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

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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.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Technical University Bergakademie Freiberg, Freiberg, Germany

    Stephan Dempe & Maria Pilecka

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  1. Stephan Dempe
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  2. Maria Pilecka
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Correspondence to Maria Pilecka.

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Dedicated to the memory of Vladimir F. Demyanov.

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Dempe, S., Pilecka, M. Necessary optimality conditions for optimistic bilevel programming problems using set-valued programming. J Glob Optim 61, 769–788 (2015). https://doi.org/10.1007/s10898-014-0200-4

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