Comparison of Alternative Strategies for Multilevel Optimization of Hierarchical Systems

Saber DorMohammadi, Masoud Rais-Rohani

Center for Advanced Vehicular Systems, Mississippi State University, Starkville, USA.

DOI: 10.4236/am.2012.330204                

Abstract

The augmented Lagrangian penalty formulation and four different coordination strategies are used to examine the nu- merical behavior of Analytical Target Cascading (ATC) for multilevel optimization of hierarchical systems. The coordination strategies considered include augmented Lagrangian using the method of multipliers and alternating direction method of multipliers, diagonal quadratic approximation, and truncated diagonal quadratic approximation. Properties examined include computational cost and solution accuracy based on the selected values for the different parameters that appear in each formulation. The different strategies are implemented using two- and three-level decomposed example problems. While the results show the interaction between the selected ATC formulation and the values of associated parameters, they clearly highlight the impact they could have on both the solution accuracy and computational cost.

Keywords

Analytical Target Cascading; Multilevel Design Optimization; Augmented Lagrangian; Method of Multipliers

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	N. F. Michelena, H. M. Kim and P. Y. Papalambros, “A System Partitioning and Optimization Approach to Target Cascading,” Proceedings of the 12th International Conference on Engineering Design, Munich, 1999.
[2]	H. M. Kim, N. F. Michelena, P. Y. Papalambros and T. Jiang, “Target Cascading in Optimal System Design,” ASME Transaction on Journal of Mechanical Design, Vol. 125, 2003, pp. 474-480. doi:10.1115/1.1582501
[3]	H. M. Kim, N. F. Michelena, P. Y. Papalambros and T. Jiang, “Target Cascading in Optimal System Design,” Proceedings of the 26th Design Automation Conference, Baltimore, 2000.
[4]	H. M. Kim, M. Kokkolaras, L. S. Louca, G. J. Delagrammatikas, N. F. Michelena, Z. S. Filipi, P. Y. Papalambros, J. L. Stein and D. N. Assanis, “Target Cascading in Automotive Vehicle Design: A Class 6 Truck Study,” International Journal of Vehicle Design, Vol. 29, No. 3, 2002, pp. 199-225. doi:10.1504/IJVD.2002.002010
[5]	H. M. Kim, D. G. Rideout, P. Y. Papalambros and J. L. Stein, “Analytical Target Cascading in Automotive Vehicle Design,” Proceedings of ASME Design Engineering Technical Conference and Computers and Information in Engineering Conference, Pittsburgh, 2001.
[6]	N. F. Michelena, H. Park and P. Y. Papalambros, “Convergence Properties of Analytical Target Cascading,” AIAA Journal, Vol. 41, No. 5, 2003, pp. 897-905. doi:10.2514/2.2025
[7]	N. Tzevelekos, M. Kokkolaras, P. Y. Papalambros, M. F. Hulshof, L. F. P. Etman and J. E. Rooda, “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, 19-23 May 2003, 6 p.
[8]	J. J. Michalek and P. Y. Papalambros, “An Efficient Weighting Update Method to Achieve Acceptable Inconsistency Deviation in Analytical Target Cascading,” Journal of Mechanical Design, Vol. 127, No. 2, 2005, pp. 206-214. doi:10.1115/1.1830046
[9]	J. J. Michalek and P. Y. Papalambros, “Weights, Norms, and Notation in Analytical Target Cascading,” Journal of Mechanical Design, Vol. 127, 2005, pp. 499-501. doi:10.1115/1.1862674
[10]	S. Tosserams, “Analytical Target Cascading: Convergence Improvement by Subproblem Post-Optimality Sensitivities,” MS Thesis, Eindhoven University of Technology, Eindhoven, 2004.
[11]	D. P. Bertsekas, “Nonlinear Programming,” 2nd Edition, Athena Scientific, Belmont, 2003.
[12]	J. B. Lassiter, M. M. Wiecek and K. R. Andrighetti, “Lagrangian Coordination and Analytical Target Cascading: Solving ATC-Decomposed Problems with LagrangianDuality,” Optimization and Engineering, Vol. 6, No. 3, 2005, pp. 361-381. doi:10.1007/s11081-005-1744-4
[13]	V. Y. Blouin, J. Lassiter, M. Wiecek and G. M. Fadel, “Augmented LagrangianCoordination for Decomposed Design Problems,” 6th World Congresses of Structural and Multidisciplinary Optimization, Rio de Janeiro, 30 May-3 June, 2005.
[14]	H. M. Kim, W. Chen and M. M. Wiecek, “Lagrangian Coordination for Enhancing the Convergence of Analytical Target Cascading,” AIAA Journal, Vol. 44, No. 10, 2006, pp. 2197-2207. doi:10.2514/1.15326
[15]	S. Tosserams, L. F. P. Etman, P. Y. Papalambros and J. E. Rooda, “An Augmented Lagrangian Relaxation for Analytical Target Cascading Using the Alternating Directions Method of Multipliers,” Structural and Multidisciplinary Optimization, Vol. 31, No. 3, 2006, pp. 176-189. doi:10.1007/s00158-005-0579-0
[16]	Y. Li, Z. Lu and J. J. Michalek, “Diagonal Quadratic Approximation for Parallelization of Analytical Target Cascading,” Journal of Mechanical Design, Vol. 130, 2008, pp. 051402-1-051402-11.
[17]	W. Wang, V. Y. Blouin, M. Gardenghi, M. M. Wiecek, G. M. Fadel and B. Sloop, “A Cutting Plane Method For Analytical Target Cascading With Augmented Lagrangian Coordination,” Proceedings of the ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE, Montreal, 2010.