Abstract
A new branch and bound method is proposed for the Bilevel Linear Programming based on a transformation of the problem into a linear program with an additional reverse convex constraint. The method exploits the separated non-convexity and a monotonic property of the reverse convex constraint. Computational experiments are reported which show the efficiency of the approach for problems in which the matrix A2 is substantially smaller than the total number of variables.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J.F. Bard and J.E. Falk: 1982, ‘An explicit solution to the multilevel programming problem’, Comput. & Ops. Res. 9, 77–100.
J.F. Bard and J.T. Moore: 1990, ‘A branch and bound algorithm for the bilevel programming problem, SIAM J. Statist. Comput. 11, 281–292.
O. Ben-Ayed and C.E. Blair: 1990, ‘Computational difficulties of bilevel linear programming, Operations Research, 38, 556–559.
J.E. Falk: 1973, ‘A Linear Max-Min Problem’, Mathematical Programming, 5, 169–188.
P. Hansen, B. Jaumard and G. Savard: 1992, ‘New branch-and-bound rules for linear bilinear programming’, SIAM J. Sci. Stat. Comput., 13, 1194–1217.
R. Horst and H. Tuy: 1993, ‘Global Optimization (Deterministic Approaches), Second Edition, Springer-Verlag, Berlin, New York.
J.J. Judice and A.M. Faustino: 1988, ‘The solution of the linear bilevel programming problem by using the linear complementarity problem, Investigacao Oper., 8, 77–95.
C.H. Papadimitriou and K. Steiglitz: 1982, Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, New Jersey.
R.T. Rockafellar: 1970, Convex Analysis, Princeton University Press.
H. Thy: 1991, `Effect of the subdivision strategy on convergence and efficiency of some globaloptimization algorithms’, Journal of Global Optimization, 1, 23–36.
H. Thy: 1992, ‘On nonconvex optimization problems with separated nonconvex variables’, Journal of Global Optimization, 2, 133–144.
H. Thy: 1994, ‘Introduction to Global Optimization’, GERAD G-9404, Ecole Polytechnique de Montréal.
H. Thy: 1995, ‘D.C. Optimization: Theory, Methods and Algorithms’ in eds. R. Horst and P.M. Pardalos Handbook on Global Optimization, Kluwer Academic Publishers, 149–216.
H. Thy, A.Migdalas and P. Värbrand: 1992, ‘A Global Optimization Approach for the Linear Two-Level Program’, Journal of Global Optimization, 3, 1–23.
H. Thy, A.Migdalas and P. Värbrand: 1994, ‘A Quasiconcave Minimization Method for Solving Linear Two-Level Programs’, Journal of Global Optimization, 4, 243–263
U.P. Wen and S.T. Hsu: 1991, ‘Linear Bilevel Programming — A Review’, J. Opl. Res. Soc., 42, 125–133.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Kluwer Academic Publishers
About this chapter
Cite this chapter
Tuy, H., Ghannadan, S. (1998). A New Branch and Bound Method for Bilevel Linear Programs. In: Migdalas, A., Pardalos, P.M., Värbrand, P. (eds) Multilevel Optimization: Algorithms and Applications. Nonconvex Optimization and Its Applications, vol 20. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0307-7_10
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0307-7_10
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7989-8
Online ISBN: 978-1-4613-0307-7
eBook Packages: Springer Book Archive