Article Outline
Keywords
Basic Difficulties
Optimality
Parametric Approach To Optimality
Duality
See also
References
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bank B, Guddat J, Klatte D, Kummer B, Tammer K (1982) Nonlinear parametric optimization. Akad Verlag, Berlin
Bard J (1984) Optimality conditions for the bilevel programming problem. Naval Res Logist Quart 31:13–26
Bard J (1988) Convex two-level optimization. Math Program 40:15–27
Ben-Ayed O, Blair C (1990) Computational difficulties of bilevel linear programming. Oper Res 38:556–560
Ben-Israel A, Ben-Tal A, Zlobec S (1981) Optimality in nonlinear programming: A feasible directions approach. Wiley/Interscience, New York
Berge C (1963) Topological spaces. Oliver and Boyd, Edinburgh
Bi Z, Calamai P (1991) Optimality conditions for a class of bilevel programming problems. Techn Report Dept Systems Design Engin, Univ Waterloo, no. 191-O-191291
Bracken J, Falk J, McGill J (1974) Equivalence of two mathematical programs with optimization problems in the constraints. Oper Res 22:1102–1104
Calamai P, Vicente LN (1993) Generating linear and linear-quadratic bilevel programming problems. SIAM J Sci Statist Comput 14:770–782
Candler W (1988) A linear bilevel programming algorithm: A comment. Comput Oper Res 15:297–298
Chen Y, Florian M (1995) The nonlinear bilevel programming problem: formulations, regularity and optimality conditions. Optim 32:193–209
Clarke P, Westerberg A (1988) A note on the optimality conditions for the bilevel programming problem. Naval Res Logist Quart 35:413–418
Dempe S (1992) A necessary and sufficient optimality condition for bilevel programming problems. Optim 25:341–354
Dempe S (1997) On the leader's dilemma and a new idea for attacking bilevel programming problems. Preprint Techn Univ Chemnitz
Floudas CA, Zlobec S (1998) Optimality and duality in parametric convex lexicographic programming. In: Pardalos PM Migdalas A and Värbrand P (eds) Multilevel Optimization: Algorithms and Applications. Kluwer, Dordrecht, pp 359–379
Harker P, Pang J-S (1988) Existence of optimal solutions to mathematical programs with equilibrium constraints. Oper Res Lett 7:61–64
Haurie A, Savard G, White D (1990) A note on: An efficient point algorithm for a linear two-stage optimization problem. Oper Res 38:553–555
Hettich R, Jongen HTh (1978) Semi-infinite programming: conditions of optimality and applications. In: Stoer J (ed) Optimization Techniques, Part 2. Lecture Notes Control Inform Sci. Springer, Berlin, pp 1–11
Ishizuka Y (1988) Optimality conditions for quasi‐differentiable programs with applications to two-level optimization. SIAM J Control Optim 26:1388–1398
Jongen HTh, Rückmann J-J, Stein O (1998) Generalized semi-infinite optimization: A first order optimality condition and examples. Math Program 83:145–158
Jongen HTh, Rückmann J-J (1998) On stability and deformation in semi-infinite optimization. In: Reemtsen R, Rückmann J-J (eds) Semi-Infinite Programming. Kluwer, Dordrecht, pp 29–67
Kolstad CD (Oct. 1985) A review of the literature on bilevel mathematical programming. Techn Report Los Alamos Nat Lab no. LA-10284-MS, UC-32
Lignola MB, Morgan J (1995) Topological existence and stability for Stackelberg problems. J Optim Th Appl 84:145–169
Lignola MB, Morgan J (1998) Existence of solutions to generalized bilevel programing problem. In: Migdalas A, Pardalos PM, Värbrand P (eds) Multilevel Optimization: Algorithms and Applications. Kluwer, Dordrecht, pp 315–332
Liu Y, Hart S (1994) Characterizing an optimal solution to the linear bilevel programming problem. Europ J Oper Res 166:164–166
Loridan P, Morgan J (1989) New results on approximate solutions in two-level optimization. Optim 20:819–836
Mallozzi L, Morgan J (1995) Weak Stackelberg problem and mixed solutions under data perturbations. Optim 32:269–290
Marcotte P, Savard G (1991) A note on Pareto optimality of solutions to the linear bilevel programming problem. Comput Oper Res 18:355–359
Migdalas A (1995) When is a Stackelberg equilibrium Pareto optimum? In: Pardalos PM, Siskos Y, Zopounidis C (eds) Advances in Multicriteria Analysis. Kluwer, Dordrecht, pp 175–181
Migdalas A, Pardalos PM (1996) Editorial: Hierarchical and bilevel programming. J Global Optim 8:209–215
Migdalas A Pardalos PM, Värbrand P (eds) (1998) Multilevel optimization: Algorithms and applications. Kluwer, Dordrecht, pp 29–67
Outrata J (1993) Necessary optimality conditions for Stackelberg problems. J Optim Th Appl 76:305–320
Savard G, Gauvin J (1994) The steepest descent direction for the nonlinear bilevel programming problem. Oper Res Lett 15:275–282
Tammer K, Rückmann J-J (1990) Relations between the Karush–Kuhn–Tucker points of a nonlinear optimization problem and of a generalized Lagrange dual. In: Sebastian H-J, Tammer K (eds) System Modelling and Optimization. Lecture Notes Control Inform Sci. Springer, Berlin
Trujillo-Cortez R (1997) LFS functions in stable bilevel programming. PhD Thesis Dept Math and Statist, McGill Univ
Trujillo-Cortez R (2000) Stable bilevel programming and applications. PhD Thesis McGill Univ, in preparation
Tuy H (1998) Bilevel linear programming, multiobjective programming, and monotonic reverse convex programming. In: Migdalas A Pardalos PM, Värbrand P (eds) Multilevel Optimization: Algorithms and Applications. Kluwer, Dordrecht, 295–314
Ünlü G (1987) A linear bilevel programming algorithm based on bicriteria programming. Comput Oper Res 14:173–179
Vicente LN, Calamai PH (1994) Bilevel and multilevel programming: A bibliography review. J Global Optim 5:291–306
Vicente LN, Calamai PH (1995) Geometry and local optimality conditions for bilevel programs with quadratic strictly convex lower levels. In: Dhu D-Z, Pardalos PM (eds) Minimax and Applications. Kluwer, Dordrecht, pp 141–151
Vicente LN, Savard G, Judice J (1994) Descent approaches for quadratic bilevel programming. J Optim Th Appl 81:379–399
Visweswaran V, Floudas CA, Ierapetritou MG, Pistikopoulos EN (1996) A decomposition-based global optimization approach for solving bilevel linear and quadratic programs. In: Floudas CA, Pardalos PM (eds) State of the Art in Global Optimization. Kluwer, Dordrecht, 139–162
Wen U-P, Hsu S-T (1989) A note on a linear bilevel programming algorithm based on bicriteria programming. Comput Oper Res 16:79–83
Wetterling W (1970) Definitheitsbedingungen für relative Extrema bei Optimierungs- und Approximationsaufgaben. Numerische Math 15:122–136
Ye J (1995) Necessary conditions for bilevel dynamic optimization problems. SIAM J Control Optim 33:1208–1223
Ye JJ, Zhu DL (1995) Optimality conditions for bilevel programming problems. Optim 33:9–27
Zlobec S (1996) Lagrange duality in partly convex programming. In: Floudas CA, Pardalos PM (eds) State of the Art in Global Optimization. Kluwer, Dordrecht, pp 1–18
Zlobec S (1998) Stable parametric programming. Optim 45:387–416
Zlobec S (2000) Parametric programming: An illustrative mini-encyclopedia. Math Commun 5:1–39
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag
About this entry
Cite this entry
Zlobec, S. (2008). Bilevel Programming: Optimality Conditions and Duality . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_47
Download citation
DOI: https://doi.org/10.1007/978-0-387-74759-0_47
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