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The Generalized Mangasarian-Fromowitz Constraint Qualification and Optimality Conditions for Bilevel Programs

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  • Stephan Dempe

    (Technische Universität Bergakademie Freiberg)

  • Alain B. Zemkoho

    (Technische Universität Bergakademie Freiberg)

Abstract

We consider the optimal value reformulation of the bilevel programming problem. It is shown that the Mangasarian-Fromowitz constraint qualification in terms of the basic generalized differentiation constructions of Mordukhovich, which is weaker than the one in terms of Clarke’s nonsmooth tools, fails without any restrictive assumption. Some weakened forms of this constraint qualification are then suggested, in order to derive Karush-Kuhn-Tucker type optimality conditions for the aforementioned problem. Considering the partial calmness, a new characterization is suggested and the link with the previous constraint qualifications is analyzed.

Suggested Citation

  • Stephan Dempe & Alain B. Zemkoho, 2011. "The Generalized Mangasarian-Fromowitz Constraint Qualification and Optimality Conditions for Bilevel Programs," Journal of Optimization Theory and Applications, Springer, vol. 148(1), pages 46-68, January.
    • Handle: RePEc:spr:joptap:v:148:y:2011:i:1:d:10.1007_s10957-010-9744-8
    DOI: 10.1007/s10957-010-9744-8
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    References listed on IDEAS

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    1. Boris S. Mordukhovich & Nguyen Mau Nam, 2005. "Variational Stability and Marginal Functions via Generalized Differentiation," Mathematics of Operations Research, INFORMS, vol. 30(4), pages 800-816, November.
    2. Stephan Dempe & Vyatcheslav V. Kalashnikov & Nataliya Kalashnykova, 2006. "Optimality conditions for bilevel programming problems," Springer Optimization and Its Applications, in: Stephan Dempe & Vyacheslav Kalashnikov (ed.), Optimization with Multivalued Mappings, pages 3-28, Springer.
    3. Stephan Dempe & Vyacheslav Kalashnikov (ed.), 2006. "Optimization with Multivalued Mappings," Springer Optimization and Its Applications, Springer, number 978-0-387-34221-4, July.
    4. S. Dempe & N. Dinh & J. Dutta, 2010. "Optimality Conditions for a Simple Convex Bilevel Programming Problem," Springer Optimization and Its Applications, in: Regina S. Burachik & Jen-Chih Yao (ed.), Variational Analysis and Generalized Differentiation in Optimization and Control, pages 149-161, Springer.
    5. A. Moldovan & L. Pellegrini, 2009. "On Regularity for Constrained Extremum Problems. Part 1: Sufficient Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 142(1), pages 147-163, July.
    6. Joydeep Dutta & Stephan Dempe, 2006. "Bilevel programming with convex lower level problems," Springer Optimization and Its Applications, in: Stephan Dempe & Vyacheslav Kalashnikov (ed.), Optimization with Multivalued Mappings, pages 51-71, Springer.
    7. J. J. Ye, 1998. "New Uniform Parametric Error Bounds," Journal of Optimization Theory and Applications, Springer, vol. 98(1), pages 197-219, July.
    8. A. Moldovan & L. Pellegrini, 2009. "On Regularity for Constrained Extremum Problems. Part 2: Necessary Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 142(1), pages 165-183, July.
    9. Jane J. Ye, 2006. "Constraint Qualifications and KKT Conditions for Bilevel Programming Problems," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 811-824, November.
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    Cited by:

    1. Alain B. Zemkoho & Shenglong Zhou, 2021. "Theoretical and numerical comparison of the Karush–Kuhn–Tucker and value function reformulations in bilevel optimization," Computational Optimization and Applications, Springer, vol. 78(2), pages 625-674, March.
    2. Lorenzo Lampariello & Simone Sagratella, 2015. "It is a matter of hierarchy: a Nash equilibrium problem perspective on bilevel programming," DIAG Technical Reports 2015-07, Department of Computer, Control and Management Engineering, Universita' degli Studi di Roma "La Sapienza".
    3. Polyxeni-M. Kleniati & Claire Adjiman, 2014. "Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part II: Convergence analysis and numerical results," Journal of Global Optimization, Springer, vol. 60(3), pages 459-481, November.
    4. Gaoxi Li & Xinmin Yang, 2021. "Convexification Method for Bilevel Programs with a Nonconvex Follower’s Problem," Journal of Optimization Theory and Applications, Springer, vol. 188(3), pages 724-743, March.
    5. Polyxeni-Margarita Kleniati & Claire Adjiman, 2014. "Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part I: Theoretical development," Journal of Global Optimization, Springer, vol. 60(3), pages 425-458, November.
    6. Jörg Fliege & Andrey Tin & Alain Zemkoho, 2021. "Gauss–Newton-type methods for bilevel optimization," Computational Optimization and Applications, Springer, vol. 78(3), pages 793-824, April.
    7. Gaoxi Li & Zhongping Wan, 2018. "On Bilevel Programs with a Convex Lower-Level Problem Violating Slater’s Constraint Qualification," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 820-837, December.
    8. S. Dempe & A. Zemkoho, 2012. "Bilevel road pricing: theoretical analysis and optimality conditions," Annals of Operations Research, Springer, vol. 196(1), pages 223-240, July.
    9. S. Dempe & N. Gadhi & A. B. Zemkoho, 2013. "New Optimality Conditions for the Semivectorial Bilevel Optimization Problem," Journal of Optimization Theory and Applications, Springer, vol. 157(1), pages 54-74, April.
    10. Lorenzo Lampariello & Simone Sagratella, 2017. "A Bridge Between Bilevel Programs and Nash Games," Journal of Optimization Theory and Applications, Springer, vol. 174(2), pages 613-635, August.

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