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Optimality conditions for bilevel programming problems

In: Optimization with Multivalued Mappings

Author

Listed:
  • Stephan Dempe

    (Technical University Bergakademie Freiberg)

  • Vyatcheslav V. Kalashnikov

    (Centro de Calidad ITESM)

  • Nataliya Kalashnykova

    (Universidad Autónoma de Nuevo León)

Abstract

Summary Focus in the paper is on optimality conditions for bilevel programming problems. We start with a general condition using tangent cones of the feasible set of the bilevel programming problem to derive such conditions for the optimistic bilevel problem. More precise conditions are obtained if the tangent cone possesses an explicit description as it is possible in the case of linear lower level problems. If the optimal solution of the lower level problem is a PC 1-function, sufficient conditions for a global optimal solution of the optimistic bilevel problem can be formulated. In the second part of the paper relations of the bilevel programming problem to set-valued optimization problems and to mathematical programs with equilibrium constraints are given which can also be used to formulate optimality conditions for the original problem. Finally, a variational inequality approach is described which works well when the involved functions are monotone. It consists in a variational re-formulation of the optimality conditions and looking for a solution of the thus obtained variational inequality among the points satisfying the initial constraints. A penalty function technique is applied to get a sequence of approximate solutions converging to a solution of the original problem with monotone operators.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:spochp:978-0-387-34221-4_1
    DOI: 10.1007/0-387-34221-4_1
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    Citations

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    Cited by:

    1. Hacopian Dolatabadi, Sarineh & Latify, Mohammad Amin & Karshenas, Hamidreza & Sharifi, Alimorad, 2024. "Demand response mechanisms: A new debate on internalizing power generation sector negative technical spillovers," Energy, Elsevier, vol. 301(C).
    2. Rihab Said & Maha Elarbi & Slim Bechikh & Lamjed Ben Said, 2022. "Solving combinatorial bi-level optimization problems using multiple populations and migration schemes," Operational Research, Springer, vol. 22(3), pages 1697-1735, July.
    3. 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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