TY - JOUR A1 - Berthold, Holger A1 - Heitsch, Holger A1 - Henrion, René A1 - Schwientek, Jan T1 - On the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints N2 - We present an adaptive grid refinement algorithm to solve probabilistic optimization problems with infinitely many random constraints. Using a bilevel approach, we iteratively aggregate inequalities that provide most information not in a geometric but in a probabilistic sense. This conceptual idea, for which a convergence proof is provided, is then adapted to an implementable algorithm. The efficiency of our approach when compared to naive methods based on uniform grid refinement is illustrated for a numerical test example as well as for a water reservoir problem with joint probabilistic filling level constraints. KW - probabilistic constraints KW - probust constraints KW - chance constraints KW - bilevel optimization KW - semi-infinite optimization Y1 - 2021 U6 - https://doi.org/10.1007/s00186-021-00764-8 ER - TY - JOUR A1 - Adam, Lukas A1 - Branda, Martin A1 - Heitsch, Holger A1 - Henrion, René T1 - Solving joint chance constrained problems using regularization and Benders' decomposition JF - Annals of Operations Research N2 - In this paper we investigate stochastic programs with joint chance constraints. We consider discrete scenario set and reformulate the problem by adding auxiliary variables. Since the resulting problem has a difficult feasible set, we regularize it. To decrease the dependence on the scenario number, we propose a numerical method by iteratively solving a master problem while adding Benders cuts. We find the solution of the slave problem (generating the Benders cuts) in a closed form and propose a heuristic method to decrease the number of cuts. We perform a numerical study by increasing the number of scenarios and compare our solution with a solution obtained by solving the same problem with continuous distribution. KW - chance constrained programming KW - optimality conditions KW - regularization KW - Benders cuts KW - gas networks Y1 - U6 - https://doi.org/10.1007/s10479-018-3091-9 VL - 292 SP - 683 EP - 709 ER - TY - INPR A1 - Heitsch, Holger A1 - Henrion, René A1 - Kleinert, Thomas A1 - Schmidt, Martin T1 - On Convex Lower-Level Black-Box Constraints in Bilevel Optimization with an Application to Gas Market Models with Chance Constraints N2 - Bilevel optimization is an increasingly important tool to model hierarchical decision making. However, the ability of modeling such settings makes bilevel problems hard to solve in theory and practice. In this paper, we add on the general difficulty of this class of problems by further incorporating convex black-box constraints in the lower level. For this setup, we develop a cutting-plane algorithm that computes approximate bilevel-feasible points. We apply this method to a bilevel model of the European gas market in which we use a joint chance constraint to model uncertain loads. Since the chance constraint is not available in closed form, this fits into the black-box setting studied before. For the applied model, we use further problem-specific insights to derive bounds on the objective value of the bilevel problem. By doing so, we are able to show that we solve the application problem to approximate global optimality. In our numerical case study we are thus able to evaluate the welfare sensitivity in dependence of the achieved safety level of uncertain load coverage. KW - Bilevel optimization KW - Black-box constraints KW - Chance constraints KW - Cutting planes KW - European gas market Y1 - 2021 ER - TY - JOUR A1 - Heitsch, Holger A1 - Henrion, René T1 - An enumerative formula for the spherical cap discrepancy N2 - The 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. KW - spherical cap discrepancy KW - uniform distribution on sphere KW - optimality conditions Y1 - 2019 U6 - https://doi.org/10.1016/j.cam.2021.113409 ER - TY - JOUR A1 - Farshbaf Shaker, Mohammad Hassan A1 - Gugat, Martin A1 - Heitsch, Holger A1 - Henrion, René T1 - Optimal Neumann boundary control of a vibrating string with uncertain initial data and probabilistic terminal constraints N2 - In optimal control problems, often initial data are required that are not known exactly in practice. In order to take into account this uncertainty, we consider optimal control problems for a system with an uncertain initial state. A finite terminal time is given. On account of the uncertainty of the initial state, it is not possible to prescribe an exact terminal state. Instead, we are looking for controls that steer the system into a given neighborhood of the desired terminal state with sufficiently high probability. This neighborhood is described in terms of an inequality for the terminal energy. The probabilistic constraint in the considered optimal control problem leads to optimal controls that are robust against the inevitable uncertainties of the initial state. We show the existence of such optimal controls. Numerical examples with optimal Neumann control of the wave equation are presented. KW - PDE constrained optimization, probabilistic constraints, uncertain initial data Y1 - U6 - https://doi.org/10.1137/19M1269944 ER - TY - JOUR A1 - Adelhütte, Dennis A1 - Aßmann, Denis A1 - Gonzàlez Grandòn, Tatiana A1 - Gugat, Martin A1 - Heitsch, Holger A1 - Liers, Frauke A1 - Henrion, René A1 - Nitsche, Sabrina A1 - Schultz, Rüdiger A1 - Stingl, Michael A1 - Wintergerst, David T1 - Joint model of probabilistic/robust (probust) constraints applied to gas network optimization N2 - Optimization tasks under uncertain conditions abound in many real-life applications. Whereas 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 used in case solutions are sought that are feasible for all realizations of uncertainties within some pre-defined 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 as its appears in optimization problems under uncertainty. 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 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. KW - robust optimization KW - chance constraints KW - optimal control KW - spheric-radial decomposition Y1 - 2017 U6 - https://doi.org/10.1007/s10013-020-00434-y ER - TY - JOUR A1 - Heitsch, Holger A1 - Strogies, Nikolai T1 - Consequences of uncertain friction for the transport of natural gas through passive networks of pipelines JF - Springer N2 - Assuming a pipe-wise constant structure of the friction coefficient in the modeling of natural gas transport through a passive network of pipes via semilinear systems of balance laws with associated linear coupling and boundary conditions, uncertainty in this parameter is quantified by a Markov chain Monte Carlo method. Information on the prior distribution is obtained from practitioners. The results are applied to the problem of validating technical feasibility under random exit demand in gas transport networks. The impact of quantified uncertainty to the probability level of technical feasible exit demand situations is studied by two example networks of small and medium size. The gas transport of the network is modeled by stationary solutions that are steady states of the time dependent semilinear problems. KW - uncertainty quantification, Markov chain Monte Carlo, reliability of gas networks, nomination validation, spheric-radial decomposition Y1 - 2019 SP - 211 EP - 238 ET - Topics in Applied Analysis and Optimisation ER - TY - JOUR A1 - Heitsch, Holger T1 - On probabilistic capacity maximization in a stationary gas network JF - Optimization N2 - The question for the capacity of a given gas network, i.e., determining the maximal amount of gas that can be transported by a given network, appears as an essential question that network operators and political administrations are regularly faced with. In that context we present a novel mathematical approach in order to assist gas network operators in managing increasing uncertainty with respect to customers gas nominations and in exposing free network capacities while reliability of transmission and supply is taken into account. The approach is based on the rigorous examination of optimization problems with nonlinear probabilistic constraints. As consequence we deal with solving a problem belonging to the class of probabilistic/robust optimization problems, which can be formulated with some joint probabilistic constraint over an infinite system of random inequalities. We will show that the inequality system can be reduced to a finite one in the situation of considering a tree network topology. A detailed study of the problem of maximizing bookable capacities in a stationary gas network is presented that comes up with an algebraic model involving Kirchhoff's first and second laws. The focus will be on both the theoretical and numerical side. The analytical part consists in introducing and validating a generalized version of the known rank two constraint qualification implying the differentiability of the considered capacity problem. The results are important in order to solve the capacity problem numerically, where function and gradient evaluations of the probabilistic constraints are performed by an approach using spheric-radial decomposition applicable for multivariate Gaussian random variables and more general distributions. KW - stationary gas networks, booked capacities, probabilistic constraints, constraint qualification, spheric-radial decomposition Y1 - 2019 U6 - https://doi.org/10.1080/02331934.2019.1625353 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 - 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 - Leövey, Hernan A1 - Heitsch, Holger A1 - Römisch, Werner T1 - Are Quasi-Monte Carlo algorithms efficient for two-stage stochastic programs? JF - Computational Optimization and Applications N2 - Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs with random right-hand side and continuous probability distribution. The latter should allow for a transformation to a distribution with independent marginals. The two-stage integrands are piecewise linear, but neither smooth nor lie in the function spaces considered for QMC error analysis. We show that under some weak geometric condition on the two-stage model all terms of their ANOVA decomposition, except the one of highest order, are continuously differentiable and that first and second order ANOVA terms have mixed first order partial derivatives and belong to L2 . Hence, randomly shifted lattice rules (SLR) may achieve the optimal rate of convergence O(n−1+δ ) with δ ∈ (0, 12 ] and a constant not depending on the dimension if the effective superposition dimension is at most two. We discuss effective dimensions and dimension reduction for two-stage integrands. The geometric condition is shown to be satisfied almost everywhere if the underlying probability distribution is normal and principal component analysis (PCA) is used for transforming the covariance matrix. Numerical experiments for a large scale two-stage stochastic production planning model with normal demand show that indeed convergence rates close to the optimal are achieved when using SLR and randomly scrambled Sobol’ point sets accompanied with PCA for dimension reduction. Y1 - 2016 U6 - https://doi.org/10.1007/s10589-016-9843-z VL - 65 IS - 3 SP - 567 EP - 603 ER - TY - JOUR A1 - Ouanes, Nesrine A1 - González Grandón, Tatiana A1 - Heitsch, Holger A1 - Henrion, René T1 - Optimizing the economic dispatch of weakly-connected mini-grids under uncertainty using joint chance constraints N2 - In this paper, we deal with a renewable-powered mini-grid, connected to an unreliable main grid, in a Joint Chance Constrained (JCC) programming setting. In many countries with low energy access rates, grid-connected mini-grid system operators contend with four different types of uncertainties: stochastic solar power and demand forecast errors; absolute uncertain national grid outage onset times; and outages duration subjected to statistical analysis. These uncertainties pose new challenges to the classical power system’s operation tasks. Two alternatives to the JCC problem are presented. In particular, we present an Individual Chance Constraint (ICC) and a purely deterministic dispatch model. The JCC model has the capability to address all four uncertainties, while the ICC covers only three of them, overlooking the uncertainty about the outage duration. In contrast, the purely deterministic model completely ignores any uncertain parameters. We illustrate the three models through a comparison of outcomes attained from a real mini-grid in Lake Victoria, Tanzania. Results show how the dispatch is modified across the models to plan the battery and diesel reserves in the chance-constrained models, with the reserves in the JCC being larger than in the ICC model. In comparison between all models, we prove that the JCC model offers the most robust results, since it can handle uncertainties about forecasting errors, on the one hand, and grid outages, on the other. The results also show that the decrease in profits due to the hedging with reserves kept in the MG is significantly small compared to the high level of reliability reached and the potential load shedding that could be avoided in the case of an outage. Y1 - 2023 ER - TY - JOUR A1 - Gugat, Martin A1 - Henrion, René A1 - Heitsch, Holger T1 - A turnpike property for optimal control problems with dynamic probabilistic constraints JF - Journal of Convex Analysis N2 - In this paper we consider systems that are governed by linear time-discrete dynamics with an initial condition and a terminal condition for the expected values. We study optimal control problems where in the objective function a term of tracking type for the expected values and a control cost appear. In addition, the feasible states have to satisfy a conservative probabilistic constraint that requires that the probability that the trajectories remain in a given set F is greater than or equal to a given lower bound. An application are optimal control problems related to storage management systems with uncertain in- and output. We give suffcient conditions that imply that the optimal expected trajectories remain close to a certain state that can be characterized as the solution of an optimal control problem without prescribed initial- and terminal condition. Hence we contribute to the study of the turnpike phenomenon that is well-known in mathematical economics. KW - Probabilistic Constraints KW - Probabilistic Robustness KW - here-and-now decision KW - Turnpike phenomenon KW - Measure turnpike Y1 - 2021 VL - 30 IS - 3 SP - 1025 EP - 1052 PB - Heldermann Verlag 2023 ER - TY - JOUR A1 - Heitsch, Holger A1 - Henrion, René T1 - On the Lipschitz continuity of the spherical cap discrepancy around generic point sets JF - Unif. Distrib. Theory N2 - The spherical cap discrepancy is a prominent measure of uniformity for sets on the d-dimensional sphere. It is particularly important for estimating the integration error for certain classes of functions on the sphere. Building on a recently proven explicit formula for the spherical discrepancy, we show as a main result of this paper that this discrepancy is Lipschitz continuous in a neighbourhood of so-called generic point sets (as they are typical outcomes of Monte-Carlo sampling). This property may have some impact (both algorithmically and theoretically for deriving necessary optimality conditions) on optimal quantization, i.e., on finding point sets of fixed size on the sphere having minimum spherical discrepancy. KW - spherical cap discrepancy KW - uniform distribution on sphere KW - Lipschitz continuity KW - necessary optimality conditions Y1 - 2025 U6 - https://doi.org/10.2478/udt-2025-0011 VL - 20 IS - 1 SP - 35 EP - 63 ER - TY - JOUR A1 - Bernhard, Daniela A1 - Heitsch, Holger A1 - Henrion, René A1 - Liers, Frauke A1 - Stingl, Michael A1 - Uihlein, Andrian A1 - Zipf, Viktor T1 - Continuous stochastic gradient and spherical radial decomposition N2 - In this paper, a new method is presented for solving chance-constrained optimization problems. The method combines the well-established Spherical-Radial Decomposition approach with the Continuous Stochastic Gradient method. While the Continuous Stochastic Gradient method has been successfully applied to chance-constrained problems in the past, only the combination with the Spherical-Radial Decomposition allows to avoid smoothing of the integrand. In this chapter, we prove this fact for a relevant class of chance-constrained problems and apply the resulting method to the capacity maximization problem for gas networks. KW - chance constraints KW - continuous stochastic gradient KW - spheric-radial decomposition Y1 - ER -