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Convex Bilevel Programs

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Bilevel Programming Problems

Part of the book series: Energy Systems ((ENERGY))

Abstract

The task to find a best point within the set of optimal solutions of a convex optimization problem is called simple bilevel optimization problem. In general, a necessary optimality condition for a convex simple bilevel problem does not need to be sufficient. An adapted necessary and sufficient optimality condition is derived using tools from cone-convex optimization and a gradient type descent method is suggested which combines the use of a convex combination of both objective functions and projection onto the feasible set. In the second section, a similar algorithm is used to find a best point within the solutions of a variational inequality.

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Authors and Affiliations

  1. Fak. für Mathematik und Informatik, TU Bergakademie Freiberg, Freiberg, Sachsen, Germany

    Stephan Dempe

  2. Department of Systems and Industrial Engineering, Tecnologico de Monterrey (ITESM), Campus Monterrey, Av. Eugenio Garza Sada 2501 Sur, 64849, Monterrey, NL, Mexico

    Vyacheslav Kalashnikov

  3. Department of Social Modeling, Central Economics and Mathematics Institute (CEMI), Russian Academy of Sciences, Nakhimovsky pr. 47, Moscow, 117418, Russian Federation

    Vyacheslav Kalashnikov

  4. Department of Computing, Sumy State University, Rimsky-Korsakov st. 2, Sumy, 40007, Ukraine

    Vyacheslav Kalashnikov

  5. Department of Industrial Economics, Norwegian University of Science and Technology (NTNU), Trondheim, Norway

    Gerardo A. Pérez-Valdés

  6. Facultad Ciencias Físico-Mathemáticas (FCFM), Universidad Autónoma de Nuevo León (UANL), San Nicolás de los Garza, Mexico

    Nataliya Kalashnykova

Authors
  1. Stephan Dempe
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  2. Vyacheslav Kalashnikov
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  3. Gerardo A. Pérez-Valdés
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  4. Nataliya Kalashnykova
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Corresponding author

Correspondence to Stephan Dempe .

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© 2015 Springer-Verlag Berlin Heidelberg

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Dempe, S., Kalashnikov, V., Pérez-Valdés, G.A., Kalashnykova, N. (2015). Convex Bilevel Programs. In: Bilevel Programming Problems. Energy Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45827-3_4

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  • DOI: https://doi.org/10.1007/978-3-662-45827-3_4

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-45826-6

  • Online ISBN: 978-3-662-45827-3

  • eBook Packages: EnergyEnergy (R0)

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