Abstract
The paper presents a decomposition based global optimization approach to bilevel linear and quadratic programming problems. By replacing the inner problem by its corresponding KKT optimality conditions, the problem is transformed to a single yet non-convex, due to the complementarity condition, mathematical program. Based on the primal-dual global optimization approach of Floudas and Visweswaran (1990, 1993), the problem is decomposed into a series of primal and relaxed-dual subproblems whose solutions provide lower and upper bounds to the global optimum. By further exploiting the special structure of the bilevel problem, new properties are established which enable the efficient implementation of the proposed algorithm. Computational results are reported for both linear and quadratic example problems.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Anandalingam G. and D.J. White (1990). A solution method for the Linear Static Stackelberg Problem Using Penalty Method.IEEE Trans. Aut. Control,35, 1170.
Aiyoshi, E and K. Shimizu (1981). Hierarchical Decentralized systems and its new solution by a Barrier Method.IEEE Trans. Systems, Man and Cybernetics,SMC-11, 444.
Aiyoshi, E and K. Shimizu (1984). A solution method for the Static Constraint Stackelberg Problem Via Penalty Method.IEEE Trans. Aut Control,AC-29. 1111.
Al-Khayyal F., R. Horst and P. Pardalos (1992). Global Optimization on Concave Functions Subject to Quadratic Constraints: An Application in nonlinear Bilevel Programming.Annals Oper. Res.,34, 125.
Bard, J.F . (1983). An Algorithm for Solving the General Bilevel Programming Problem.Math. Oper. Res.,8, 260.
Bard, J.F . (1984). Optimality Conditions for the Bilevel Programming Problem.Nav. Res. Log. Quart.,31, 13.
Bard, J.F . (1988). Convex Two-Level Optimization.Math. Prog.,40, 15.
Bard, J.F. and J.E. Falk (1982). An Explicit Solution to the Multi-Level programming Problem.Comp. Oper. Res.,9, 77.
Bard, J.F. and J.T. Moore (1990). A Branch and Bound Algorithm for the Bilevel Programming Problem.SIAM J. Seien. Stat. Comp.,11, 281.
Ben-Ayed O. and C. E. Blair (1989). Computational Difficulties of Bilivel Linear Programming.Oper. Res. Tech. Notes, 556.
Bialas, W.F. and M.H. Karwan (1984). Two-Level Linear Programming.Mang. Sci.,30. 1004.
Candler, W. and R. Townsley (1982). A Linear Two-Level Programming Problem.Comp. Oper. Res.,9, 59.
Clark, P.A. and A.W. Westerberg (1988). A Note on the Optimality Conditions for the Bilevel Programming Problem.Nav. Res. Log.,35, 413.
Floudas, C.A. and V. Visweswaran (1990). A Global Optimization Algorithm (GOP) for Certain Classes of Nonconvex NLPs-I. Theory.Comput. Chem. Engng.,14, 1397.
Floudas, C.A. and V. Visweswaran (1993). Primal-Relaxed Dual Global Optimization Approach.JOTA,78, 187.
Fortuny-Amat, J. and B. McCarl (1981). A representation and Economic Interpretation of a Two-Level Programming problem.J. Oper. Res. Soc.,32, 783.
Hansen, P. B. Jaumard and G. Savard (1990). New Branching and Bounding Rules for Linear Bilevel Programming.SIAM J. Sei. Stat. Comp.
Haurie, A., G. Savard and D.J. White (1990). A Note on: An Efficient Point Algorithm For a Linear Two-Stage Optimization Problem.Oper. Res. Tech. Notes, 553.
Judice J.J. and A.M. Faustino (1992). A sequential LCP method for Bilevel Linear Programming.Annals Oper. Res.,34, 89.
Shimizu, K. and E. Aiyoshi, E (1981). A New Computational Method for Stackelberg and Min-Max Problems by Use of a Penalty Method.IEEE Trans, on Aut Control,AC-26, 460.
Tuy, H., A. Migdalas and P. Varbrand (1993a). A Global Optimization Approach for the Linear Two-Level Program.J. Global Opt.,3, 1.
Tuy, H., A. Migdalas and P. Varbrand (1993b). A Quasiconcave Minimization Method for Solving Linear Two-Level Programs.J. Global Opt.,4, 243.
White D.J. and G. Anandalingam (1993). A Penalty Function Approach for Solving Bi-Level Linear Programs.J. Global Opt.,3, 397.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Kluwer Academic Publishers
About this chapter
Cite this chapter
Visweswaran, V., Floudas, C.A., Ierapetritou, M.G., Pistikopoulos, E.N. (1996). A Decomposition-Based Global Optimization Approach for Solving Bilevel Linear and Quadratic Programs. In: Floudas, C.A., Pardalos, P.M. (eds) State of the Art in Global Optimization. Nonconvex Optimization and Its Applications, vol 7. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3437-8_10
Download citation
DOI: https://doi.org/10.1007/978-1-4613-3437-8_10
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-3439-2
Online ISBN: 978-1-4613-3437-8
eBook Packages: Springer Book Archive