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Technical Briefs

Cutting Plane Methods for Analytical Target Cascading With Augmented Lagrangian Coordination

[+] Author and Article Information
Wenshan Wang

Research Assistant
Department of Mechanical Engineering,
Clemson University,
Clemson, SC 29634
e-mail: wenshaw@clemson.edu

Vincent Y. Blouin

Assistant Professor
School of Materials Science and Engineering,
Clemson University,
Clemson, SC 29634
e-mail: vblouin@clemson.edu

Melissa K. Gardenghi

Associate Professor
Division of Mathematical Sciences,
Bob Jones University,
Greenville, SC 29614
e-mail: mgardeng@bju.edu

Georges M. Fadel

Professor
Department of Mechanical Engineering,
Clemson University,
Clemson, SC 29634
e-mail: fgeorge@clemson.edu

Margaret M. Wiecek

Professor
e-mail: wmalgor@clemson.edu

Benjamin C. Sloop

Research Assistant
e-mail: bsloop@clemson.edu
Department of Mathematical Sciences,
Clemson University,
Clemson, SC 29634

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the Journal of Mechanical Design. Manuscript received November 17, 2011; final manuscript received May 14, 2013; published online August 7, 2013. Assoc. Editor: Wei Chen.

J. Mech. Des 135(10), 104502 (Aug 07, 2013) (6 pages) Paper No: MD-11-1471; doi: 10.1115/1.4024847 History: Received November 17, 2011; Revised May 14, 2013

Analytical target cascading (ATC), a hierarchical, multilevel, multidisciplinary coordination method, has proven to be an effective decomposition approach for large-scale engineering optimization problems. In recent years, augmented Lagrangian relaxation methods have received renewed interest as dual update methods for solving ATC decomposed problems. These problems can be solved using the subgradient optimization algorithm, the application of which includes three schemes for updating dual variables. To address the convergence efficiency disadvantages of the existing dual update schemes, this paper investigates two new schemes, the linear and the proximal cutting plane methods, which are implemented in conjunction with augmented Lagrangian coordination for ATC-decomposed problems. Three nonconvex nonlinear example problems are used to show that these two cutting plane methods can significantly reduce the number of iterations and the number of function evaluations when compared to the traditional subgradient update methods. In addition, these methods are also compared to the method of multipliers and its variants, showing similar performance.

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References

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]
Kim, H. M., 2001, “Target Cascading in Optimal System Design,” Ph.D. thesis, University of Michigan, Ann Arbor.
Sobieszczanski-Sobieski, J., 1988, “Optimization by Decomposition: A Step From Hierarchic to Non-Hierarchic Systems,” Proceedings of the 2nd NASA Air Force Symposium on Advances in Multidisciplinary Analysis and Optimization.
Braun, R. D., 1996, Collaborative Optimization: an Architecture for Large-Scale Distributed Design,” Ph.D. thesis, Stanford University, Stanford, CA.
Sobieszczanski-Sobieski, J., Agte, J. S., and Sandusky, R. R., Jr., 2000, “Bi-Level Integrated System Synthesis,” AIAA J., 38(1), pp. 164–172. [CrossRef]
Haftka, R. T., and Watson, L. T., 2005, “Multidisciplinary Design Optimization With Quasi-Separable Subsystems,” Optim. Eng., 6(1), pp. 9–20. [CrossRef]
Demiguel, A. V., and Murray, W., 2006, “A Local Convergence Analysis of Bi-Level Decomposition Algorithms,” Optim. Eng., 7(2), pp. 99–133. [CrossRef]
Lassiter, J. B., Wiecek, M. M., and Andrighetti, K. R., 2005, “Lagrangian Coordination and Analytical Target Cascading: Solving ATC-Decomposed Problems With Lagrangian Duality,” Optim. Eng., 6(3), pp. 361–381. [CrossRef]
Li, Y., Lu, Z., and Michalek, J. J., 2008, “Diagonal Quadratic Approximation for Parallelization of Analytical Target Cascading,” ASME J. Mech. Des., 130(5), p. 0514021. [CrossRef]
Bertsekas, D. P., 2003, Nonlinear Programming, 2nd ed., Athena Scientific, Nashua, NH.
Bazaraa, M. S., Sherali, H. D., and Shetty, C. M., 2006, Nonlinear Programming: Theory and Algorithms, 3rd ed., Wiley, Hoboken, NJ.
Blouin, V. Y., Lassiter, J., Wiecek, M. M., and Fadel, G. M., 2005, “Augmented Lagrangian Coordination for Decomposed Design Problems,” Proceedings of the 6th World Congress on Structural and Multidisciplinary Optimization.
Kim, H. M., Chen, W., and Wiecek, M. M., 2006, “Lagrangian Coordination for Enhancing the Convergence of Analytical Target Cascading,” AIAA J., 44(10), pp. 2197–2207. [CrossRef]
Blouin, V. Y., Samuels, H. B., Fadel, G. M., Haque, I. U., and Wagner, J. R., 2004, “Continuously Variable Transmission Design for Optimum Vehicle Performance by Analytical Target Cascading,” Int. J. Heavy Veh. Syst., Spec. Issue Adv. Ground Veh. Simul., 11(2/3), pp. 327–348. [CrossRef]
Michalek, J. J., and Papalambros, P. Y., 2005, “Technical Brief: Weights, Norms, and Notation in Analytical Target Cascading,” ASME J. Mech. Des., 127(2), pp. 499–501. [CrossRef]
Michalek, J. J., and Papalambros, P. Y., 2005, “An Efficient Weighting Update Method to Achieve Acceptable Inconsistency Deviation in Analytical Target Cascading,” ASME J. Mech. Des., 127(3), pp. 206–214. [CrossRef]
Tosserams, S., Etman, L. F. P., Papalambros, P. Y., and Rooda, J. E., 2006, “An Augmented Lagrangian Relaxation for Analytical Target Cascading Using the Alternating Directions Method of Multipliers,” Struct. Multidiscip. Optim., 31(3), pp. 176–189. [CrossRef]
Gasimov, R. N., 2002, “Augmented Lagrangian Duality and Nondifferentiable Optimization Methods in Non-Convex Programming,” J. Global Optim., 24, pp. 187–203. [CrossRef]
Tosserams, S., Etman, L. F. P., and Rooda, J. E., 2007, “An Augmented Lagrangian Decomposition Method for Quasi-Separable Problems in MDO,” Struct. Multidiscip. Optim., 34(3), pp. 211–227. [CrossRef]
Tzevelekos, N., Kokkolaras, M., Papalambros, P. Y., Hulshof, M. F., Etman, L. 7E. P., and RoodaJ. E., 2003, “An Empirical Local Convergence Study of Alternative Coordination Schemes in Analytical Target Cascading,” Proceedings of the 5th World Congress on Structural and Multidisciplinary Optimization, Lido di Jesolo, Venice, Italy.
Golinski, J., 1970, “Optimal Synthesis Problems Solved by Means of Nonlinear Programming and Random Methods,” J. Mech., 5(3), pp. 287–309. [CrossRef]
Padula, S. L., Alexandrov, N., and Green, L. L., 1996, “MDO Test Suite at NASA Langley Research Center,” Technical Report No. 96-4028, Bellevue, Washington, September 4–6.
Allison, J. T., Kokkolaras, M., Zawislak, M., and Papalambros, P. Y., 2005, On the Use of Analytical Target Cascading and Collaborative Optimisation for Complex System Design,” Proceedings of the 6th World Congress on Structural and Multidisciplinary Optimization, Rio de Janeiro, Brazil.
Bertsekas, D. P., 1999, Nonlinear Programming, Athena Scientific, Nashua, NH.
Goffin, J. L., 1977, “On Convergence Rates of Subgradient Optimization Methods,” Math. Program., 13(1), pp. 329–347. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Decomposition structures

Grahic Jump Location
Fig. 2

Example 1: number of iterations and number of function evaluations versus solution error

Grahic Jump Location
Fig. 3

Example 2: number of iterations and number of function evaluations versus solution error

Grahic Jump Location
Fig. 4

Example 3: number of iterations and number of function evaluations versus solution error

Grahic Jump Location
Fig. 5

CPU time versus solution error

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