TY - JOUR A1 - Hantoute, Abderrahim A1 - Henrion, Rene A1 - Perez-Aros, Pedro T1 - Subdifferential characterization of probability functions under Gaussian distribution N2 - Probability functions figure prominently in optimization problems of engineering. They may be nonsmooth even if all input data are smooth. This fact motivates the consideration of subdifferentials for such typically just continuous functions. The aim of this paper is to provide subdifferential formulae of such functions in the case of Gaussian distributions for possibly infinite-dimensional decision variables and nonsmooth (locally Lipschitzian) input data. These formulae are based on the spheric-radial decomposition of Gaussian random vectors on the one hand and on a cone of directions of moderate growth on the other. By successively adding additional hypotheses, conditions are satisfied under which the probability function is locally Lipschitzian or even differentiable. Y1 - 2018 U6 - https://doi.org/10.1007/s10107-018-1237-9 ER - TY - JOUR A1 - Farshbaf-Shaker, M. Hassan A1 - Henrion, Rene A1 - Hömberg, Dietmar T1 - Properties of Chance Constraints in Infinite Dimensions with an Application to PDE Constrained Optimization JF - Set-Valued and Variational Analysis N2 - Chance constraints represent a popular tool for finding decisions that enforce the 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 finite dimensions. The aim of this paper is to generalize some of these well-known semi-continuity and convexity properties as well as a stability result to an infinite dimensional setting. The abstract results are applied to a simple PDE constrained control problem subject to (uniform) state chance constraints. KW - Chance constraints KW - Probabilistic constraints KW - PDE constrained optimization Y1 - 2018 U6 - https://doi.org/doi:10.1007/s11228-017-0452-5 ER - TY - JOUR A1 - Gonzalez Grandon, Tatiana A1 - Heitsch, Holger A1 - Henrion, Rene T1 - A joint model of probabilistic/robust constraints for gas transport management in stationary networks JF - Computational Management Science N2 - We 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 non-stochastic 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. KW - chance constraint KW - robust constraint KW - uncertainty set KW - spheric-radial decomposition Y1 - 2017 U6 - https://doi.org/10.1007/s10287-017-0284-7 VL - 14 SP - 443 EP - 460 ER - TY - JOUR A1 - Adam, Lukas A1 - Henrion, Rene A1 - Outrata, Jiri T1 - On M-stationarity conditions in MPECs and the associated qualification conditions JF - Mathematical Programming N2 - Depending on whether a mathematical program with equilibrium constraints (MPEC) is considered in its original or its enhanced (via KKT conditions) form, the assumed qualification conditions as well as the derived necessary optimality conditions may differ significantly. In this paper, we study this issue when imposing one of the weakest possible qualification conditions, namely the calmness of the perturbation mapping associated with the respective generalized equations in both forms of the MPEC. It is well known that the calmness property allows one to derive the so-called M-stationarity conditions. The restrictiveness of assumptions and the strength of conclusions in the two forms of the MPEC is also strongly related to the qualification conditions on the “lower level”. For instance, even under the Linear Independence Constraint Qualification (LICQ) for a lower level feasible set described by C 1 functions, the calmness properties of the original and the enhanced perturbation mapping are drastically different. When passing to C 1,1 data, this difference still remains true under the weaker Mangasarian-Fromovitz Constraint Qualification, whereas under LICQ both the calmness assumption and the derived optimality conditions are fully equivalent for the original and the enhanced form of the MPEC. After clarifying these relations, we provide a compilation of practically relevant consequences of our analysis in the derivation of necessary optimality conditions. The obtained results are finally applied to MPECs with structured equilibria. KW - equilibrium constraints KW - optimality conditions KW - constraint qualification KW - calmness KW - perturbation mapping Y1 - 2017 ER - TY - JOUR A1 - Gotzes, Claudia A1 - Heitsch, Holger A1 - Henrion, Rene A1 - Schultz, Rüdiger T1 - On the quantification of nomination feasibility in stationary gas networks with random load JF - Mathematical Methods of Operations Research N2 - The 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. Y1 - 2016 U6 - https://doi.org/10.1007/s00186-016-0564-y VL - 84 IS - 2 SP - 427 EP - 457 ER - TY - JOUR A1 - van Ackooij, Wim A1 - Henrion, Rene T1 - (Sub-) Gradient formulae for probability functions of random inequality systems under Gaussian distribution JF - SIAM/ASA J. Uncertainty Quantification N2 - We consider probability functions of parameter-dependent random inequality systems under Gaussian distribution. As a main result, we provide an upper estimate for the Clarke subdifferential of such probability functions without imposing compactness conditions. A constraint qualification ensuring continuous differentiability is formulated. Explicit formulae are derived from the general result in case of linear random inequality systems. In the case of a constant coefficient matrix an upper estimate for even the smaller Mordukhovich subdifferential is proven. Y1 - 2017 U6 - https://doi.org/10.1137/16M1061308 VL - 5 SP - 63 EP - 87 ER - TY - JOUR A1 - Guigues, Vincent A1 - Henrion, Rene T1 - Joint dynamic probabilistic constraints with projected linear decision rules JF - Optimization Methods and Software N2 - We 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. Y1 - 2016 U6 - https://doi.org/10.1080/10556788.2016.1233972 VL - 32 SP - 1006 EP - 1032 ER - TY - JOUR A1 - Diniz, Andre Luiz A1 - Henrion, Rene T1 - On probabilistic constraints with multivariate truncated Gaussian and lognormal distributions JF - Energy Systems N2 - Many engineering problems with uncertain data, notably arising in power management, can be formulated as optimization problems subject to probabilistic constraints. While dealing with such constraints under continuous distributions of the underlying random parameter remains a difficult task in general both from the numerical and theoretical point of view, quite some progress has been made in the special case of multivariate Gaussian distributions. These are not perfectly adequate, however, in many circumstances, in particular not, when modeling uncertain inflows to hydro reservoirs or uncertain demands in gas networks. Interesting alternatives are offered by truncations of multivariate Gaussian distributions to polyhedra or by multivariate lognormal distributions. The paper discusses the applicability of such distributions in the context of a simple joint linear probabilistic constraint putting the emphasis on the numerical approximation of probabilities and their gradients (w.r.t. decisions to be optimized) as well as on the convexity of the set of feasible decisions. Y1 - 2016 U6 - https://doi.org/10.1007/s12667-015-0180-6 VL - 8 SP - 149 EP - 167 ER - TY - JOUR A1 - Geiersbach, Caroline A1 - Henrion, René T1 - Optimality conditions in control problems with random state constraints in probabilistic or almost-sure form N2 - In this paper, we discuss optimality conditions for optimization problems {involving} random state constraints, which are modeled in probabilistic or almost sure form. While the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation (PDE) with random inputs. In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them to a model based on robust constraints with respect to the (compact) support of the given distribution. Y1 - 2023 ER - TY - JOUR A1 - Geiersbach, Caroline A1 - Henrion, René A1 - Pérez-Aros, Pedro T1 - Numerical solution of an optimal control problem with probabilistic and almost sure state constraints N2 - We consider the optimal control of a PDE with random source term subject to probabilistic or almost sure state constraints. In the main theoretical result, we provide an exact formula for the Clarke subdifferential of the probability function without a restrictive assumption made in an earlier paper. The focus of the paper is on numerical solution algorithms. As for probabilistic constraints, we apply the method of spherical radial decomposition. Almost sure constraints are dealt with a Moreau--Yosida smoothing of the constraint function accompanied by Monte Carlo sampling of the given distribution or its support or even just the boundary of its support. Moreover, one can understand the almost sure constraint as a probabilistic constraint with safety level one which offers yet another perspective. Finally, robust optimization can be applied efficiently when the support is sufficiently simple. A comparative study of these five different methodologies is carried out and illustrated. Y1 - 2023 ER -