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search.filters.applied.f.access: openAccess × Author: Henrion, René × Author: Surowiec, Thomas × Type: Text × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik ×

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    On regular coderivatives in parametric equilibria with non-unique multipliers
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Henrion, René; Outrata, Jiří; Surowiec, Thomas
    This paper deals with the computation of regular coderivatives of solution maps associated with a frequently arising class of generalized equations. The constraint sets are given by (not necessarily convex) inequalities, and we do not assume linear independence of gradients to active constraints. The achieved results enable us to state several versions of sharp necessary optimality conditions in optimization problems with equilibria governed by such generalized equations. The advantages are illustrated by means of examples.
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    Analysis of M-stationary points to an EPEC modeling oligopolistic competition in an electricity spot market
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Henrion, René; Outrata, Jií̌; Surowiec, Thomas
    We consider an equilibrium problem with equilibrium constraints (EPEC) as it arises from modeling competition in an electricity spot market (under ISO regulation). For a characterization of equilibrium solutions, so-called M-stationarity conditions are derived. This requires a structural analysis of the problem first (constraint qualifications, strong regularity). Second, the calmness property of a certain multifunction has to be verified in order to justify M-stationarity. Third, for stating the stationarity conditions, the co-derivative of a normal cone mapping has to be calculated. Finally, the obtained necessary conditions are made fully explicit in terms of the problem data for one typical constellation. A simple two-settlements example serves as an illustration.
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    On calmness conditions in convex bilevel programming
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Henrion, René; Surowiec, Thomas
    In this article we compare two different calmness conditions which are widely used in the literature on bilevel programming and on mathematical programs with equilibrium constraints. In order to do so, we consider convex bilevel programming as a kind of intersection between both research areas. The so-called partial calmness concept is based on the function value approach for describing the lower level solution set. Alternatively, calmness in the sense of multifunctions may be considered for perturbations of the generalized equation representing the same lower level solution set. Both concepts allow to derive first order necessary optimality conditions via tools of generalized differentiation introduced by Mordukhovich. They are very different, however, concerning their range of applicability and the form of optimality conditions obtained. The results of this paper seem to suggest that partial calmness is considerably more restrictive than calmness of the perturbed generalized equation. This fact is also illustrated by means of a dicretized obstacle control problem.
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    Strong stationary solutions to equilibrium problems with equilibrium constraints with applications to an electricity spot market model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Henrion, René; Outrata, Jiří; Surowiec, Thomas
    Literaturverz. S. 26 In this paper, we consider the characterization of strong stationary solutions to equilibrium problems with equilibrium constraints (EPECs). Assuming that the underlying generalized equation satisfies strong regularity in the sense of Robinson, an explicit multiplier-based stationarity condition can be derived. This is applied then to an equilibrium model arising from ISO-regulated electricity spot markets.
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    On the co-derivative of normal cone mappings to inequality systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Henrion, René; Outrata, Jiří; Surowiec, Thomas
    The paper deals with co-derivative formulae for normal cone mappings to smooth inequality systems. Both, the regular (Linear Independence Constraint Qualification satisfied) and nonregular (Mangasarian-Fromovitz Constraint Qualification satisfied) case are considered. A major part of the results relies on general transformation formulae previously obtained by Mordukhovich and Outrata. This allows to derive exact formulae for general smooth, regular and polyhedral, possibly nonregular systems. In the nonregular, nonpolyhedral case a generalized transformation formula by Mordukhovich and Outrata applies, however a major difficulty consists in checking a calmness condition of a certain multivalued mapping. The paper provides a translation of this condition in terms of much easier to verify constraint qualifications. A series of examples illustrates the use and comparison of the presented formulae.