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
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aiyoshi E, Shimizu K (1981) Hierarchical decentralized system and its new solution by a barrier method. IEEE Trans Syst, Man Cybern 11:444–449
Alexandrov NM (1993) Multilevel algorithms for nonlinear equations and equality constrained optimization. PhD Thesis Rice Univ.
Alexandrov NM (1996) Multilevel and multiobjective optimization in multidisciplinary design. AIAA Paper no. 96-4122
Alexandrov NM (2000) A trust-region algorithm for nonlinear bilevel optimization. in preparation
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
Alexandrov NM, Hussaini MY (eds) (1997) Multidisciplinary design optimization: State of the art. SIAM, Philadelphia
Alexandrov NM, Kodiyalam S (1998) Initial results of an MDO method evaluation study. AIAA Paper no. 98-4884
Alexandrov NM, Lewis RM (2000) Algorithmic perspectives on problem formulations in MDO. AIAA Paper no. 2000-4719
Alexandrov NM, Lewis RM (2002) Analytical and computational aspects of collaborative optimization for multidisciplinary design. AIAA J 40:301–309
Alexandrov NM, Lewis RM (2000) Analytical and computational properties of distributed approaches to MDO. AIAA Paper no. 2000-4718
Anadalingam G, Friesz TL (1992) Hierarchical optimization: An introduction. Ann Oper Res 34:1–11
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
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)
Bard JF (1983) An algorithm for solving the general bilevel programming program. Math Oper Res 8:260–272
Bard JF, Falk JE (1982) An explicit solution to the multi-level programming problem. Comput Oper Res 9:77–100
Barthelemy J-FM (1988) Engineering applications of heuristic multilevel optimization methods. NASA TM-101504
Barthelemy J-FM, Riley MF (1988) Improved multilevel optimization approach for the design of complex engineering systems. AIAA J 26:353–360
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
Benders JF (1962) Partitioning procedures for solving mixed variables programming problems. Numerische Math 4:238–252
Bialas WF, Karwan MH (1982) On two-level optimization. IEEE Trans Autom Control AC-27:211–214
Bracken J, McGill J (1973) Mathematical programs with optimization problems in the constraints. Oper Res 21:37–44
Braun RD (1996) Collaborative optimization: An architecture for large-scale distributed design. PhD Thesis Stanford Univ.
Braun RD, Moore AA, Kroo IM (1997) Collaborative approach to launch vehicle design. J Spacecraft and Rockets 34:478–486
Burke JV (1991) Calmness and exact penalization. SIAM J Control Optim 29:493–497
Burke JV (1991) An exact penalization viewpoint of constrained optimization. SIAM J Control Optim 29:968–998
Calamai PH, Vicente LN (1993) Generating linear and linear-quadratic bilevel programming problems. SIAM J Sci Comput 14:770–782
Candler W, Townsley R (1982) A linear two-level programming problem. Comput Oper Res 9:59–76
Chen Y (1994) Bilevel programming problems: Analysis, algorithms and applications. PhD Thesis Univ. Montreal
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
Clarke FH (ed) (1990) Optimization and nonsmooth analysis. SIAM, Philadelphia
Danzig GB, Wolfe P (1960) Decomposition principle for linear programming. Oper Res 8:101–111
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
De Miguel W, Murray W (2006) A Local Convergence Analysis of Bilevel Decomposition Algorithms. Optim Eng 7:99–133
De Silva AH, McCormick GP (1992) Implicitly defined optimization problems. Ann Oper Res 34:107–124
Dennis JE Jr, Schnabel RB (1983) Numerical methods for unconstrained optimization and nonlinear equations. Prentice-Hall, Englewood Cliffs, NJ
Dirickx YMI, Jennergren LP (1979) Systems analysis by multilevel methods. Wiley, New York
Edmunds TA, Bard JF (1991) Algorithm for nonlinear bilevel mathematical programs. IEEE Trans Syst, Man Cybern 21:83–89
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
Falk JE, Liu J (1995) On bilevel programming, Part I: General nonlinear cases. Math Program 70:47–72
Fiacco AV (ed) (1983) Introduction to sensitivity and stability analysis in nonlinear programming. Acad. Press, New York
Fiacco AV, McCormick GP (eds) (1990) Nonlinear programming, sequential unconstrained minimization techniques. SIAM, Philadelphia
Flippo OE (1989) Stability, duality and decomposition in general mathematical programming. PhD Thesis, Erasmus Univ. Rotterdam
Flippo OE, Rinnooy Kan AHG (1993) Decomposition in general mathematical programming. Math Program 60:361–382
Fortuny-Amat J, McCarl B (1981) A representation of a two-level programming problem. J Oper Res Soc 32:783–792
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
Gahutu DWH, Looze DP (1985) Parametric coordination in hierarchical control. Large Scale Systems 8:33–45
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
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
Geoffrion AM (1972) Generalized Benders decomposition. J Optim Th Appl 10:237–260
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
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
Haimes YY (1977) Hierarchical analyses of water resources systems. McGraw-Hill, New York
Haimes YY, Tarvainen K, Shima T, Thadathil J (1990) Hierarchical multiobjective analysis of large-scale systems. Hemisphere, Washington, DC
Hafka RT, Watson LT (2005) Multidisciplinary Design Optimization Problems with Quasiseparable Subsystems. Optim Eng 6:9–20
Hafka RT, Watson LT (2006) Decomposition Theory for Multidisciplinary Design Optimization Problems with Mixed Integer Quasiseparable Subsystems. Optim Eng 7:135–149
Ishizuka Y, Aiyoshi E (1992) Double penalty method for bilevel optimization problems. Ann Oper Res 34:73–88
Kirsch U (1989) Improved optimum structural design by passive control. Engin with Comput 5:13–22
Kodiyalam S (1998) Evaluation of methods for multidisciplinary design optimization (MDO), phase I. NASA Contractor Report
Kolstad CD, Lasdon LS (1990) Derivative evaluation and computational experience with large bilevel mathematical programs. J Optim Th Appl 65:485–499
Lasdon LS (1970) Optimization theory for large systems. MacMillan, New York
Loridan P, Morgan J (1988) Quasiconvex lower level problems and applications in two-level optimization. Techn Report Univ Montreal CRM-1578
Mahmoudi MS (1977) Multilevel systems control and applications: A survey. IEEE Trans Syst, Man Cybern SMC-7:125–143
Marcotte P (1985) A new algorithm for solving variational inequalities with application to the traffic assignment problem. Math Program 33:339–351
Marcotte P, Zhu DL (1996) Exact and inexact penalty methods for the generalized bilevel programming problem. Math Program 74:141–157
Mesarović MD, Macko D, Takahara Y (1970) Theory of hierarchical, multilevel, systems. Acad. Press, New York
Migdalas A, Pardalos PM, Värbrand P (eds) (1998) Multilevel optimization: Algorithms and applications. Kluwer, Dordrecht
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
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
Nachane DM (1984) Optimization methods in multilevel systems: A methodological survey. Europ J Oper Res 21:25–38
Nicholls MG (1995) Aluminium production modelling – a non-linear bi-level programming approach. Oper Res 43:208–218
Outrata J (1994) On optimization problems with variational inequality constraints. SIAM J Optim 4:340–357
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
Padula SL, Young KC (1986) Simulator for multilevel optimization research. NASA Techn Memorandum TM-87751
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.
Rao JRJ, Chidambaram B (1993) Parametric deformation and model optimality in concurrent design. Techn Report Univ Houston UH-ME-SDL-93-01
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
Rockafellar RT (1984) Directional differentiability of the optimal value function in a nonlinear programming problem. Math Program Stud 21:312–226
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.
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
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
Schmitt LA Jr, Mehrinfar M (1982) Multilevel optimum design of structures with fiber-composite stiffened panel components. AiAA J 20(1):138–147
Schmitt LA Jr, Ramanathan RK (1978) Multilevel approach to minimum weight design inclusing buckling constraints. AiAA J 16(2):97–104
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
Simaan M, Cruz JB Jr (1973) On the Stackelberg strategy in nonzero-sum games. J Optim Th Appl 11(5):535–555
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
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
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
Sobieszczanski-Sobieski J (1993) Two alternative ways for solving the coordination problem in multilevel optimization. Structural Optim 6:205–215
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.
Sobieszczanski-Sobieski J, Haftka RT (1997) Multidisciplinary aerospace design optimization: survey of recent developments. Structural Optim 14:1–23
Sobieszczanski-Sobieski J, James BB, Dovi AR (1985) Structural optimization by generalized multilevel optimization. AIAA J 23:1775–1782
Stackelberg H (ed) (1952) The theory of the market economy. Oxford Univ. Press, Oxford
Tappeta RV, Renaud JE (1997) Multiobjective collaborative optimization. J Mechanical Design 119:403–411
Vicente LN, Calamai PH (1994) Bilevel and multilevel programming: A bibliographic review. J Global Optim 5:291–306
Wagner TC (1993) A general decomposition methodology for optimal system design. PhD Thesis, Univ. Michigan
Walsh JL, LaMarsh WJ, Adelman HM (1992) Fully integrated aerodynamic/dynamic/structural optimization of helicopter rotor blades. NASA TM-104226
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
Wismer DA (ed) (1971) Optimization methods for large-scale systems. McGraw-Hill, New York
Ye JJ, Zhu DL, Zhu QJ (1997) Exact penalization and necessary optimality conditions for generalized bilevel programming problems. SIAM J Optim 7:481–507
Zhang R (1994) Problems of hierarchical optimization in finite dimensions. SIAM J Optim 4:521–536
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag
About this entry
Cite this entry
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
Download citation
DOI: https://doi.org/10.1007/978-0-387-74759-0_416
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-74758-3
Online ISBN: 978-0-387-74759-0
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering