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Multilevel Methods for Optimal Design

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Encyclopedia of Optimization

Article Outline

Keywords

Problem Formulation

Bilevel Optimization

  Penalty-Based Methods

  KKT-Based Methods

  Descent-Based Methods

Examples: Collaborative Optimization

Example: Multilevel Algorithms

Summary

See also

References

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References

  1. Aiyoshi E, Shimizu K (1981) Hierarchical decentralized system and its new solution by a barrier method. IEEE Trans Syst, Man Cybern 11:444–449

    Article  MathSciNet  Google Scholar 

  2. Alexandrov NM (1993) Multilevel algorithms for nonlinear equations and equality constrained optimization. PhD Thesis Rice Univ.

    Google Scholar 

  3. Alexandrov NM (1996) Multilevel and multiobjective optimization in multidisciplinary design. AIAA Paper no. 96-4122

    Google Scholar 

  4. Alexandrov NM (2000) A trust-region algorithm for nonlinear bilevel optimization. in preparation

    Google Scholar 

  5. Alexandrov NM, Dennis JE (1995) Multilevel algorithms for nonlinear optimization. In: Borggaard J, Burkardt J, Gunzburger M, Peterson J (eds) Optimal Design and Control. Birkhäuser, Basel, pp 1–22

    Google Scholar 

  6. Alexandrov NM, Hussaini MY (eds) (1997) Multidisciplinary design optimization: State of the art. SIAM, Philadelphia

    Google Scholar 

  7. Alexandrov NM, Kodiyalam S (1998) Initial results of an MDO method evaluation study. AIAA Paper no. 98-4884

    Google Scholar 

  8. Alexandrov NM, Lewis RM (2000) Algorithmic perspectives on problem formulations in MDO. AIAA Paper no. 2000-4719

    Google Scholar 

  9. Alexandrov NM, Lewis RM (2002) Analytical and computational aspects of collaborative optimization for multidisciplinary design. AIAA J 40:301–309

    Article  Google Scholar 

  10. Alexandrov NM, Lewis RM (2000) Analytical and computational properties of distributed approaches to MDO. AIAA Paper no. 2000-4718

    Google Scholar 

  11. Anadalingam G, Friesz TL (1992) Hierarchical optimization: An introduction. Ann Oper Res 34:1–11

    Article  MathSciNet  Google Scholar 

  12. Badhrinath K, Rao JRJ (1994) Bilevel models for optimum designs which are insensitive to perturbations in variables and parameters. Techn Report Univ Houston UH-ME-SDL-94-03

    Google Scholar 

  13. Balling RJ, Sobieszczanski-Sobieski J (1994) An algorithm for solving the system-level problem in multilevel optimization. In: Fifth AIAA/USAF/NASA/ISSMO Symp. Multidisciplinary Analysis and Optimization (Panama Beach, Florida, Sept. 7-9, 1994), AIAA Paper no. 94-4333 (1994)

    Google Scholar 

  14. Bard JF (1983) An algorithm for solving the general bilevel programming program. Math Oper Res 8:260–272

    Article  MathSciNet  MATH  Google Scholar 

  15. Bard JF, Falk JE (1982) An explicit solution to the multi-level programming problem. Comput Oper Res 9:77–100

    Article  MathSciNet  Google Scholar 

  16. Barthelemy J-FM (1988) Engineering applications of heuristic multilevel optimization methods. NASA TM-101504

    Google Scholar 

  17. Barthelemy J-FM, Riley MF (1988) Improved multilevel optimization approach for the design of complex engineering systems. AIAA J 26:353–360

    Article  Google Scholar 

  18. Basar T, Selbuz H (1979) Closed loop Stackelberg strategies with applications in optimal control of multilevel systems. IEEE Trans Autom Control AC-24:166–178

    Article  MathSciNet  Google Scholar 

  19. Benders JF (1962) Partitioning procedures for solving mixed variables programming problems. Numerische Math 4:238–252

    Article  MathSciNet  MATH  Google Scholar 

  20. Bialas WF, Karwan MH (1982) On two-level optimization. IEEE Trans Autom Control AC-27:211–214

    Article  Google Scholar 

  21. Bracken J, McGill J (1973) Mathematical programs with optimization problems in the constraints. Oper Res 21:37–44

    Article  MathSciNet  MATH  Google Scholar 

  22. Braun RD (1996) Collaborative optimization: An architecture for large-scale distributed design. PhD Thesis Stanford Univ.

    Google Scholar 

  23. Braun RD, Moore AA, Kroo IM (1997) Collaborative approach to launch vehicle design. J Spacecraft and Rockets 34:478–486

    Article  Google Scholar 

  24. Burke JV (1991) Calmness and exact penalization. SIAM J Control Optim 29:493–497

    Article  MathSciNet  MATH  Google Scholar 

  25. Burke JV (1991) An exact penalization viewpoint of constrained optimization. SIAM J Control Optim 29:968–998

    Article  MathSciNet  MATH  Google Scholar 

  26. Calamai PH, Vicente LN (1993) Generating linear and linear-quadratic bilevel programming problems. SIAM J Sci Comput 14:770–782

    Article  MathSciNet  MATH  Google Scholar 

  27. Candler W, Townsley R (1982) A linear two-level programming problem. Comput Oper Res 9:59–76

    Article  MathSciNet  Google Scholar 

  28. Chen Y (1994) Bilevel programming problems: Analysis, algorithms and applications. PhD Thesis Univ. Montreal

    Google Scholar 

  29. Chidambaram B, Rao JRJ (1994) A study of constraint activity in bilevel models of optimal design. Techn Report Univ Houston UH-ME-SDL-94-01

    Google Scholar 

  30. Clarke FH (ed) (1990) Optimization and nonsmooth analysis. SIAM, Philadelphia

    MATH  Google Scholar 

  31. Danzig GB, Wolfe P (1960) Decomposition principle for linear programming. Oper Res 8:101–111

    Article  Google Scholar 

  32. De Luca A, Di Pillo G (1987) Exact augmented Lagrangian approach to multilevel optimization of large-scale systems. Internat J Syst Sci 18:157–176

    Article  MathSciNet  MATH  Google Scholar 

  33. De Miguel W, Murray W (2006) A Local Convergence Analysis of Bilevel Decomposition Algorithms. Optim Eng 7:99–133

    Article  MathSciNet  Google Scholar 

  34. De Silva AH, McCormick GP (1992) Implicitly defined optimization problems. Ann Oper Res 34:107–124

    Article  MathSciNet  MATH  Google Scholar 

  35. Dennis JE Jr, Schnabel RB (1983) Numerical methods for unconstrained optimization and nonlinear equations. Prentice-Hall, Englewood Cliffs, NJ

    MATH  Google Scholar 

  36. Dirickx YMI, Jennergren LP (1979) Systems analysis by multilevel methods. Wiley, New York

    MATH  Google Scholar 

  37. Edmunds TA, Bard JF (1991) Algorithm for nonlinear bilevel mathematical programs. IEEE Trans Syst, Man Cybern 21:83–89

    Article  MathSciNet  Google Scholar 

  38. El-Alem MM (1991) A global convergence theory for the {Celis–Dennis–Tapia} trust region algorithm for constrained optimization. SIAM J Numer Anal 28:266–290

    Article  MathSciNet  MATH  Google Scholar 

  39. Falk JE, Liu J (1995) On bilevel programming, Part I: General nonlinear cases. Math Program 70:47–72

    MathSciNet  Google Scholar 

  40. Fiacco AV (ed) (1983) Introduction to sensitivity and stability analysis in nonlinear programming. Acad. Press, New York

    MATH  Google Scholar 

  41. Fiacco AV, McCormick GP (eds) (1990) Nonlinear programming, sequential unconstrained minimization techniques. SIAM, Philadelphia

    MATH  Google Scholar 

  42. Flippo OE (1989) Stability, duality and decomposition in general mathematical programming. PhD Thesis, Erasmus Univ. Rotterdam

    Google Scholar 

  43. Flippo OE, Rinnooy Kan AHG (1993) Decomposition in general mathematical programming. Math Program 60:361–382

    Article  MathSciNet  Google Scholar 

  44. Fortuny-Amat J, McCarl B (1981) A representation of a two-level programming problem. J Oper Res Soc 32:783–792

    Article  MathSciNet  MATH  Google Scholar 

  45. Friesz T, Tobin R, Cho H, Mehta N (1990) Sensitivity analysis based heuristic algorithms for mathematical programs with variational inequality constraints. Math Program 48:265–284

    Article  MathSciNet  MATH  Google Scholar 

  46. Gahutu DWH, Looze DP (1985) Parametric coordination in hierarchical control. Large Scale Systems 8:33–45

    MathSciNet  MATH  Google Scholar 

  47. Gauvin J (1978) The method of parametric decomposition in mathematical programming: the nonconvex case. In: Lemarechal C, Griffin R (eds) Nonsmooth optimization. Pergamon, Oxford, pp 131–149

    Google Scholar 

  48. Gauvin J, Dubeau F (1983) Some examples and counterexamples for stability analysis of nonlinear programming problems. In: Fiacco AV (ed) Math. Program. Stud., vol 21. North-Holland, Amsterdam, pp 69–78

    Google Scholar 

  49. Geoffrion AM (1972) Generalized Benders decomposition. J Optim Th Appl 10:237–260

    Article  MathSciNet  MATH  Google Scholar 

  50. Goulbeck B, Brdys M, Orr CH, Rance JP (1988) A hierarchical approach to optimized control of water distribution systems: Part I, decomposition. Optimal Control Appl Meth 9:51–61

    Google Scholar 

  51. Goulbeck B, Brdys M, Orr CH, Rance JP (1988) A hierarchical approach to optimized control of water distribution systems: part II, lower-level algorithm. Optimal Control Appl Meth 9:109–126

    Google Scholar 

  52. Haimes YY (1977) Hierarchical analyses of water resources systems. McGraw-Hill, New York

    Google Scholar 

  53. Haimes YY, Tarvainen K, Shima T, Thadathil J (1990) Hierarchical multiobjective analysis of large-scale systems. Hemisphere, Washington, DC

    Google Scholar 

  54. Hafka RT, Watson LT (2005) Multidisciplinary Design Optimization Problems with Quasiseparable Subsystems. Optim Eng 6:9–20

    Article  MathSciNet  Google Scholar 

  55. Hafka RT, Watson LT (2006) Decomposition Theory for Multidisciplinary Design Optimization Problems with Mixed Integer Quasiseparable Subsystems. Optim Eng 7:135–149

    Article  MathSciNet  Google Scholar 

  56. Ishizuka Y, Aiyoshi E (1992) Double penalty method for bilevel optimization problems. Ann Oper Res 34:73–88

    Article  MathSciNet  MATH  Google Scholar 

  57. Kirsch U (1989) Improved optimum structural design by passive control. Engin with Comput 5:13–22

    Article  Google Scholar 

  58. Kodiyalam S (1998) Evaluation of methods for multidisciplinary design optimization (MDO), phase I. NASA Contractor Report

    Google Scholar 

  59. Kolstad CD, Lasdon LS (1990) Derivative evaluation and computational experience with large bilevel mathematical programs. J Optim Th Appl 65:485–499

    Article  MathSciNet  MATH  Google Scholar 

  60. Lasdon LS (1970) Optimization theory for large systems. MacMillan, New York

    MATH  Google Scholar 

  61. Loridan P, Morgan J (1988) Quasiconvex lower level problems and applications in two-level optimization. Techn Report Univ Montreal CRM-1578

    Google Scholar 

  62. Mahmoudi MS (1977) Multilevel systems control and applications: A survey. IEEE Trans Syst, Man Cybern SMC-7:125–143

    Article  Google Scholar 

  63. Marcotte P (1985) A new algorithm for solving variational inequalities with application to the traffic assignment problem. Math Program 33:339–351

    Article  MathSciNet  Google Scholar 

  64. Marcotte P, Zhu DL (1996) Exact and inexact penalty methods for the generalized bilevel programming problem. Math Program 74:141–157

    MathSciNet  Google Scholar 

  65. Mesarović MD, Macko D, Takahara Y (1970) Theory of hierarchical, multilevel, systems. Acad. Press, New York

    MATH  Google Scholar 

  66. Migdalas A, Pardalos PM, Värbrand P (eds) (1998) Multilevel optimization: Algorithms and applications. Kluwer, Dordrecht

    MATH  Google Scholar 

  67. Moré JJ (1991) Recent developments in software for trust region methods. In: Bachem A, Grötschel M, Korte B (eds) Mathematical Programming: The State of the Art. Springer, Berlin, pp 266–290

    Google Scholar 

  68. Muralidhar R, Rao JRJ, Badhrinath K, Kalagatla A (1995) Multilevel formulations and limit analysis and design of structures with bilateral contact constraints. Techn Report Univ Houston UH-ME-SDL-95-02

    Google Scholar 

  69. Nachane DM (1984) Optimization methods in multilevel systems: A methodological survey. Europ J Oper Res 21:25–38

    Article  MathSciNet  Google Scholar 

  70. Nicholls MG (1995) Aluminium production modelling – a non-linear bi-level programming approach. Oper Res 43:208–218

    Article  MATH  Google Scholar 

  71. Outrata J (1994) On optimization problems with variational inequality constraints. SIAM J Optim 4:340–357

    Article  MathSciNet  MATH  Google Scholar 

  72. Padula SL, Alexandrov NM, Green LL (1996) MDO test suite at NASA Langley Research Center. In: Proc. Sixth AIAA/NASA/ISSMO Symp. Multidisciplinary Analysis and Optimization, AIAA

    Google Scholar 

  73. Padula SL, Young KC (1986) Simulator for multilevel optimization research. NASA Techn Memorandum TM-87751

    Google Scholar 

  74. Rao JRJ, Badhrinath K (1996) Solution of multilevel structural design problems using a nonsmooth algorithm. In: Proc. Sixth AIAA/NASA/ISSMO Symp. Multidisciplinary Analysis and Optimization, AIAA, AIAA-96-3986-CR.

    Google Scholar 

  75. Rao JRJ, Chidambaram B (1993) Parametric deformation and model optimality in concurrent design. Techn Report Univ Houston UH-ME-SDL-93-01

    Google Scholar 

  76. Reddy SY, Fertig KW, Smith DE (Aug. 1996) Constraint management methodology for conceptual design tradeoff studies. In: Proc. DETC’96, ASME Paper 96-DETC/DTM-1228

    Google Scholar 

  77. Rockafellar RT (1984) Directional differentiability of the optimal value function in a nonlinear programming problem. Math Program Stud 21:312–226

    MathSciNet  Google Scholar 

  78. Rogers JL (1996) DeMAID/GA – An enhanced design manager's aid for intelligent decomposition. Proc. Sixth AIAA/NASA/ISSMO Symp. Multidisciplinary Analysis and Optimization,), AIAA Paper 96-4157.

    Google Scholar 

  79. Rogers JL (1996) DeMAID/GA user's guide – Design manager's aid for intelligent decomposition with a genetic algorithm. NASA Langley Res Center TM-110241

    Google Scholar 

  80. Schmitt LA, Chang KJ (1984) A multilevel method for structural synthesis. In: A Collection of Technical Papers: AIAA/ASME/ASCE/AHS 25th Structures, Structural Dynamics and Materials Conf., AIAA

    Google Scholar 

  81. Schmitt LA Jr, Mehrinfar M (1982) Multilevel optimum design of structures with fiber-composite stiffened panel components. AiAA J 20(1):138–147

    Article  Google Scholar 

  82. Schmitt LA Jr, Ramanathan RK (1978) Multilevel approach to minimum weight design inclusing buckling constraints. AiAA J 16(2):97–104

    Article  Google Scholar 

  83. Shimizu K, Aiyoshi E (1981) A new computational method for Stackelberg and min-max problems by use of a penalty method. IEEE Trans Autom Control AC-26:460–466

    Article  MathSciNet  Google Scholar 

  84. Simaan M, Cruz JB Jr (1973) On the Stackelberg strategy in nonzero-sum games. J Optim Th Appl 11(5):535–555

    Article  Google Scholar 

  85. Singh MG, Mahmoud MS, Titli A (1981) A survey of recent developments in hierarchical optimization and control. Proc. IFAC Control Sci. and Techn. 8th Triennial World Congress, Kyoto, Japan, IFAC

    Google Scholar 

  86. Sobieski I, Kroo I (1996) Aircraft design using collaborative optimization. In: AIAA paper 96-0715 Presented at the 34th AIAA Aerospace Sci. Meeting, Reno, Nevada, Jan. 15-18, 1996, AIAA

    Google Scholar 

  87. Sobieszczanski-Sobieski J (1993) Optimization by decomposition. In: Kamat MP (ed) Structural Optimization: Status and Promise. Progress in Astronautics and Aeronautics, vol 150. AIAA, pp 487–515

    Google Scholar 

  88. Sobieszczanski-Sobieski J (1993) Two alternative ways for solving the coordination problem in multilevel optimization. Structural Optim 6:205–215

    Article  Google Scholar 

  89. Sobieszczanski-Sobieski J (1996) Multidisciplinary aerospace design optimization: Survey of recent developments. In: Proc. 34-th Aerospace Sci. Meeting and Exhibit, Reno, Nevada, AIAA, AIAA paper 96-0711.

    Google Scholar 

  90. Sobieszczanski-Sobieski J, Haftka RT (1997) Multidisciplinary aerospace design optimization: survey of recent developments. Structural Optim 14:1–23

    Article  Google Scholar 

  91. Sobieszczanski-Sobieski J, James BB, Dovi AR (1985) Structural optimization by generalized multilevel optimization. AIAA J 23:1775–1782

    Article  MathSciNet  MATH  Google Scholar 

  92. Stackelberg H (ed) (1952) The theory of the market economy. Oxford Univ. Press, Oxford

    Google Scholar 

  93. Tappeta RV, Renaud JE (1997) Multiobjective collaborative optimization. J Mechanical Design 119:403–411

    Article  Google Scholar 

  94. Vicente LN, Calamai PH (1994) Bilevel and multilevel programming: A bibliographic review. J Global Optim 5:291–306

    Article  MathSciNet  MATH  Google Scholar 

  95. Wagner TC (1993) A general decomposition methodology for optimal system design. PhD Thesis, Univ. Michigan

    Google Scholar 

  96. Walsh JL, LaMarsh WJ, Adelman HM (1992) Fully integrated aerodynamic/dynamic/structural optimization of helicopter rotor blades. NASA TM-104226

    Google Scholar 

  97. Walsh JL, Young KC, Pritchard JI, Adelman HM, Mantay WR (1995) Integrated aerodynamic/dynamic/structural optimization of helicopter rotor blades using multilevel decomposition. NASA TP-3465

    Google Scholar 

  98. Wismer DA (ed) (1971) Optimization methods for large-scale systems. McGraw-Hill, New York

    MATH  Google Scholar 

  99. Ye JJ, Zhu DL, Zhu QJ (1997) Exact penalization and necessary optimality conditions for generalized bilevel programming problems. SIAM J Optim 7:481–507

    Article  MathSciNet  MATH  Google Scholar 

  100. Zhang R (1994) Problems of hierarchical optimization in finite dimensions. SIAM J Optim 4:521–536

    Article  MathSciNet  MATH  Google Scholar 

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Alexandrov, N.M. (2008). Multilevel Methods for Optimal Design . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_416

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