Abstract
Bilevel programming involves two optimization problems where the constraint region of the upper level problem is implicitly determined by another optimization problem. In this paper we focus on bilevel problems over polyhedra with upper level constraints involving lower level variables. On the one hand, under the uniqueness of the optimal solution of the lower level problem, we prove that the fact that the objective functions of both levels are quasiconcave characterizes the property of the existence of an extreme point of the polyhedron defined by the whole set of constraints which is an optimal solution of the bilevel problem. An example is used to show that this property is in general violated if the optimal solution of the lower level problem is not unique. On the other hand, if the lower level objective function is not quasiconcave but convex quadratic, assuming the optimistic approach we prove that the optimal solution is attained at an extreme point of an ‘enlarged’ polyhedron.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Audet C., Haddad J., Savard G.: A note on the definition of a linear bilevel programming solution. Appl. Math. Comput. 181, 351–355 (2006)
Bard J.: Practical Bilevel Optimization. Algorithms and applications. Kluwer Academic Publishers, Dordrecht (1998)
Calvete H., Galé C.: On the quasiconcave bilevel programming problem. J. Optim. Theory Appl. 98(3), 613–622 (1998)
Calvete H., Galé C.: Linear bilevel multi-follower programming with independent followers. J. Global Optim. 39(3), 409–417 (2007)
Calvete, H., Galé, C., Mateo, P.: A new approach for solving linear bilevel problems using genetic algorithms. To appear in Eur. J. Oper. Res. (2007). doi:10.1016/j.ejor.2007.03.034
Chinchuluun, A., Huang, H., Pardalos, P. : Multilevel (hierarchical) optimization: complexity issues, optimality conditions, algorithms. In: Gao, D., Sherali, H. (eds.) Advances in Applied Mathematics and Global Optimization, pp. 197–221. Springer, New York (2009)
Dantzig G., Folkman J., Shapiro N.: On the continuity of the minimum set of a continuous function. J. Math. Anal. Appl. 17, 519–548 (1967)
Dempe S.: Foundations of Bilevel Programming. Kluwer Academic Publishers, Dordrecht Boston London (2002)
Dempe S.: Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52, 333–359 (2003)
Dempe S., Dutta J.: Is bilevel programming a special case of a mathematical program with complementary constraints? Technical report, Department of Mathematics and Computer Science. TU Bergakademie, Freiberg (2008)
Gauvin J.: A necessary and sufficient regularity condition to have multipliers in nonconvex programming. Math. Program. 12, 136–139 (1977)
Martos B.: The direct power of adjacent vertex programming problems. Manag. Sci. 12(3), 241–252 (1965)
Mersha A., Dempe S.: Linear bilevel programming with upper level constraints depending on the lower level solution. Appl. Math. Comput. 180, 247–254 (2006)
Migdalas, A., Pardalos, P.M. (eds): Hierarchical and bilevel programming. J. Global Optim. 8(3), 209–215 (1996)
Migdalas, A., Pardalos, P., Värbrand, P. (eds): Multilevel Optimization Algorithms and Applications. Kluwer Academic Publishers, Dordrecht (1998)
Rockafellar R.: Convex Analysis. Princeton University Press, Princeton, New Jersey (1970)
Savard, G.: Contribution à la programmation mathématique à deux niveaux.Ph.D. thesis, Ecole Polytechnique de Montréal, Université de Montréal, Montréal, QC, Canada (1989)
Shimizu K., Ishizuka Y., Bard J.: Nondifferentiable and Two-level Mathematical Programming. Kluwer Academic Publishers, Boston (1997)
Vicente L., Calamai P.: Bilevel and multilevel programming: a bibliography review. J. Global Optim. 5, 291–306 (1994)
Xu Z.: Deriving the properties of linear bilevel programming via a penalty function approach. J. Optim. Theory Appl. 103(2), 441–456 (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research work has been partially funded by the Spanish Ministry of Education and Science under grant MTM2007-66893.
Rights and permissions
About this article
Cite this article
Calvete, H.I., Galé, C., Dempe, S. et al. Bilevel problems over polyhedra with extreme point optimal solutions. J Glob Optim 53, 573–586 (2012). https://doi.org/10.1007/s10898-011-9762-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-011-9762-6