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Optimality Conditions for a Simple Convex Bilevel Programming Problem

In: Variational Analysis and Generalized Differentiation in Optimization and Control

Author

Listed:
  • S. Dempe

    (TU Bergakademie Freiberg)

  • N. Dinh

    (International University, Vietnam National University of Ho Chi Minh city)

  • J. Dutta

    (Indian Institute of Technology)

Abstract

The problem to find a best solution within the set of optimal solutions of a convex optimization problem is modeled as a bilevel programming problem. It is shown that regularity conditions like Slater’s constraint qualification are never satisfied for this problem. If the lower-level problem is replaced with its (necessary and sufficient) optimality conditions, it is possible to derive a necessary optimality condition for the resulting problem. An example is used to show that this condition in not sufficient even if the initial problem is a convex one. If the lower-level problem is replaced using its optimal value, it is possible to obtain an optimality condition that is both necessary and sufficient in the convex case.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:spochp:978-1-4419-0437-9_7
    DOI: 10.1007/978-1-4419-0437-9_7
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    Citations

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    Cited by:

    1. Francesco Caruso & M. Beatrice Lignola & Jacqueline Morgan, 2020. "Regularization and Approximation Methods in Stackelberg Games and Bilevel Optimization," Springer Optimization and Its Applications, in: Stephan Dempe & Alain Zemkoho (ed.), Bilevel Optimization, chapter 0, pages 77-138, Springer.
    2. Francesco Cesarone & Lorenzo Lampariello & Davide Merolla & Jacopo Maria Ricci & Simone Sagratella & Valerio Giuseppe Sasso, 2023. "A bilevel approach to ESG multi-portfolio selection," Computational Management Science, Springer, vol. 20(1), pages 1-23, December.
    3. Lampariello, Lorenzo & Neumann, Christoph & Ricci, Jacopo M. & Sagratella, Simone & Stein, Oliver, 2021. "Equilibrium selection for multi-portfolio optimization," European Journal of Operational Research, Elsevier, vol. 295(1), pages 363-373.
    4. 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.

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