Hostname: page-component-89b8bd64d-ktprf Total loading time: 0 Render date: 2026-05-07T08:08:23.535Z Has data issue: false hasContentIssue false
  • English
  • Français

A regularization method for ill-posed bilevel optimization problems

Published online by Cambridge University Press:  01 July 2006

Maitine Bergounioux  and
Mounir Haddou
Maitine Bergounioux
Affiliation:
MAPMO-UMR 6628, Fédération Denis Poisson, Université d'Orléans, BP 6759, 45067 Orléans Cedex 2, France; maitine.bergounioux@univ-orleans.fr, mounir.haddou@univ-orleans.fr
Mounir Haddou
Affiliation:
MAPMO-UMR 6628, Fédération Denis Poisson, Université d'Orléans, BP 6759, 45067 Orléans Cedex 2, France; maitine.bergounioux@univ-orleans.fr, mounir.haddou@univ-orleans.fr
Get access

Abstract

We present a regularization method to approach a solution of the pessimistic formulation of ill-posed bilevel problems. This allows to overcome the difficulty arising from the non uniqueness of the lower level problems solutions and responses. We prove existence of approximated solutions, give convergence result using Hoffman-like assumptions. We end with objective value error estimates.

Information

Type
Research Article
Information
RAIRO - Operations Research , Volume 40 , Issue 1 , January 2006 , pp. 19 - 35
Copyright
© EDP Sciences, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

A. Auslender and M. Teboulle, Asymptotic cones and functions in optimization and variational inequalities. Springer Monographs in Mathematics. Springer-Verlag, New York (2003).
Azé, D. and Corvellec, J.N., Characterizations of error bounds for lower semicontinuous functions on metric spaces. ESAIM: COCV 10 (2004) 409–425. CrossRef
Azé, D. and Rahmouni, A., On primal-dual stability in convex optimization. J. Convex Anal. 3 (1996) 309–327.
J.F. Bonnans and A. Shapiro, Perturbation analysis of optimization problems. Springer Series in Operations Research, Springer-Verlag, New York (2000).
Brotcorne, L., Labbé, M., Marcotte, P. and Savard, G., A bilevel model for toll optimization on a multicommodity transportation network. Transportation Sci. 35 (2001) 1–14. CrossRef
S. Dempe, Foundations of bilevel programming, in Nonconvex optimization and its applications. Kluwer Acad. Publ., Dordrecht (2002).
M. Labbé, P. Marcotte and G. Savard, On a class of bilevel programs, in Nonlinear optimization and related topics, (Erice, 1998). Appl. Optim. 36 (2000) 183–206.
Lewis, A.S. and Pang, J.-S., Error bounds for convex inequality systems, in Generalized convexity, generalized monotonicity: recent results (Luminy, 1996). Nonconvex Optim. Appl. 27 (1998) 75–110. CrossRef
Abadie's, W. Li constraint qualification, metric regularity, and error bounds for differentiable convex inequalities. SIAM J. Optim. 7 (1997) 966–978.
Loridan, P. and Morgan, J., New results on approximate solutions in two-level optimization. Optimization 20 (1989) 819–836. CrossRef
Loridan, P. and Morgan, J., Regularizations for two-level optimization problems. Advances in optimization. Lect. Notes Econ. Math. 382 (1992) 239–255. CrossRef
P. Marcotte and G. Savard, A bilevel programming approach to Price Setting in: Decision and Control in Management Science, in Essays in Honor of Alain Haurie, edited by G. Zaccour. Kluwer Academic Publishers (2002) 97–117.
K.F. Ng and X.Y. Zheng, Global error bounds with fractional exponents. Math. Program. Ser. B 88 (2000) 357–370.
Nguyen, T.Q., Bouhtou, M. and Lutton, J.-L., D.C. approach to bilevel bilinear programming problem: application in telecommunication pricing, in Optimization and optimal control (Ulaanbaatar, 2002). Ser. Comput. Oper. Res. 1 (2003) 211–231. CrossRef
Z. Wu and J. Ye, On error bounds for lower semicontinuous functions. Math. Program. Ser. A (2002).
Zhao, J., The lower semicontinuity of optimal solution sets. J. Math. Anal. Appl. 207 (1997) 240–254. CrossRef