Abstract
In this paper, we are concerned with a bilevel optimization problem \(\left( P\right) \). Reducing it into a one-level nonlinear and nonsmooth program using an exact separation theorem we give necessary optimality conditions in terms of Fréchet subdifferentials and Fréchet normal cones.
Similar content being viewed by others
References
Babahadda, H., Gadhi, N.: Necessary optimality conditions for bilevel optimization problems using convexificators. J. Glob. Optim. 34, 535–549 (2006)
Bard, J.F.: Some properities of the bilevel programming problem. J. Optim. Theory Appl. 68, 371–378 (1991)
Dempe, S.: A necessary and a sufficient optimality condition for bilevel programming problem. Optimization 25, 341–354 (1992)
Dempe, S., Gadhi, N.: Necessary optimality conditions for bilevel set optimization problems. J. Glob. Optim. 39, 529–542 (2007)
Dempe, S., Gadhi, N.: A new equivalent single-level problem for bilevel problems. Optimization 63(5), 789–798 (2014)
Kruger, A.: Ya, On Fréchet subdifferentials. J. Math. Sci. 116(3), 3325–3358 (2003)
Mangasarian, O.L., Fromovitz, S.: The Fritz John necessary optimality conditions in the presence of equality and inequality constraints. J. Math. Anal. Appl. 17, 37–47 (1967)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Grundlehren Series (Fundamental Principles of Mathematical Sciences), vol. 330. Springer, Berlin (2006)
Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysis in Asplund spaces. Trans. Am. Math. Soc. 348, 1235–1280 (1996)
Outrata, J.V.: On necessary optimality conditions for Stackelberg problems. J. Optim. Theory Appl. 76, 306–320 (1993)
Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33, 9–27 (1995)
Zhang, R.: Problems of hierarchical optimization in finite dimensions. SIAM J. Optim. 4, 521–536 (1993)
Zheng, X.Y., Yang, Z., Zou, J.: Exact separation theorem for closed sets in Asplund spaces. Optimization 66(7), 1065–1077 (2017)
Acknowledgements
This work has been supported by the Alexander-von-Humboldt foundation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Gadhi, N., Dempe, S. & El Idrissi, M. New optimality conditions for bilevel programs by using an exact separation principle. Optim Lett 14, 1381–1392 (2020). https://doi.org/10.1007/s11590-019-01443-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-019-01443-8