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KKT Reformulation and Necessary Conditions for Optimality in Nonsmooth Bilevel Optimization

Stephan Dempe (dempe***at***tu-freiberg.de)
Alain B. Zemkoho (a.zemkoho***at***bham.ac.uk)

Abstract: For a long time, the bilevel programming problem has essentially been considered as a special case of mathematical programs with equilibrium constraints (MPECs), in particular when the so-called KKT reformulation is in question. Recently though, this widespread believe was shown to be false in general. In this paper, other aspects of the difference between both problems are revealed as we consider the KKT approach for the nonsmooth bilevel program. In fact, we discover that the main difficulty in handling the latter problem is not where one would usually expect, that is, in the complementarity constraints. It rather turns out that the new inclusion (constraint) which appears as a consequence of the partial subdifferential of the lower-level Lagrangian (PSLLL) places the KKT reformulation of the bilevel program in a new class of mathematical program with both set-valued and complementarity constraints. We attempt here to establish the link between this problem and the standard optimistic bilevel program. Moreover, we discuss possible natural extensions for C-, M-, and S-stationarity concepts. Most of the results rely on a coderivative estimate for the PSLLL that we also provide in this paper.

Keywords: nonsmooth bilevel optimization, parametric optimization, coderivative, variational analysis, constraint qualifications, stationarity conditions

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization )

Citation: Preprint 2013-02, Department of Mathematics and Computer Science, TU Bergakademie Freiberg, 2013

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Entry Submitted: 06/07/2013
Entry Accepted: 06/07/2013
Entry Last Modified: 06/29/2014

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