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Research Papers

Advanced Robust Optimization With Interval Uncertainty Using a Single-Looped Structure and Sequential Quadratic Programming

[+] Author and Article Information
Mian Li

e-mail: mianli@sjtu.edu.cn
University of Michigan-Shanghai Jiao Tong
University Joint Institute,
Shanghai Jiao Tong University,
Shanghai 200240, China

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received April 10, 2013; final manuscript received October 11, 2013; published online December 11, 2013. Assoc. Editor: Michael Kokkolaras.

J. Mech. Des 136(2), 021008 (Dec 11, 2013) (11 pages) Paper No: MD-13-1159; doi: 10.1115/1.4025963 History: Received April 10, 2013; Revised October 11, 2013

Uncertainty is inevitable and has to be taken into consideration in engineering optimization; otherwise, the obtained optimal solution may become infeasible or its performance can degrade significantly. Robust optimization (RO) approaches have been proposed to deal with this issue. Most existing RO algorithms use double-looped structures in which a large amount of computational efforts have been spent in the inner loop optimization to determine the robustness of candidate solutions. In this paper, an advanced approach is presented where no optimization run is required for robustness evaluation in the inner loop. Instead, a concept of Utopian point is proposed and the corresponding maximum variable/parameter variation will be obtained just by performing matrix operations. The obtained robust optimal solution from the new approach may be conservative, but the deviation from the true robust optimal solution is small enough and acceptable given the significant improvement in the computational efficiency. Six numerical and engineering examples are tested to show the applicability and efficiency of the proposed approach, whose solutions and computational efforts are compared to those from a previously proposed double-looped approach, sequential quadratic program-robust optimization (SQP-RO).

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References

Taguchi, G., 1978, “Performance Analysis Design,” Int. J. Prod. Res., 16, pp. 521–530. [CrossRef]
Ben-Tal, A., and Nemirovski, A., 2002, “Robust Optimization-Methodology and Applications,” Math. Program., Ser. B, 92(3), pp. 453–480. [CrossRef]
Park, G.-J., Lee, T.-H., Lee, K. H., and Hwang, K.-H., 2006, “Robust Design: An Overview,” AIAA J., 44(1), pp. 181–191. [CrossRef]
Beyer, H.-G., and Sendhoff, B., 2007, “Robust Optimization—A Comprehensive Survey,” Comput. Methods Appl. Mech. Eng., 196(33–34), pp. 3190–3218. [CrossRef]
Bertsimas, D., Brown, D. B., Caramanis, C., 2010, “Theory and Applications of Robust Optimization,” Math. Program., Ser. B, 107, pp. 464–501.
Hora, S. C., 1996, “Aleatory and Epistemic Uncertainty in Probability Elicitation With an Example From Hazardous Waste Management,” Reliab. Eng. Syst. Saf., 54, pp. 217–223. [CrossRef]
Oberkampf, W. L., Diegert, K. V., Alvin, K. F., Rutherford, B. M., 1998, “Variability, Uncertainty, and Error in Computational Simulation,” ASME Proceedings of the 7th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, Albuquerque, New Mexico, 357(2), pp. 259–272.
Hofer, E., Kloos, M., Krzykacz-Hausmann, B., Peschke, J., and Woltereck, M., 2002, “An Approximate Epistemic Uncertainty Analysis Approach in the Presence of Epistemic and Aleatory Uncertainties,” Reliab. Eng. Syst. Saf., 77, pp. 229–238. [CrossRef]
Agarwal, H., Renaud, J. E., Preston, E. L., and Padmanabhan, D., 2004, “Uncertainty Quantification Using Evidence Theory in Multidisciplinary Design Optimization,” Reliab. Eng. Syst. Saf., 85, pp.281–294. [CrossRef]
Helton, J. C., Johnson, J. D., Oberkampf, W. L., and Storlie, C. B., 2007, “A Sampling-Based Computational Strategy For The Representation of Epistemic Uncertainty in Model Predictions With Evidence Theory,” Comput. Methods Appl. Mech. Eng., 196, pp.3980–3998. [CrossRef]
Haldar, A., and MahadevanS., 2000, Probability, Reliability and Statistical Methods in Engineering Design, Wiley, New York.
Stacy, L. J., Lin, X., Floudas, C. A., 2007, “A New Robust Optimization Approach for Scheduling Under Uncertainty: II. Uncertainty With Known Probability Distribution,” Comput. Chem. Eng., 31(3), pp. 171–195. [CrossRef]
Shapiro, A., 2008, “Stochastic Programming Approach to Optimization Under Uncertainty,” Math. Program., Ser. B, 112, pp. 183–220. [CrossRef]
Lin, X., Stacy, L. J., and Floudas, C. A., 2003, “A New Robust Optimization Approach for Scheduling Under Uncertainty: I. Bounded Uncertainty,” Comput. Chem. Eng., 28(6–7), pp. 1069–1085. [CrossRef]
Li, M., and Azarm, S., 2008, “Multiobjective Collaborative Robust Optimization With Interval Uncertainty and Interdisciplinary Uncertainty Propagation,” ASME J. Mech. Des., 130(8), p. 081402. [CrossRef]
Hu, W., Li, M., Azarm, S., and Almansoori, A., 2011, “Multi-Objective Robust Optimization Under Interval Uncertainty Using Online Approximation and Constraint Cuts,” ASME J. Mech. Des., 133(6), p. 061002. [CrossRef]
Ben-Tal, A., and Nemirovski, A., 1999, “Robust Solutions of Uncertain Linear Programs,” Oper. Res. Lett., 25, pp. 1–13. [CrossRef]
Ben-Tal, A., and Nemirovski, A., 2000, “Robust Solutions of Linear Programming Problems Contaminated With Uncertain Data,” Math. Program., Ser. A, 88(3), pp. 411–424. [CrossRef]
Bertsimas, D., Pachamanova, D., and Sim, M., 2004, “Robust Linear Optimization Under General Norms,” Oper. Res. Lett., 32(6), pp. 510–516. [CrossRef]
Ben-Tal, A., and Nemirovski, A., 1998, “Robust Convex Optimization,” Math. Oper. Res., 23, pp. 769–805. [CrossRef]
Lobo, M., 2000, “Robust and Convex Optimization With Applications in Finance,” Ph.D. thesis, Department of Electrical Engineering, Stanford University, Stanford, CA.
Teo, K. M., 2007, “Nonconvex Robust Optimization,” Ph.D. thesis, Operations Research Center, Massachusetts Institute of Technology, Cambridge, MA.
Zhou, J. H., Cheng, S., and Li, M., 2012, “Sequential Quadratic Programming for Robust Optimization With Interval Uncertainty,” ASME J. Mech. Des., 134(10), p. 100913. [CrossRef]
Li, M., 2007, “Robust Optimization and Sensitivity Analysis With Multi-Objective Genetic Algorithms: Single- and Multi-Disciplinary Applications,” Ph.D. thesis, Department of Mechanical Engineering, UMD, College Park, MD.
Liang, J., Mourelatos, Z. P., and Nikolaidis, E., 2007. “A Single-Loop Approach for System Reliability-Based Design Optimization,” ASME J. Mech. Des., 129(12), pp. 1215–1224. [CrossRef]
Li, M., Gabriel, S. A., Shim, Y., and Azarm, S., 2011, “Interval Uncertainty-Based Robust Optimization for Convex and Non-Convex Quadratic Programs With Applications in Network Infrastructure Planning,” Netw. Spatial Econ., 11(1), pp. 159–191. [CrossRef]
Siddiqui, S., Azarm, S., and Gabriel, S., 2011, “A Modified Benders Decomposition Method for Efficient Robust Optimization Under Interval Uncertainty,” Struct. Multidiscip. Optim., 44(2), pp. 259–275. [CrossRef]
Floudas, C. A., and Visweswaran, V., 1995, “Quadratic optimization,” Handbook of Global Optimization, R.Horst and P. M.Pardalos, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 217–269.
Gao, D. Y., and Ruan, N., 2010, “On the Solutions to Quadratic Minimization Problems With Box and Integer Constraints,” J. Global Optim., 47(3), pp. 463–484. [CrossRef]
Friedlander, A., Martinez, J. M., and Raydan, M., 1995, “A New Method for Large-Scale Box Constrained Convex Quadratic Minimization Problems,” Optim. Methods Software, 5, pp. 57–74. [CrossRef]
Dembo, R. S., and Tulowitzki, U., 1984, “On the Minimization of Quadratic Functions Subject to Box Constraints,” Working Paper No.71, Series B, School of Organization and Management, Yale University (New Haven, CT, 1983).
Vandenbussche, D., and Nemhauser, G. L., 2005, “A Branch-and-Cut Algorithm for Nonconvex Quadratic Programs With Box Constraints,” Math. Program, Ser. A, 102, pp. 559–575. [CrossRef]
YangE. K., and Tolle, J. W., 1991, “A Class of Methods for Solving Large, Convex Quadratic Programs Subject to Box Constraints,” Math. Programing, 51, pp. 223–228. [CrossRef]
Ben-Tal, A., Nemirovski, A., and Roos, C., 2002, “Robust Solutions of Uncertain Quadratic and Conic-Quadratic Problems,” SIAM, J. Optim., 13(2), pp. 535–360.
Gunawan, S., and Azarm, S., 2004, “Non-Gradient Based Parameter Sensitivity Estimation for Single Objective Robust Design Optimization,” ASME J. Mech. Des., 126(3), pp. 395–402. [CrossRef]

Figures

Grahic Jump Location
Fig. 2

Illustration of true solution and Utopian solution

Grahic Jump Location
Fig. 3

Shapes of a parabolic function

Grahic Jump Location
Fig. 4

Concave parabola for constraints

Grahic Jump Location
Fig. 5

Solution comparison of SQP-RO and A-SQP-RO: (a) Constraint contour comparison and (b) final solution comparison

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