Browsing by Author "Henrion, René"
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- ItemAnalysis 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, ThomasWe 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.
- ItemChance constraints in PDE constrained optimization(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Farshbaf-Shaker, M. Hassan; Henrion, René; Hömberg, DietmarChance constraints represent a popular tool for finding decisions that enforce a robust satisfaction of random inequality systems in terms of probability. They are widely used in optimization problems subject to uncertain parameters as they arise in many engineering applications. Most structural results of chance constraints (e.g., closedness, convexity, Lipschitz continuity, differentiability etc.) have been formulated in a finite-dimensional setting. The aim of this paper is to generalize some of these well-known semi-continuity and convexity properties to a setting of control problems subject to (uniform) state chance constraints.
- ItemCondition number and eccentricity of a closed convex cone(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Henrion, René; Seeger, AlbertoWe discuss some extremality issues concerning the circumradius, the inradius, and the condition number of a closed convex cone in $mathbbR^n$. The condition number refers to the ratio between the circumradius and the inradius. We also study the eccentricity of a closed convex cone, which is a coefficient that measures to which extent the circumcenter differs from the incenter.
- ItemConditioning of linear-quadratic two-stage stochastic optimization problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Emich, Konstantin; Henrion, René; Römisch, WernerIn this paper a condition number for linear-quadratic two-stage stochastic optimization problems is introduced as the Lipschitz modulus of the multifunction assigning to a (discrete) probability distribution the solution set of the problem. Being the outer norm of the Mordukhovich coderivative of this multifunction, the condition number can be estimated from above explicitly in terms of the problem data by applying appropriate calculus rules. Here, a chain rule for the extended partial second-order subdifferential recently proved by Mordukhovich and Rockafellar plays a crucial role. The obtained results are illustrated for the example of two-stage stochastic optimization problems with simple recourse.
- ItemControlled polyhedral sweeping processes: Existence, stability, and optimality conditions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Henrion, René; Jourani, Abderrahim; Mordukhovich, Boris S.This paper is mainly devoted to the study of controlled sweeping processes with polyhedral moving sets in Hilbert spaces. Based on a detailed analysis of truncated Hausdorff distances between moving polyhedra, we derive new existence and uniqueness theorems for sweeping trajectories corresponding to various classes of control functions acting in moving sets. Then we establish quantitative stability results, which provide efficient estimates on the sweeping trajectory dependence on controls and initial values. Our final topic, accomplished in finite-dimensional state spaces, is deriving new necessary optimality and suboptimality conditions for sweeping control systems with endpoint constrains by using constructive discrete approximations.
- ItemConvexity of chance constraints with independent random variables(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Henrion, René; Strugarek, CyrilleWe 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.
- ItemCritical objective size and calmness modulus in linear programming(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Cánovas, Maria J.; Henrion, René; Parra, Juan; Toledo, F. JavierThis paper introduces the concept of critical objective size associated with a linear program in order to provide operative point-based formulas (only involving the nominal data, and not data in a neighborhood) for computing or estimating the calmness modulus of the optimal set (argmin) mapping under uniqueness of nominal optimal solution and perturbations of all coefficients. Our starting point is an upper bound on this modulus given in [4]. In this paper we prove that this upper bound is attained if and only if the norm of the objective function coefficient vector is less than or equal to the critical objective size. This concept also allows us to obtain operative lower bounds on the calmness modulus. We analyze in detail an illustrative example in order to xplore some strategies that can improve the referred upper and lower bounds.
- ItemDiscrepancy distances and scenario reduction in two-stage stochastic integer programming(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Henrion, René; Küchler, Christian; Römisch, WernerPolyhedral 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.
- ItemDynamic probabilistic constraints under continuous random distributions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) González Grandón, Tatiana; Henrion, René; Pérez-Aros, PedroThe paper investigates analytical properties of dynamic probabilistic constraints (chance constraints). The underlying random distribution is supposed to be continuous. In the first part, a general multistage model with decision rules depending on past observations of the random process is analyzed. Basic properties like (weak sequential) (semi-) continuity of the probability function or existence of solutions are studied. It turns out that the results differ significantly according to whether decision rules are embedded into Lebesgue or Sobolev spaces. In the second part, the simplest meaningful two-stage model with decision rules from L 2 is investigated. More specific properties like Lipschitz continuity and differentiability of the probability function are considered. Explicitly verifiable conditions for these properties are provided along with explicit gradient formulae in the Gaussian case. The application of such formulae in the context of necessary optimality conditions is discussed and a concrete identification of solutions presented.
- ItemAn enumerative formula for the spherical cap discrepancy(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Heitsch, Holger; Henrion, RenéThe spherical cap discrepancy is a widely used measure for how uniformly a sample of points on the sphere is distributed. Being hard to compute, this discrepancy measure is typically replaced by some lower or upper estimates when designing optimal sampling schemes for the uniform distribution on the sphere. In this paper, we provide a fully explicit, easy to implement enumerative formula for the spherical cap discrepancy. Not surprisingly, this formula is of combinatorial nature and, thus, its application is limited to spheres of small dimension and moderate sample sizes. Nonetheless, it may serve as a useful calibrating tool for testing the efficiency of sampling schemes and its explicit character might be useful also to establish necessary optimality conditions when minimizing the discrepancy with respect to a sample of given size.
- ItemFeasibility of nominations in stationary gas networks with random load(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Gotzes, Claudia; Heitsch, Holger; Henrion, René; Schultz, RüdigerThe paper considers the computation of the probability of feasible load constellations in a stationary gas network with uncertain demand. More precisely, a network with a single entry and several exits with uncertain loads is studied. Feasibility of a load constellation is understood in the sense of an existing flow meeting these loads along with given pressure bounds in the pipes. In a first step, feasibility of deterministic exit loads is characterized algebraically and these general conditions are specified to networks involving at most one cycle. This prerequisite is essential for determining probabilities in a stochastic setting when exit loads are assumed to follow some (joint) Gaussian distribution when modeling uncertain customer demand. The key of our approach is the application of the spheric-radial decomposition of Gaussian random vectors coupled with Quasi Monte-Carlo sampling. This approach requires an efficient algorithmic treatment of the mentioned algebraic relations moreover depending on a scalar parameter. Numerical results are illustrated for different network examples and demonstrate a clear superiority in terms of precision over simple generic Monte-Carlo sampling. They lead to fairly accurate probability values even for moderate sample size.
- ItemGeneralized gradients for probabilistic/robust (probust) constraints(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Ackooij, Wim van; Henrion, René; Pérez-Aros, PedroProbability functions are a powerful modelling tool when seeking to account for uncertainty in optimization problems. In practice, such uncertainty may result from different sources for which unequal information is available. A convenient combination with ideas from robust optimization then leads to probust functions, i.e., probability functions acting on generalized semi-infinite inequality systems. In this paper we employ the powerful variational tools developed by Boris Mordukhovich to study generalized differentiation of such probust functions. We also provide explicit outer estimates of the generalized subdifferentials in terms of nominal data.
- ItemA gradient formula for linear chance constraints under Gaussian distribution(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Henrion, René; Möller, AndrisWe provide an explicit gradient formula for linear chance constraints under a (possibly singular) multivariate Gaussian distribution. This formula allows one to reduce the calculus of gradients to the calculus of values of the same type of chance constraints (in smaller dimension and with different distribution parameters). This is an important aspect for the numerical solution of stochastic optimization problems because existing efficient codes for e.g., calculating singular Gaussian distributions or regular Gaussian probabilities of polyhedra can be employed to calculate gradients at the same time. Moreover, the precision of gradients can be controlled by that of function values which is a great advantage over using finite difference approximations. Finally, higher order derivatives are easily derived explicitly. The use of the obtained formula is illustrated for an example of a transportation network with stochastic demands.
- ItemGradient formulae for nonlinear probabilistic constraints with Gaussian and aussian-like distributions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) van Ackooij, Wim; Henrion, RenéProbabilistic constraints represent a major model of stochastic optimization. A possible approach for solving probabilistically constrained optimization problems consists in applying nonlinear programming methods. In order to do so, one has to provide sufficiently precise approximations for values and gradients of probability functions. For linear probabilistic constraints under Gaussian distribution this can be successfully done by analytically reducing these values and gradients to values of Gaussian distribution functions and computing the latter, for instance, by Genz’ code. For nonlinear models one may fall back on the spherical-radial decomposition of Gaussian random vectors and apply, for instance, Deák’s sampling scheme for the uniform distribution on the sphere in order to compute values of corresponding probability functions. The present paper demonstrates how the same sampling scheme can be used in order to simultaneously compute gradients of these probability functions. More precisely, we prove a formula representing these gradients in the Gaussian case as a certain integral over the sphere again. Later, the result is extended to alternative distributions with an emphasis on the multivariate Student (or T-) distribution.
- ItemInradius and circumradius of various convex cones arising in applications(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Henrion, René; Seeger, AlbertoThis note addresses the issue of computing the inradius and the circumradius of a convex cone in a Euclidean space. It deals also with the related problem of finding the incenter and the circumcenter of the cone. We work out various examples of convex cones arising in applications.
- ItemJoint dynamic probabilistic constraints with projected linear decision rules(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Guigues, Vincent; Henrion, RenéWe consider multistage stochastic linear optimization problems combining joint dynamic probabilistic constraints with hard constraints. We develop a method for projecting decision rules onto hard constraints of wait-and-see type. We establish the relation between the original (infinite dimensional) problem and approximating problems working with projections from different subclasses of decision policies. Considering the subclass of linear decision rules and a generalized linear model for the underlying stochastic process with noises that are Gaussian or truncated Gaussian, we show that the value and gradient of the objective and constraint functions of the approximating problems can be computed analytically.
- ItemJoint Model of Probabilistic-Robust (Probust) Constraints Applied to Gas Network Optimization(Singapore : Springer, 2020) Adelhütte, Dennis; Aßmann, Denis; Grandòn, Tatiana Gonzàlez; Gugat, Martin; Heitsch, Holger; Henrion, René; Liers, Frauke; Nitsche, Sabrina; Schultz, Rüdiger; Stingl, Michael; Wintergerst, DavidOptimization problems under uncertain conditions abound in many real-life applications. While solution approaches for probabilistic constraints are often developed in case the uncertainties can be assumed to follow a certain probability distribution, robust approaches are usually applied in case solutions are sought that are feasible for all realizations of uncertainties within some predefined uncertainty set. As many applications contain different types of uncertainties that require robust as well as probabilistic treatments, we deal with a class of joint probabilistic/robust constraints. Focusing on complex uncertain gas network optimization problems, we show the relevance of this class of problems for the task of maximizing free booked capacities in an algebraic model for a stationary gas network. We furthermore present approaches for finding their solution. Finally, we study the problem of controlling a transient system that is governed by the wave equation. The task consists in determining controls such that a certain robustness measure remains below some given upper bound with high probability.
- ItemJoint model of probabilistic-robust (probust) constraints with application to gas network optimization(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Adelhütte, Dennis; Aßmann, Denis; Grandón, Tatiana González; Gugat, Martin; Heitsch, Holger; Henrion, René; Liers, Frauke; Nitsche, Sabrina; Schultz, Rüdiger; Stingl, Michael; Wintergerst, DavidOptimization problems under uncertain conditions abound in many real-life applications. While solution approaches for probabilistic constraints are often developed in case the uncertainties can be assumed to follow a certain probability distribution, robust approaches are usually applied in case solutions are sought that are feasible for all realizations of uncertainties within some predefined uncertainty set. As many applications contain different types of uncertainties that require robust as well as probabilistic treatments, we introduce a class of joint probabilistic/robust constraints. Focusing on complex uncertain gas network optimization problems, we show the relevance of this class of problems for the task of maximizing free booked capacities in an algebraic model for a stationary gas network. We furthermore present approaches for finding their solution. Finally, we study the problem of controlling a transient system that is governed by the wave equation. The task consists in determining controls such that a certain robustness measure remains below some given upper bound with high probability.
- ItemA joint model of probabilistic/robust constraints for gas transport management in stationary networks(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Grandón, Tatiana González; Heitsch, Holger; Henrion, RenéWe present a novel mathematical algorithm to assist gas network operators in managing uncertainty, while increasing reliability of transmission and supply. As a result, we solve an optimization problem with a joint probabilistic constraint over an infinite system of random inequalities. Such models arise in the presence of uncertain parameters having partially stochastic and partially nonstochastic character. The application that drives this new approach is a stationary network with uncertain demand (which are stochastic due to the possibility of fitting statistical distributions based on historical measurements) and with uncertain roughness coefficients in the pipes (which are uncertain but non-stochastic due to a lack of attainable measurements). We study the sensitivity of local uncertainties in the roughness coefficients and their impact on a highly reliable network operation. In particular, we are going to answer the question, what is the maximum uncertainty that is allowed (shaping a maximal uncertainty set) around nominal roughness coefficients, such that random demands in a stationary gas network can be satisfied at given high probability level for no matter which realization of true roughness coefficients within the uncertainty set. One ends up with a constraint, which is probabilistic with respect to the load of gas and robust with respect to the roughness coefficients. We demonstrate how such constraints can be dealt with in the framework of the so-called spheric-radial decomposition of multivariate Gaussian distributions. The numerical solution of a corresponding optimization problem is illustrated. The results might assist the network operator with the implementation of cost-intensive roughness measurements.
- ItemLipschitz lower semicontinuity moduli for linear inequality systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Cánovas, Maria Josefa; Gisbert, María Jesús; Henrion, René; Parra, JuanThe paper is focussed on the Lipschitz lower semicontinuity of the feasible set mapping for linear (finite and infinite) inequality systems in three different perturbation frameworks: full, right-hand side and left-hand side perturbations. Inspired by [14], we introduce the Lipschitz lower semicontinuity-star as an intermediate notion between the Lipschitz lower semicontinuity and the well-known Aubin property. We provide explicit point-based formulae for the moduli (best constants) of all three Lipschitz properties in all three perturbation settings.
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