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Necessary optimality conditions for optimistic bilevel programming problems using set-valued programming

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  • Stephan Dempe
  • Maria Pilecka

Abstract

In this paper we adapt the main results from Amahroq and Gadhi (J Glob Optim 21:435–443, 2001 ) for a general set-valued optimization problem to an optimistic bilevel programming problem as an optimization problem with implicitly given set-valued constraint. Since this constraint is assumed to be upper but not lower semicontinuous in the sense of Berge, we need to deal with a lower semicontinuous distance function to this mapping. In order to approximate the gradient of the distance function, we introduce a new concept for a directional convexificator. Some calculus rules for this new tool are adapted and several properties are characterized. The main result presents optimality conditions for an optimistic bilevel programming problem using a convexificator constructed with the aid of the directional convexificator. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Stephan Dempe & Maria Pilecka, 2015. "Necessary optimality conditions for optimistic bilevel programming problems using set-valued programming," Journal of Global Optimization, Springer, vol. 61(4), pages 769-788, April.
    • Handle: RePEc:spr:jglopt:v:61:y:2015:i:4:p:769-788
    DOI: 10.1007/s10898-014-0200-4
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    References listed on IDEAS

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    1. 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.
    2. V. Jeyakumar & D.T. LUC, 2008. "Nonsmooth Vector Functions and Continuous Optimization," Springer Optimization and Its Applications, Springer, number 978-0-387-73717-1, June.
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    Cited by:

    1. Leonardo Lozano & J. Cole Smith, 2017. "A Value-Function-Based Exact Approach for the Bilevel Mixed-Integer Programming Problem," Operations Research, INFORMS, vol. 65(3), pages 768-786, June.
    2. Maryam Esmaeili & Habibe Sadeghi, 2018. "An Investigation of the Optimistic Solution to the Linear Trilevel Programming Problem," Mathematics, MDPI, vol. 6(10), pages 1-11, September.
    3. S. Dempe & S. Franke, 2016. "On the solution of convex bilevel optimization problems," Computational Optimization and Applications, Springer, vol. 63(3), pages 685-703, April.

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