TY - INPR A1 - Gugat, Martin A1 - Habermann, Jens A1 - Hintermüller, Michael A1 - Huber, Olivier T1 - Constrained exact boundary controllability of a semilinear model for pipeline gas flow N2 - While the quasilinear isothermal Euler equations are an excellent model for gas pipeline flow, the operation of the pipeline flow with high pressure and small Mach numbers allows us to obtain approximate solutions by a simpler semilinear model. We provide a derivation of the semilinear model that shows that the semilinear model is valid for sufficiently low Mach numbers and sufficiently high pressures. We prove an existence result for continuous solutions of the semilinear model that takes into account lower and upper bounds for the pressure and an upper bound for the magnitude of the Mach number of the gas flow. These state constraints are important both in the operation of gas pipelines and to guarantee that the solution remains in the set where the model is physically valid. We show the constrained exact boundary controllability of the system with the same pressure and Mach number constraints. Y1 - 2021 ER - TY - INPR A1 - Grübel, Julia A1 - Huber, Olivier A1 - Hümbs, Lukas A1 - Klimm, Max A1 - Schmidt, Martin A1 - Schwartz, Alexandra T1 - Nonconvex Equilibrium Models for Energy Markets: Exploiting Price Information to Determine the Existence of an Equilibrium N2 - Motivated by examples from the energy sector, we consider market equilibrium problems (MEPs) involving players with nonconvex strategy spaces or objective functions, where the latter are assumed to be linear in market prices. We propose an algorithm that determines if an equilibrium of such an MEP exists and that computes an equilibrium in case of existence. Three key prerequisites have to be met. First, appropriate bounds on market prices have to be derived from necessary optimality conditions of some players. Second, a technical assumption is required for those prices that are not uniquely determined by the derived bounds. Third, nonconvex optimization problems have to be solved to global optimality. We test the algorithm on well-known instances from the power and gas literature that meet these three prerequisites. There, nonconvexities arise from considering the transmission system operator as an additional player besides producers and consumers who, e.g., switches lines or faces nonlinear physical laws. Our numerical results indicate that equilibria often exist, especially for the case of continuous nonconvexities in the context of gas market problems. KW - Energy markets KW - Nonconvex games KW - Existence KW - Equilibrium computation KW - Perfect competition Y1 - 2021 ER - 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 - INPR A1 - Branda, Martin A1 - Henrion, René A1 - Pištěk, Miroslav T1 - Value at risk approach to producer's best response in electricity market with uncertain demand N2 - We deal with several sources of uncertainty in electricity markets. The independent system operator (ISO) maximizes the social welfare using chance constraints to hedge against discrepancies between the estimated and real electricity demand. We find an explicit solution of the ISO problem, and use it to tackle the problem of a producer. In our model, production as well as income of a producer are determined based on the estimated electricity demand predicted by the ISO, that is unknown to producers. Thus, each producer is hedging against the uncertainty of prediction of the demand using the value-at-risk approach. To illustrate our results, a numerical study of a producer's best response given a historical distribution of both estimated and real electricity demand is provided. KW - electricity market KW - multi-leader-common-follower game KW - stochastic demand KW - day-ahead bidding KW - chance constraints Y1 - 2021 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 - JOUR A1 - Branda, Martin A1 - Henrion, René A1 - Pistek, Miroslav T1 - Producer’s Best Response in Pay-as-clear Day-ahead Electricity Market with Uncertain Demand N2 - We deal with several sources of uncertainty in electricity markets. The independent system operator (ISO) maximizes the social welfare using chance constraints to hedge against discrepancies between the estimated and real electricity demand. We find an explicit solution of the ISO problem, and use it to tackle the problem of a producer. In our model, production as well as income of a producer are determined based on the estimated electricity demand predicted by the ISO, that is unknown to producers. Thus, each producer is hedging against the uncertainty of prediction of the demand using the value-at-risk approach. To illustrate our results, a numerical study of a producer’s best response given a historical distribution of both estimated and real electricity demand is provided. KW - electricity market KW - multi-leader-common-follower game KW - stochastic demand KW - day-ahead bidding KW - chance constraints Y1 - 2020 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 - INPR A1 - Schmidt, Martin A1 - Hiller, Benjamin A1 - Koch, Thorsten A1 - Pfetsch, Marc A1 - Geißler, Björn A1 - Henrion, René A1 - Joormann, Imke A1 - Martin, Alexander A1 - Morsi, Antonio A1 - Römisch, Werner A1 - Schewe, Lars A1 - Schultz, Rüdiger A1 - Steinbach, Marc C. T1 - Capacity Evaluation for Large-Scale Gas Networks N2 - Natural gas is important for the energy turnaround in many countries like in Germany, where it serves as a "bridging energy" towards a fossil-free energy supply in the future. About 20% of the total German energy demand is provided by natural gas, which is transported through a complex pipeline network with a total length of about 30000 km and the efficient use of the given transport infrastructure for natural gas is of political, economic, and societal importance. As a consequence of the liberalization of the European gas market in the last decades, gas trading and transport have been decoupled. This has led to new challenges for gas transport companies, and mathematical optimization is perfectly suited for tackling many of these challenges. However, the underlying mathematical problems are by far too hard to be solved by today's general-purpose software so that novel mathematical theory and algorithms are needed. The industrial research project "ForNe: Research Cooperation Network Optimization" has been initiated and funded by Open Grid Europe in 2009 and brought together experts in mathematical optimization from seven German universities and research institutes, which cover almost the entire range of mathematical optimization: integer and nonlinear optimization as well as optimization under uncertainty. The mathematical research results have been put together in a software package that has been delivered to Open Grid Europe at the end of the project. Moreover, the research is still continuing - e.g., in the Collaborative Research Center/Transregio 154 "Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks" funded by the German Research Foundation. Y1 - 2019 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 - Hintermüller, Michael A1 - Strogies, Nikolai T1 - Identification of the friction function in a semilinear system for gas transport through a network JF - Optimization Methods and Software N2 - An identification problem for the friction parameter in a semilinear system of balance laws, describing the transport of gas through a passive network of pipelines, is considered. The existence of broad solutions to the state system is proven and sensitivity results for the corresponding solution operator are obtained. The existence of solutions to the output least squares formulation of the identification problem, based on noisy measurements over time at fixed spatial positions is established. Finally, numerical experiments validate the theoretical findings. Y1 - 2017 VL - 35 SP - 576 EP - 617 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 - 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 - Hajian, Soheil A1 - Hintermüller, Michael A1 - Schillings, Claudia A1 - Strogies, Nikolai T1 - A Bayesian approach to parameter identification in gas networks JF - Control and Cybernetics N2 - The inverse problem of identifying the friction coefficient in an isothermal semilinear Euler system is considered. Adopting a Bayesian approach, the goal is to identify the distribution of the quantity of interest based on a finite number of noisy measurements of the pressure at the boundaries of the domain. First well-posedness of the underlying non-linear PDE system is shown using semigroup theory, and then Lipschitz continuity of the solution operator with respect to the friction coefficient is established. Based on the Lipschitz property, well-posedness of the resulting Bayesian inverse problem for the identification of the friction coefficient is inferred. Numerical tests for scalar and distributed parameters are performed to validate the theoretical results. Y1 - 2018 VL - 48 SP - 377 EP - 402 ER - TY - JOUR A1 - Hajian, Soheil A1 - Hintermüller, Michael A1 - Ulbrich, Stefan T1 - Total variation diminishing schemes in optimal control of scalar conservation laws JF - IMA Journal of Numerical Analysis N2 - In this paper, optimal control problems subject to a nonlinear scalar conservation law are studied. Such optimal control problems are challenging both at the continuous and at the discrete level since the control-to-state operator poses difficulties as it is, e.g., not differentiable. Therefore discretization of the underlying optimal control problem should be designed with care. Here the discretize-then-optimize approach is employed where first the full discretization of the objective function as well as the underlying PDE is considered. Then, the derivative of the reduced objective is obtained by using an adjoint calculus. In this paper total variation diminishing Runge-Kutta (TVD-RK) methods for the time discretization of such problems are studied. TVD-RK methods, also called strong stability preserving (SSP), are originally designed to preserve total variation of the discrete solution. It is proven in this paper that providing an SSP state scheme, is enough to ensure stability of the discrete adjoint. However requiring SSP for both discrete state and adjoint is too strong. Also approximation properties that the discrete adjoint inherits from the discretization of the state equation are studied. Moreover order conditions are derived. In addition, optimal choices with respect to CFL constant are discussed and numerical experiments are presented. Y1 - 2017 U6 - https://doi.org/10.20347/WIAS.PREPRINT.2383 VL - 39 SP - 105 EP - 140 ER - TY - JOUR A1 - Papafitsoros, Konstantinos A1 - Hintermüller, Michael A1 - Rautenberg, Carlos T1 - Analytical aspects of spatially adapted total variation regularisation JF - Journal of Mathematical Analysis and Applications N2 - In this paper we study the structure of solutions of the one dimensional weighted total variation regularisation problem, motivated by its application in signal recovery tasks. We study in depth the relationship between the weight function and the creation of new discontinuities in the solution. A partial semigroup property relating the weight function and the solution is shown and analytic solutions for simply data functions are computed. We prove that the weighted total variation minimisation problem is well-posed even in the case of vanishing weight function, despite the lack of coercivity. This is based on the fact that the total variation of the solution is bounded by the total variation of the data, a result that it also shown here. Finally the relationship to the corresponding weighted fidelity problem is explored, showing that the two problems can produce completely different solutions even for very simple data functions. Y1 - 2017 VL - 454 SP - 891 EP - 935 ER - TY - JOUR A1 - Keil, Tobias A1 - Hintermüller, Michael A1 - Wegner, Donat T1 - Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system with non-matched fluid densities JF - SIAM Journal on Control and Optimization N2 - This paper is concerned with the distributed optimal control of a time-discrete Cahn– Hilliard/Navier–Stokes system with variable densities. It focuses on the double-obstacle potential which yields an optimal control problem for a family of coupled systems in each time instant of a variational inequality of fourth order and the Navier–Stokes equation. By proposing a suitable time- discretization, energy estimates are proved and the existence of solutions to the primal system and of optimal controls is established for the original problem as well as for a family of regularized problems. The latter correspond to Moreau–Yosida type approximations of the double-obstacle potential. The consistency of these approximations is shown and first order optimality conditions for the regularized problems are derived. Through a limit process with respect to the regularization parameter, a stationarity system for the original problem is established. The resulting system corresponds to a function space version of C-stationarity which is a special notion of stationarity for MPECs. Y1 - 2017 VL - 55 SP - 1954 EP - 1989 ER - TY - JOUR A1 - Hintermüller, Michael A1 - Rautenberg, Carlos A1 - Mohammadi, Masoumeh A1 - Kanitsar, Martin T1 - Optimal sensor placement: A robust approach JF - SIAM Journal on Control and Optimization N2 - We address the problem of optimally placing sensor networks for convection-diffusion processes where the convective part is perturbed. The problem is formulated as an optimal control problem where the integral Riccati equation is a constraint and the design variables are sensor locations. The objective functional involves a term associated to the trace of the solution to the Riccati equation and a term given by a constrained optimization problem for the directional derivative of the previous quantity over a set of admissible perturbations. The paper addresses the existence of the derivative with respect to the convective part of the solution to the Riccati equation, the well-posedness of the optimization problem and finalizes with a range of numerical tests. Y1 - 2017 VL - 55 SP - 3609 EP - 3639 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 - Hintermüller, Michael A1 - Strogies, Nikolai T1 - On the consistency of Runge--Kutta methods up to order three applied to the optimal control of scalar conservation laws JF - Numerical Analysis and Optimization N2 - Higher-order Runge-Kutta (RK) time discretization methods for the optimal control of scalar conservation laws are analyzed and numerically tested. The hyperbolic nature of the state system introduces specific requirements on discretization schemes such that the discrete adjoint states associated with the control problem converge as well. Moreover, conditions on the RK-coefficients are derived that coincide with those characterizing strong stability preserving Runge-Kutta methods. As a consequence, the optimal order for the adjoint state is limited, e.g., to two even in the case where the conservation law is discretized by a third-order method. Finally, numerical tests for controlling Burgers equation validate the theoretical results. Y1 - 2017 U6 - https://doi.org/10.20347/WIAS.PREPRINT.2442 SP - 119 EP - 154 ER - TY - JOUR A1 - Hintermüller, Michael A1 - Rautenberg, Carlos A1 - Strogies, Nikolai T1 - Dissipative and Non-dissipative Evolutionary Quasi-variational Inequalities with Gradient Constraints JF - Set-Valued and Variational Analysis N2 - Evolutionary quasi-variational inequality (QVI) problems of dissipative and non-dissipative nature with pointwise constraints on the gradient are studied. A semi-discretization in time is employed for the study of the problems and the derivation of a numerical solution scheme, respectively. Convergence of the discretization procedure is proven and properties of the original infinite dimensional problem, such as existence, extra regularity and non-decrease in time, are derived. The proposed numerical solver reduces to a finite number of gradient-constrained convex optimization problems which can be solved rather efficiently. The paper ends with a report on numerical tests obtained by a variable splitting algorithm involving different nonlinearities and types of constraints. Y1 - 2017 VL - 27 SP - 433 EP - 468 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 - 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 - Egger, Herbert A1 - Kugler, Thomas A1 - Strogies, Nikolai T1 - Parameter identification in a semilinear hyperbolic system JF - Inverse Problems N2 - We consider the identification of a nonlinear friction law in a one-dimensional damped wave equation from additional boundary measurements. Well-posedness of the governing semilinear hyperbolic system is established via semigroup theory and con- traction arguments. We then investigte the inverse problem of recovering the unknown nonlinear damping law from additional boundary measurements of the pressure drop along the pipe. This coefficient inverse problem is shown to be ill-posed and a varia- tional regularization method is considered for its stable solution. We prove existence of minimizers for the Tikhonov functional and discuss the convergence of the regularized so- lutions under an approximate source condition. The meaning of this condition and some arguments for its validity are discussed in detail and numerical results are presented for illustration of the theoretical findings Y1 - 2016 VL - 33 IS - 055022 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 - INPR A1 - Gahururu, Deborah A1 - Hintermüller, Michael A1 - Surowiec, Thomas T1 - Risk-Neutral PDE-Constrained Generalized Nash Equilibrium Problems N2 - A class of risk-neutral PDE-constrained generalized Nash equilibrium problems is introduced in which the feasible strategy set of each player is subject to a common linear elliptic partial differential equation with random inputs. In addition, each player’s actions are taken from a bounded, closed, and convex set on the individual strategies and a bound constraint on the common state variable. Existence of Nash equilibria and first-order optimality conditions are derived by exploiting higher integrability and regularity of the random field state variables and a specially tailored constraint qualification for GNEPs with the assumed structure. A relaxation scheme based on the Moreau-Yosida approximation of the bound constraint is proposed, which ultimately leads to numerical algorithms for the individual player problems as well as the GNEP as a whole. The relaxation scheme is related to probability constraints and the viability of the proposed numerical algorithms are demonstrated via several examples. Y1 - 2021 ER - TY - INPR A1 - Geiersbach, Caroline A1 - Hintermüller, Michael T1 - Optimality conditions and Moreau–Yosida regularization for almost sure state constraints N2 - We analyze a potentially risk-averse convex stochastic optimization problem, where the control is deterministic and the state is a Banach-valued essentially bounded random variable. We obtain strong forms of necessary and sufficient optimality conditions for problems subject to equality and conical constraints. We propose a Moreau–Yosida regularization for the conical constraint and show consistency of the optimality conditions for the regularized problem as the regularization parameter is taken to infinity. Y1 - 2021 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 - Henrion, Rene T1 - On M-stationarity conditions for probabilistic MPECs N2 - We consider Mathematical Programs with Equilibrium Constraints with proba- bilistic constraints (PMPECs). Such models have proven to be useful in modeling electricity or gas markets subject to random parameters. Our main interest is the derivation of Mordukhovich (M-) stationarity conditions under suitable constraint quali...cations ensuring the calmness of the canonically perturbed generalized equation. Applying recent results from deterministic MPECs, we identify the needed properties of the probability function in order to derive explicit M-stationarity conditions. The results are applied to a simple stochastic bilevel problem in an economic context. KW - Mathematical Programs with Equilibrium Constraints, probabilistic con- straints, M-stationarity conditions, calmness Y1 - 2021 ER - TY - INPR A1 - Bongarti, Marcelo A1 - Hintermüller, T1 - Optimal boundary control of the isothermal semilinear Euler equation for gas dynamics on a network N2 - The analysis and boundary optimal control of the nonlinear transport of gas on a network of pipelines is considered. The evolution of the gas distribution on a given pipe is modeled by an isothermal semilinear compressible Euler system in one space dimension. On the network, solutions satisfying (at nodes) the Kirchhoff flux continuity conditions are shown to exist in a neighborhood of an equilibrium state. The associated nonlinear optimization problem then aims at steering such dynamics to a given target distribution by means of suitable (network) boundary controls while keeping the distribution within given (state) constraints. The existence of local optimal controls is established and a corresponding Karush-Kuhn-Tucker (KKT) stationarity system with an almost surely non-singular Lagrange multiplier is derived. KW - optimal boundary control KW - gas dynamics KW - gas networks KW - isothermal Euler equation KW - compressible fluid dynamics Y1 - 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 - TY - JOUR A1 - Henrion, René A1 - Schmidt, Martin T1 - Chance-Constrained Linear Complementarity Problems N2 - We study linear complementarity problems (LCPs) under uncer- tainty, which we model using chance constraints. Since the complementarity condition of the LCP is an equality constraint, it is required to consider relax- ations, which naturally leads to optimization problems in which the relaxation parameters are minimized for given probability levels. We focus on these optimization problems and first study the continuity of the related probability functions and the compactness of the feasible sets. This leads to existence results for both types of models: one with a joint chance constraint and one with separate chance constraints for both uncertainty-affected conditions of the LCP. For both, we prove the differentiability of all probability functions and derive respective gradient formulae. For the separate case, we prove con- vexity of the respective optimization problem and use the gradient formulae to derive necessary and sufficient optimality conditions. In a small case study regarding a Cournot oligopoly among energy producers, we finally illustrate the applicability of our theoretical findings. KW - Linear complementarity problems KW - Chance constraints KW - Existence KW - Convexity KW - Optimality conditions Y1 - 2026 ER -