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search.filters.applied.f.access: openAccess × Author: Henrion, René × End date: 2009 × Start date: 2006 × WGL Institute: WIAS ×

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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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    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 M-stationary points for a stochastic equilibrium problem under equilibrium constraints in electricity spot market modeling
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Henrion, René; Römisch, Werner
    Modeling several competitive leaders and followers acting in an electricity market leads to coupled systems of mathematical programs with equilibrium constraints, called equilibrium problems with equilibrium constraints (EPECs). We consider a simplified model for competition in electricity markets under uncertainty of demand in an electricity network as a (stochastic) multi-leader-follower game. First order necessary conditions are developed for the corresponding stochastic EPEC based on a result of Outrata [17]. For applying the general result an explicit representation of the co-derivative of the normal cone mapping to a polyhedron is derived (Proposition 3.2). Later the co-derivative formula is used for verifying constraint qualifications and for identifying M-stationary solutions of the stochastic EPEC if the demand is represented by a finite number of scenarios.
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    Discrepancy distances and scenario reduction in two-stage stochastic integer programming
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Henrion, René; Küchler, Christian; Römisch, Werner
    Polyhedral discrepancies are relevant for the quantitative stability of mixed-integer two-stage and chance constrained stochastic programs. We study the problem of optimal scenario reduction for a discrete probability distribution with respect to certain polyhedral discrepancies and develop algorithms for determining the optimally reduced distribution approximately. Encouraging numerical experience for optimal scenario reduction is provided.
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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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    Convexity of chance constraints with independent random variables
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Henrion, René; Strugarek, Cyrille
    We investigate the convexity of chance constraints with independent random variables. It will be shown, how concavity properties of the mapping related to the decision vector have to be combined with a suitable property of decrease for the marginal densities in order to arrive at convexity of the feasible set for large enough probability levels. It turns out that the required decrease can be verified for most prominent density functions. The results are applied then, to derive convexity of linear chance constraints with normally distributed stochastic coefficients when assuming independence of the rows of the coefficient matrix.
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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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    Uniform boundedness of norms of convex and nonconvex processes
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Henrion, René; Seeger, Alberto
    The lower limit of a sequence of closed convex processes is again a closed convex process. In this note we prove the following uniform boundedness principle: if the lower limit is nonempty-valued everywhere, then, starting from a certain index, the given sequence is uniformly norm-bounded. As shown with an example, the uniform boundedness principle is not true if one drops convexity. By way of illustration, we consider an application to the controllability analysis of differential inclusions.
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    Stability and sensitivity of optimization problems with first order stochastic dominance constraints
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Dentcheva, Darinka; Henrion, René; Ruszczynski, Andrzej
    We analyze the stability and sensitivity of stochastic optimization problems with stochastic dominance constraints of first order. We consider general perturbations of the underlying probability measures in the space of regular measures equipped with a suitable discrepancy distance. We show that the graph of the feasible set mapping is closed under rather general assumptions. We obtain conditions for the continuity of the optimal value and upper-semicontinuity of the optimal solutions, as well as quantitative stability estimates of Lipschitz type. Furthermore, we analyze the sensitivity of the optimal value and obtain upper and lower bounds for the directional derivatives of the optimal value. The estimates are formulated in terms of the dual utility functions associated with the dominance constraints.
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    Scenario reduction in stochastic programming with respect to discrepancy distances
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Henrion, René; Küchler, Christian; Römisch, Werner
    Discrete approximations to chance constraints and mixed-integertwo-stage stochastic programs require moderately sized scenario sets. The relevant distances of (multivariate) probability distributions for deriving quantitative stability results for such stochastic programs are B-discrepancies, where the class B of Borel sets depends on their structural properties. Hence, the optimal scenario reduction problem for such models is stated with respect to B-discrepancies. In this paper, upper and lower bounds, and some explicit solutions for optimal scenario reduction problems are derived. In addition, we develop heuristic algorithms for determining nearly optimally reduced probability measures, discuss the case of the cell discrepancy (or Kolmogorov metric) in some detail and provide some numerical experience.