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Author: Henrion, René × Author: Outrata, Jiří × Type: report ×

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Now showing 1 - 5 of 5
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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.
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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 calculating the normal cone to a finite union of convex polyhedra
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Henrion, René; Outrata, Jiří
    The paper provides formulae for calculating the limiting normal cone introduced by Mordukhovich to a finite union of convex polyhedra. In the first part, special cases of independent interest are considered (almost disjoint cones, half spaces, orthants). The second part focusses on unions of general polyhedra. Due to the local nature of the normal cone, one may restrict considerations without loss of generality to finite unions of polyhedral cones. First, an explicit formula for the normal cone is provided in the situation of two cones. An algorithmic approach is presented along with a refined, more efficient formula. Afterwards, a general formula for the union of N cones is derived. Finally, an application to the stability analysis of a special type of probabilistic constraints is provided.
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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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    Some remarks on stability of generalized equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Henrion, René; Kruger, Alexander; Outrata, Jiří
    The paper concerns the computation of the graphical derivative and the regular (Fréchet) coderivative of the solution map to a class of generalized equations, where the multi-valued term amounts to the regular normal cone to a (possibly nonconvex) set given by C2 inequalities. Instead of the Linear Independence qualification condition, standardly used in this context, one assumes a combination of the Mangasarian-Fromovitz and the Constant Rank qualification conditions. On the basis of the obtained generalized derivatives, new optimality conditions for a class of mathematical programs with equilibrium constrains are derived, and a workable characterization of the isolated calmness of the considered solution map is provided.