expensive Black-Box Mixed-Integer Nonlinear Constrained Optimization,” Eng. Optim., 45(2), pp. 185–206. [CrossRef]
Basudhar, A. , Dribusch, C. , Lacaze, S. , and Missoum, S. , 2012, “ Constrained Efficient Global Optimization With Support Vector Machines,” Struct. Multidiscip. Optim., 46(2), pp. 201–221. [CrossRef]
Kramer, O. , 2010, “ A Review of Constraint-Handling Techniques for Evolution Strategies,” Appl. Comput. Intell. Soft Comput., 2010, pp. 1–11. [CrossRef]
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., 191(11–12), pp. 1245–1287. [CrossRef]
Lucken, C. , Baran, B. , and Brizuela, C. , 2014, “ A Survey on Multi-Objective Evolutionary Algorithms for Many-Objective Problems,” Comput. Optim. Appl., 58(3), pp. 707–756.
Kreisselmeier, G. , and Steinhauser, R. , 1979, “ Systematic Control Design by Optimizing a Vector Performance Index,” IFAC Proceedings Volumes, 12(7), pp. 113–117.
Powell, M. , 1987, Radial Basis Functions for Multivariable Interpolation: A Review of Algorithms for Approximation, 3rd ed., Clarendon Press, Oxford, UK.
Jin, R. , Chen, W. , and Simpson, T. , 2001, “ Comparative Studies of Metamodeling Techniques Under Multiple Modeling Criteria,” Struct. Multidiscip. Optim., 23(1), pp. 1–13. [CrossRef]
Chen, V. , Tsui, K. L. , Barton, R. R. , and Meckesheimer, M. , 2006, “ A Review on Design, Modeling and Applications of Computer Experiments,” IIE Trans., 38(4), pp. 273–291. [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]
Raspanti, C. , Bandoni, J. , and Biegler, L. , 2000, “ New Strategies for Flexibility Analysis and Design Under Uncertainty,” J. Comput. Chem. Eng., 24(9–10), pp. 2193–2209. [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]
Vanderplaats, G. , 1973, “ CONMIN—A Fortran Program for Constrained Function Minimization,” NASA Ames Research Center, Moffett Field, CA, Technical Memorandum No. TM X-62282.
Liuzzi, G. , Lucidi, S. , and Sciandrone, M. , 2010, “ Sequential Penalty Derivative-Free Methods for Nonlinear Constrained Optimization,” SIAM J. Optim., 20(5), pp. 2614–2635. [CrossRef]
Powell, M. , 1994, “ A Direct Search Optimization Method That Models the Objective and Constraint Functions by Linear Interpolation,” Advances in Optimization and Numerical Analysis, Springer, Dordrecht, The Netherlands, pp. 51–67. [CrossRef]
Jansen, P. , and Perez, R. , 2011, “ Constrained Structural Design Optimization Via a Parallel Augmented Lagrangian Particle Swarm Optimization Approach,” Int. J. Comput. Struct., 89(13–14), pp. 1352–1366. [CrossRef]
Perez, R. , Jansen, P. , and Martins, J. , 2012, “ pyOpt: A Python-Based Object-Oriented Framework for Nonlinear Constrained Optimization,” Struct. Multidiscip. Optim., 45(1), pp. 101–118. [CrossRef]
Gjernes, T. , 2014, “ Optimization of Centrifugal Slurry Pumps Through Computational Fluid Dynamics,” Masters thesis, Simon Fraser University, Burnaby, BC, Canada. http://summit.sfu.ca/item/14468
Grzina, A. , Roudnev, A. , and Burgess, K. E. , 2002, Slurry Pumping Manual, Warman International, Glasgow, Scotland.
Shah, S. , Jain, S. , Patel, R. , and Lakhera, V. , 2012, “ CFD for Centrifugal Pumps: A Review of the State-of-the-Art,” Chemical, Civil, and Mechanical Engineering Tracks of Third Nirma University International Conference, Ahmedabad, India, Dec. 6--8, pp. 715–720. http://www.nuicone.org/nuicone/pdf/Nuicone2012.pdf
Roudnev, A. , 1999, “ Some Aspects of Slurry Pump Design,” World Pumps, 1999(389), pp. 58–61. [CrossRef]
Wilde, D. , 1978, Globally Optimal Design, Wiley, New York.
Arora, J. , 2004, Introduction to Optimum Design, Elsevier Academic Press, San Diego, CA.
Hock, W. , and Schittkowski, K. , 1981, Test Examples for Nonlinear Programming Codes, Springer-Verlag, Secaucus, NJ. [CrossRef]
Hsu, Y. , Wang, S. , and Yu, C. , 2003, “ A Sequential Approximation Method Using Neural Networks for Engineering Design Optimization Problems,” Eng. Optim., 35(5), pp. 489–511. [CrossRef]
Thanedar, P. , 1995, “ Survey of Discrete Variable Optimization for Structural Design,” J. Struct. Eng., 121(2), p. 301. [CrossRef]
Hajikolaei, K. H. , Pirmoradi, Z. , Cheng, G. H. , and Wang, G. G. , 2014, “ Decomposition Based on Quantified Variable Correlations Uncovered by Metamodeling for Large Scale Global Optimization,” Engineering Optimization, 47(4), pp. 429–452.
Schutte, J. , and Haftka, R. , 2010, “ Global Structural Optimization of a Stepped Cantilever Beam Using Quasi-Separable Decomposition,” Eng. Optim., 42(4), pp. 347–367. [CrossRef]
Sagar, A. , and Gadhvi, B. , 2017, Constrained Nonlinear Optimization Problems: Formulation and Solution, Simon Fraser University, Surrey, BC, Canada.


Grahic Jump Location
Fig. 1

Kreisselmeier and Steinhauser function of two inequality constraints for increasing ρ

Grahic Jump Location
Fig. 2

Constraint classification for the P116 (left), P118 (middle), and beam (right) problems (filled = independent, blank = aggregated)

Grahic Jump Location
Fig. 3

Comparison of aggregation level across iterations for P106 given different n values

Grahic Jump Location
Fig. 4

Comparison of aggregation level across iterations for P118 given different n values

Grahic Jump Location
Fig. 5

Situational adaptive Kreisselmeier and Steinhauser-trust region optimizer flowchart

Grahic Jump Location
Fig. 6

Fluid domain geometry (left) and single vane mesh (right)

Grahic Jump Location
Fig. 7

Head drop comparison between KS-TRO optimum and baseline

Grahic Jump Location
Fig. 8

Efficiency comparison between KS-TRO optimum and baseline

Grahic Jump Location
Fig. 9

Constraint classification for impeller optimization (blue = independent, white = aggregated)



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In