@unpublished{ThueraufGruebelSchmidt2024, author = {Th{\"u}rauf, Johannes and Gr{\"u}bel, Julia and Schmidt, Martin}, title = {Adjustable Robust Nonlinear Network Design under Demand Uncertainties}, year = {2024}, abstract = {We study network design problems for nonlinear and nonconvex flow models under demand uncertainties. To this end, we apply the concept of adjustable robust optimization to compute a network design that admits a feasible transport for all, possibly infinitely many, demand scenarios within a given uncertainty set. For solving the corresponding adjustable robust mixed-integer nonlinear optimization problem, we show that a given network design is robust feasible, i.e., it admits a feasible transport for all demand uncertainties, if and only if a finite number of worst-case demand scenarios can be routed through the network. We compute these worst-case scenarios by solving polynomially many nonlinear optimization problems. Embedding this result for robust feasibility in an adversarial approach leads to an exact algorithm that computes an optimal robust network design in a finite number of iterations. Since all of the results are valid for general potential-based flows, the approach can be applied to different utility networks such as gas, hydrogen, or water networks. We finally demonstrate the applicability of the method by computing robust gas networks that are protected from future demand fluctuations.}, language = {en} } @unpublished{WilkaLang, author = {Wilka, Hendrik and Lang, Jens}, title = {Adaptive hp-Polynomial Based Sparse Grid Collocation Algorithms for Piecewise Smooth Functions with Kinks}, abstract = {High-dimensional interpolation problems appear in various applications of uncertainty quantification, stochastic optimization and machine learning. Such problems are computationally expensive and request the use of adaptive grid generation strategies like anisotropic sparse grids to mitigate the curse of dimensionality. However, it is well known that the standard dimension-adaptive sparse grid method converges very slowly or even fails in the case of non-smooth functions. For piecewise smooth functions with kinks, we construct two novel hp-adaptive sparse grid collocation algorithms that combine low-order basis functions with local support in parts of the domain with less regularity and variable-order basis functions elsewhere. Spatial refinement is realized by means of a hierarchical multivariate knot tree which allows the construction of localised hierarchical basis functions with varying order. Hierarchical surplus is used as an error indicator to automatically detect the non-smooth region and adaptively refine the collocation points there. The local polynomial degrees are optionally selected by a greedy approach or a kink detection procedure. Three numerical benchmark examples with different dimensions are discussed and comparison with locally linear and highest degree basis functions are given to show the efficiency and accuracy of the proposed methods.}, language = {en} } @article{LangSchmitt, author = {Lang, Jens and Schmitt, Bernhard A.}, title = {A Stiff MOL Boundary Control Problem for the 1D Heat Equation with Exact Discrete Solution}, volume = {Journal of Optimization Theory and Applications}, number = {Vol. 196}, doi = {https://doi.org/10.1007/s10957-022-02154-4}, pages = {1106 -- 1118}, abstract = {Method-of-lines discretizations are demanding test problems for stiff inte- gration methods. However, for PDE problems with known analytic solution the presence of space discretization errors or the need to use codes to compute reference solutions may limit the validity of numerical test results. To over- come these drawbacks we present in this short note a simple test problem with boundary control, a situation where one-step methods may suffer from order reduction. We derive exact formulas for the solution of an optimal boundary control problem governed by a one-dimensional discrete heat equation and an objective function that measures the distance of the final state from the target and the control costs. This analytical setting is used to compare the numeri- cally observed convergence orders for selected implicit Runge-Kutta and Peer two-step methods of classical order four which are suitable for optimal control problems.}, language = {en} } @article{StrelowGerischLangetal., author = {Strelow, Erik Laurin and Gerisch, Alf and Lang, Jens and Pfetsch, Marc E.}, title = {Physics-Informed Neural Networks: A Case Study for Gas Transport Problems}, volume = {Journal of Computational Physics}, number = {Vol. 481}, pages = {112041}, abstract = {Physics informed neural networks have been recently proposed and offer a new promising method to solve differential equations. They have been adapted to many more scenarios and different variations of the original method have been proposed. In this case study we review many of these variations. We focus on variants that can compensate for imbalances in the loss function and perform a comprehensive numerical comparison of these variants with application to gas transport problems. Our case study includes different formulations of the loss function, different algorithmic loss balancing methods, different optimization schemes and different numbers of parameters and sampling points. We conclude that the original PINN approach with specifically chosen constant weights in the loss function gives the best results in our tests. These weights have been obtained by a computationally expensive random-search scheme. We further conclude for our test case that loss balancing methods which were developed for other differential equations have no benefit for gas transport problems, that the control volume physics informed formulation has no benefit against the initial formulation and that the best optimization strategy is the L-BFGS method.}, language = {en} } @article{LangSchmitt, author = {Lang, Jens and Schmitt, Bernhard A.}, title = {Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems}, volume = {Algorithms}, number = {Vol. 15}, doi = {https://doi.org/10.3390/a15090310}, abstract = {This paper is concerned with the construction and convergence analysis of novel implicit Peer triplets of two-step nature with four stages for nonlinear ODE constrained optimal control problems. We combine the property of superconvergence of some standard Peer method for inner grid points with carefully designed starting and end methods to achieve order four for the state variables and order three for the adjoint variables in a first-discretize-then-optimize approach together with A-stability. The notion triplets emphasizes that these three different Peer methods have to satisfy additional matching conditions. Four such Peer triplets of practical interest are constructed. Also as a benchmark method, the well-known backward differentiation formula BDF4, which is only A(73.35)-stable, is extended to a special Peer triplet to supply an adjoint consistent method of higher order and BDF type with equidistant nodes. Within the class of Peer triplets, we found a diagonally implicit A(84)-stable method with nodes symmetric in [0,1] to a common center that performs equally well. Numerical tests with three well established optimal control problems confirm the theoretical findings also concerning A-stability.}, language = {en} } @unpublished{DomschkeGiesselmannLangetal.2023, author = {Domschke, Pia and Giesselmann, Jan and Lang, Jens and Breiten, Tobias and Mehrmann, Volker and Morandin, Riccardo and Hiller, Benjamin and Tischendorf, Caren}, title = {Gas Network Modeling: An Overview (Extended English Version)}, pages = {54}, year = {2023}, abstract = {With this overview we want to provide a compilation of different models for the description of gas flow in networks in order to facilitate the introduction to the topic. Special attention is paid to the hierarchical structure inherent to the modeling, and the detailed description of individual components such as valves and compressors. Also included are network model classes based on purely algebraic relations, and energy-based port-Hamiltonian models. A short overview of basic numerical methods and concepts for the treatment of hyperbolic balance equations is also given. We do not claim completeness and refer in many places to the existing literature.}, language = {en} } @unpublished{LangSchmitt2023, author = {Lang, Jens and Schmitt, Bernhard A.}, title = {Implicit Peer Triplets in Gradient-Based Solution Algorithms for ODE Constrained Optimal Control}, volume = {http://arxiv.org/abs/2303.18180}, year = {2023}, abstract = {It is common practice to apply gradient-based optimization algorithms to numerically solve large-scale ODE constrained optimal control problems. Gradients of the objective function are most efficiently computed by approximate adjoint variables. High accuracy with moderate computing time can be achieved by such time integration methods that satisfy a sufficiently large number of adjoint order conditions and supply gradients with higher orders of consistency. In this paper, we upgrade our former implicit two-step Peer triplets constructed in [Algorithms, 15:310, 2022] to meet those new requirements. Since Peer methods use several stages of the same high stage order, a decisive advantage is their lack of order reduction as for semi-discretized PDE problems with boundary control. Additional order conditions for the control and certain positivity requirements now intensify the demands on the Peer triplet. We discuss the construction of 4-stage methods with order pairs (4,3) and (3,3) in detail and provide three Peer triplets of practical interest. We prove convergence for s-stage methods, for instance, order s for the state variables even if the adjoint method and the control satisfy the conditions for order s-1, only. Numerical tests show the expected order of convergence for the new Peer triplets.}, language = {en} } @unpublished{GraserKreimeierWalther2024, author = {Graser, Gertrud and Kreimeier, Timo and Walther, Andrea}, title = {Solving Linear Generalized Nash Games Using an Active Signature Method}, year = {2024}, abstract = {We propose a method to solve linear generalized Nash equilibrium problems (LGNEPs). For this purpose, a reformulation of the LGNEPs as piecewise linear problems is considered. This requires the calculation of all vertices for a special kind of unbounded convex polyhedra. Then the active signature method for constrained abs-linear problems can be used to determine the Nash equilibria. We analyse the computational effort for the resulting solution procedure. This includes also the verification of suitable optimality conditions. Finally, we present and analyse numerical results for some test problems.}, language = {en} } @unpublished{AlldredgeFrankGiesselmann2023, author = {Alldredge, Graham and Frank, Martin and Giesselmann, Jan}, title = {On the convergence of the regularized entropy-based moment method for kinetic equations}, volume = {9}, doi = {https://doi.org/10.5802/smai-jcm.93}, pages = {1-29}, year = {2023}, abstract = {The entropy-based moment method is a well-known discretization for the velocity variable in kinetic equations which has many desirable theoretical properties but is difficult to implement with high-order numerical methods. The regularized entropy-based moment method was recently introduced to remove one of the main challenges in the implementation of the entropy-based moment method, namely the requirement of the realizability of the numerical solution. In this work we use the method of relative entropy to prove the convergence of the regularized method to the original method as the regularization parameter goes to zero and give convergence rates. Our main assumptions are the boundedness of the velocity domain and that the original moment solution is Lipschitz continuous in space and bounded away from the boundary of realizability. We provide results from numerical simulations showing that the convergence rates we prove are optimal.}, language = {en} } @unpublished{GiesselmannKolbe2023, author = {Giesselmann, Jan and Kolbe, Niklas}, title = {A posteriori error analysis of a positivity preserving scheme for the power-law diffusion Keller-Segel model}, year = {2023}, abstract = {We study a finite volume scheme approximating a parabolic-elliptic Keller-Segel system with power law diffusion with exponent γ∈[1,3] and periodic boundary conditions. We derive conditional a posteriori bounds for the error measured in the L∞(0,T;H1(Ω)) norm for the chemoattractant and by a quasi-norm-like quantity for the density. These results are based on stability estimates and suitable conforming reconstructions of the numerical solution. We perform numerical experiments showing that our error bounds are linear in mesh width and elucidating the behaviour of the error estimator under changes of γ.}, language = {en} } @unpublished{GiesselmannKwon2023, author = {Giesselmann, Jan and Kwon, Kiwoong}, title = {A posteriori error control for a Discontinuous Galerkin approximation of a Keller-Segel model}, year = {2023}, abstract = {We provide a posteriori error estimates for a discontinuous Galerkin scheme for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The estimates are conditional, in the sense that an a posteriori computable quantity needs to be small enough - which can be ensured by mesh refinement - and optimal in the sense that the error estimator decays with the same order as the error under mesh refinement. A specific feature of our error estimator is that it can be used to prove existence of a weak solution up to a certain time based on numerical results.}, language = {en} } @unpublished{GiesselmannKrupa2023, author = {Giesselmann, Jan and Krupa, Sam}, title = {Theory of shifts, shocks, and the intimate connections to L2-type a posteriori error analysis of numerical schemes for hyperbolic problems}, year = {2023}, abstract = {In this paper, we develop reliable a posteriori error estimates for numerical approximations of scalar hyperbolic conservation laws in one space dimension. Our methods have no inherent small-data limitations and are a step towards error control of numerical schemes for systems. We are careful not to appeal to the Kruzhkov theory for scalar conservation laws. Instead, we derive novel quantitative stability estimates that extend the theory of shifts, and in particular, the framework for proving stability first developed by the second author and Vasseur. This is the first time this methodology has been used for quantitative estimates. We work entirely within the context of the theory of shifts and a-contraction, techniques which adapt well to systems. In fact, the stability framework by the second author and Vasseur has itself recently been pushed to systems [Chen-Krupa-Vasseur. Uniqueness and weak-BV stability for 2×2 conservation laws. Arch. Ration. Mech. Anal., 246(1):299--332, 2022]. Our theoretical findings are complemented by a numerical implementation in MATLAB and numerical experiments.}, language = {en} } @unpublished{EggerGiesselmann2023, author = {Egger, Herbert and Giesselmann, Jan}, title = {Regularity and long time behavior of a doubly nonlinear parabolic problem and its discretization}, year = {2023}, abstract = {We study a doubly nonlinear parabolic problem arising in the modeling of gas transport in pipelines. Using convexity arguments and relative entropy estimates we show uniform bounds and exponential stability of discrete approximations obtained by a finite element method and implicit time stepping. Due to convergence of the approximations to weak solutions of the problem, our results also imply regularity, uniqueness, and long time stability of weak solutions of the continuous problem.}, language = {en} } @unpublished{GugatGiesselmann2023, author = {Gugat, Martin and Giesselmann, Jan}, title = {An Observer for pipeline flow with hydrogen blending in gas networks: exponential synchronization}, year = {2023}, abstract = {We consider a state estimation problem for gas flows in pipeline networks where hydrogen is blended into the natural gas. The flow is modeled by the quasi-linear isothermal Euler equations coupled to an advection equation on a graph. The flow through the vertices where the pipes are connected is governed by algebraic node conditions. The state is approximated by an observer system that uses nodal measurements. We prove that the state of the observer system converges to the original system state exponentially fast in the L2-norm if the measurements are exact. If measurement errors are present we show that the observer state approximates the original system state up to an error that is proportional to the maximal measurement error. The proof of the synchronization result uses Lyapunov functions with exponential weights.}, language = {en} } @unpublished{BernhardLiersStingl, author = {Bernhard, Daniela and Liers, Frauke and Stingl, Michael}, title = {Single and joint chance-constrained optimization with continuous distributions}, abstract = {The input parameters of an optimization problem are often affected by uncertainties. Chance constraints are a common way to model stochastic uncertainties in the constraints. Typically, algorithms for solving chance-constrained problems require convex functions or discrete probability distributions. In this work, we go one step further and allow non-convexities as well as continuous distributions. We propose a gradient-based approach to approximately solve joint chance-constrained models. We approximate the original problem by smoothing indicator functions. Then, the smoothed chance constraints are relaxed by penalizing their violation in the objective function. The approximation problem is solved with the Continuous Stochastic Gradient method that is an enhanced version of the stochastic gradient descent and has recently been introduced in the literature. We present a convergence theory for the smoothing and penalty approximations. Under very mild assumptions, our approach is applicable to a wide range of chance-constrained optimization problems. As an example, we illustrate its computational efficiency on difficult practical problems arising in the operation of gas networks. The numerical experiments demonstrate that the approach quickly finds nearly feasible solutions for joint chance-constrained problems with non-convex constraint functions and continuous distributions, even for realistically-sized instances.}, language = {en} } @article{GeiersbachHenrion2023, author = {Geiersbach, Caroline and Henrion, Ren{\´e}}, title = {Optimality conditions in control problems with random state constraints in probabilistic or almost-sure form}, year = {2023}, abstract = {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.}, language = {en} } @article{GeiersbachHenrionPerezAros2023, author = {Geiersbach, Caroline and Henrion, Ren{\´e} and P{\´e}rez-Aros, Pedro}, title = {Numerical solution of an optimal control problem with probabilistic and almost sure state constraints}, year = {2023}, abstract = {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.}, language = {en} } @unpublished{HanteKuchler, author = {Hante, Falk and Kuchler, Christian}, title = {Indirect methods for optimal control of hybrid PDE-dynamical / switching systems using relaxation}, abstract = {We propose a novel algorithmic approach to computationally solve optimal control problems governed by linear evolution-type PDEs including a state-dependent control-regime switching mechanism. We introduce an equivalent mixed-integer formulation featuring vanishing constraints arising by methods of disjunctive programming. We embed the problem into the class of equilibrium constraints by introduction of an additional slack variable. Based on theoretical results associated with Sum-Up-Rounding strategies, we proceed with the solution of the related relaxed formulation by an indirect approach. In order to obtain a computationally tractable optimality system, we apply a Moreau-Yosida type penalty approach of the vanishing constraints. After the theoretical discussion, we introduce and exert the algorithmic framework founded on a semismooth Newton method. Finally, we communicate computational experiments based on our approach.}, language = {en} } @unpublished{GrimmGruebelSchmidtetal.2023, author = {Grimm, Veronika and Gr{\"u}bel, Julia and Schmidt, Martin and Schwartz, Alexandra and Wiertz, Ann-Kathrin and Z{\"o}ttl, Gregor}, title = {On a Tractable Single-Level Reformulation of a Multilevel Model of the European Entry-Exit Gas Market with Market Power}, pages = {26}, year = {2023}, abstract = {We propose a framework that allows to quantitatively analyze the interplay of the different agents involved in gas trade and transport in the context of the European entry-exit system. While previous contributions focus on the case of perfectly competitive buyers and sellers of gas, our novel framework considers the mathematically more challenging case of a strategic and monopolistic gas seller. We present a multilevel framework that is suitable to capture the sequential nature of the decisions taken. We then derive sufficient conditions that allow for reformulating the challenging four-level model as a computationally tractable single-level reformulation. We prove the correctness of this reformulation and use it for solving several test instances to illustrate the applicability of our approach.}, language = {en} } @article{ShyshkanovaZaytsevaZhushmanetal.2023, author = {Shyshkanova, Ganna and Zaytseva, Tetyana and Zhushman, V and Levchenko, Ntaliia and Korotunova, Olena}, title = {Solving three-dimensional contact problems for foundation design in green building}, doi = {10.1088/1742-6596/2609/1/012001}, year = {2023}, abstract = {Design of foundations on an elastic base is carried out using the solution of three-dimensional problems of contact interaction. Improving the accuracy of engineering calculations is necessary to ensure economic efficiency and increase energy savings in green building. The problems of indentation of punches with a flat base bounded by doubly connected close to polygonal contact areas are researched in the present work. Small parameter method is used to obtain explicit analytical expressions for the contact pressure distribution and the punch displacement dependence in a simplified form, which is convenient for engineering practice. The found load-displacement dependence satisfies the known inequalities that are valid for an arbitrary contact domain. Also a numerical-analytical method is in consideration. It uses the simple layer potential expansion and successive approximations for the problems accounting roughness of the elastic half-space. Roughness coefficient is considered as a parameter of regularization of the integral equation for the smooth contact problem. The results of both methods coincide with sufficient accuracy.}, language = {en} } @phdthesis{Krug2023, author = {Krug, Richard}, title = {Decomposition Methods for Time-Dependent Mixed-Integer Nonlinear Optimization Problems on Graphs}, year = {2023}, abstract = {Decomposition can be the method of choice to deal with optimization problems that contain hard to solve model structures or that are of large scale. The main idea is to decompose the problematic aspects of the problem into multiple smaller blocks that can be solved more easily. Here, the challenge is to combine the single pieces to a solution that is not only feasible but maybe even optimal for the original problem. In many cases, this can be done by introducing an iteration that eventually converges to a desired solution. In this cumulative dissertation, we present several iterative decomposition methods that are tailored to different types of optimization models and use distinct approaches to split up the problems. Our main motivation for this originates from the optimization of gas transport networks, where we encounter partial differential equations as well as discrete control decisions. Additionally, we engage in the related field of district heating network optimization to study the challenges arising from large-scale and fully discretized systems as well as undesirable model features such as, e.g., complementarity constraints. Here, we introduce two temperature mixing models that are well suited for optimization and a number of techniques to speed up the solution process, which are applied in numerical experiments. As a next step, we develop an iterative time-domain decomposition method that is applied to optimal control problems subject to semilinear hyperbolic systems of partial differential equations. For this, we derive first-order optimality conditions that are then split using a non-overlapping decomposition of the time horizon. We exploit the fact that the resulting systems have a primal interpretation as so-called virtual control problems. We prove the convergence of the iterative method and develop a posteriori error estimates. Later, we extend the scheme to systems of ordinary differential equations with mixed- integer controls by using Pontryagin's maximum principle. We again show the convergence and conduct a numerical case study. Moreover, we use a consensus-based version of the classic penalty alternating direction method to solve tailored reformulations of transient gas network problems that allow us to minimize the number of coupling constraints between sub-problems. Here, we utilize the quasi-separable structure of the network to decompose it into sub-networks with more desirable properties. We also discuss different decomposition strategies and test them in a numerical case study. Finally, we present a successive linear relaxation method for mixed-integer nonlinear problems with multivariate Lipschitz continuous nonlinearities. The distinguishing feature of this algorithm is that it exploits no properties of the nonlinearities besides the Lipschitz constants. Therefore, the method is applicable for problems with non-convex or even non-differentiable constraints. The nonlinearities do not even need to be given in a closed form, which allows us to integrate black-box constraints into the model. We prove that the algorithm converges to an approximate global optimum and we provide a worst-case estimate for the number of iterations. The iterative method is applied to stationary gas transport problems, where implicitly given solutions of the differential equations are modeled via black-box constraints.}, language = {en} } @article{OuanesGonzálezGrandónHeitschetal.2023, author = {Ouanes, Nesrine and González Grandón, Tatiana and Heitsch, Holger and Henrion, René}, title = {Optimizing the economic dispatch of weakly-connected mini-grids under uncertainty using joint chance constraints}, year = {2023}, abstract = {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.}, language = {en} } @article{ShyshkanovaWalther2023, author = {Shyshkanova, Ganna and Walther, Andrea}, title = {Optimization of a punch shape with a doubly connected contact domain}, year = {2023}, abstract = {The objective is to optimize the pressure distribution under a rigid punch having a doubly connected contact domain close to a circular ring and interacting with an elastic half-space. The required design variable is the punch shape. The functional to be minimized is the root-mean-square deviation of the pressure distribution from some given distribution. An analytical technique is developed for solving the problem for the punches with doubly connected shape, by reducing to a sequence of similar problems for the circular ring punches using expansions of the simple layer potential. The method of expansion in terms of a small parameter is used. The simple layer potential expansion is proposed when mapping a doubly connected integration domain onto a circular ring by transforming the integration variables and transforming the coordinates of the pole of the kernel. As a result, a sequence of similar problems was obtained for a circular ring to determine the functions characterizing the distribution of normal pressure under the punch in the form of a non-circular ring, as well as the normal displacements, from where the optimal punch shape is determined.}, language = {en} } @unpublished{GoerigkKurtzSchmidtetal.2023, author = {Goerigk, Marc and Kurtz, Jannis and Schmidt, Martin and Th{\"u}rauf, Johannes}, title = {Connections and Reformulations between Robust and Bilevel Optimization}, pages = {20}, year = {2023}, abstract = {Robust and bilevel optimization share the common feature that they involve a certain multilevel structure. Hence, although they model something rather different when used in practice, they seem to have a similar mathematical structure. In this paper, we analyze the connections between different types of robust problems (strictly robust problems with and without decision-dependence of their uncertainty sets, min-max-regret problems, and two-stage robust problems) as well as of bilevel problems (optimistic problems, pessimistic problems, and robust bilevel problems). It turns out that bilevel optimization seems to be more general in the sense that for most types of robust problems, one can find proper reformulations as bilevel problems but not necessarily the other way around. We hope that these results pave the way for a stronger connection between the two fields - in particular to use both theory and algorithms from one field in the other and vice versa.}, language = {en} } @unpublished{GugatQianSokolowski2023, author = {Gugat, Martin and Qian, Meizhi and Sokolowski, Jan}, title = {Topological derivative method for control of wave equation on networks}, year = {2023}, abstract = {The dynamical, boundary optimal control problems on networks are considered. The domain of definition for the distributed parameter system is given by a graph G. The optimal cost function for control problem is further optimized with respect to the shape and topology of the graph Ω. The small cycle is introduced and the topological derivative of the cost with respect to the size of the cycle is determined. In this way, the singular perturbations of the graph can be analyzed in order to change the topology Ω. The topological derivative method in shape and topology optimization is a new tool which can be used to minimize the shape functionals under the Partial Differential Equations (PDEs) constraints. The topological derivative is used as well for solution of optimum design problems for graphs. In optimal control problems the topological derivative is used for optimum design of the domain of integration of the state equation. As an example, optimal control problems are considered on a cross with a small cycle. The state equation is the wave equation on the graph. The boundary control problem by Neumann conditions at a boundary vertex is solved for a tracking cost function. The shape functional is given by the optimal value of the control cost. The topological derivative of the shape functional is determined for the steady state model with the size of a cycle ε → 0. Numerical results for a model problem are presented.}, language = {en} } @unpublished{KannanKreimeierWalther2023, author = {Kannan, Aswin and Kreimeier, Timo and Walther, Andrea}, title = {On Solving Nonsmooth Retail Portfolio Maximization Problems Using Active Signature Methods}, year = {2023}, language = {en} } @unpublished{HanteKuchler2023, author = {Hante, Falk and Kuchler, Christian}, title = {An Algorithmic Framework for Optimal Control of Hybrid Dynamical System with Parabolic PDEs}, year = {2023}, abstract = {We present an algorithmic approach for the computational solution of optimal control problems with hybrid nature governed by linear parabolic PDEs featuring implicit switches. We propose a stepwise reformulation of the original formulation into a more tractable setting via application of methods from disjunctive programming and a time transformation method. After removal of the implicit switching rule at the cost of the introduction of explicit switching variables and vanishing constraints, the connection of the resulting formulation to problems with equilibrium constraints is established and studied. The previous steps in combination with smoothening and a Moreau-Yosida type penalty approach allow the derivation of necessary first order optimality conditions to characterize candidates for optimality to the original system. Following the discussion of each individual reformulation step, we introduce the algorithmic framework founded on a semismooth Newton method. Finally, we report on computational of the proposed framework.}, language = {en} } @unpublished{GoessMartinPokuttaetal., author = {G{\"o}ß, Adrian and Martin, Alexander and Pokutta, Sebastian and Sharma, Kartikey}, title = {Norm-induced Cuts: Optimization with Lipschitzian Black-box Functions}, abstract = {Optimal control problems usually involve constraints which model physical states and their possible transitions. These are represented by ordinary or partial differential equations (ODEs/PDEs) which add a component of infinite dimension to the problem. In recent literature, one method to simulate such ODEs/PDEs are physics-informed neural networks. Typically, neural networks are highly non-linear which makes their addition to optimization problems challenging. Hence, we leverage their often available Lipschitz property on a compact domain. The respective Lipschitz constants have to be computed only once and are accessible thereafter. We present a method that, based on this property, iteratively adds cuts involving the violation of the constraints by the current incumbent and the Lipschitz constant. Hereby, the "shape" of a cut depends on the norm used. We prove the correctness of the method by showing that it either returns an optimal solution when terminating or creates a sequence with optimal accumulation points. This is complemented by a discussion about the termination in the infeasible case, as well as an analysis of the problem complexity. For the analysis, we show that the lower and upper iteration bound asymptotically coincide when the relative approximation error goes to zero. In the end, we visualize the method on a small example based on a two-dimensional non-convex optimization problem, as well as stress the necessity of having a globally optimal oracle for the sub-problems by another example.}, language = {en} } @unpublished{BongartiHintermueller2023, author = {Bongarti, Marcelo and Hinterm{\"u}ller,}, title = {Optimal boundary control of the isothermal semilinear Euler equation for gas dynamics on a network}, year = {2023}, abstract = {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.}, language = {en} } @inproceedings{GugatSchuster2022, author = {Gugat, Martin and Schuster, Michael}, title = {Max-p optimal boundary control of gas flow}, volume = {Extended Abstracts of the 25th International Symposium on Mathematical Theory of Networks and Systems Bayreuth, Germany, 12-16 September 2022}, doi = {https://doi.org/10.15495/EPub_UBT_00006809}, pages = {4}, year = {2022}, abstract = {In the transition to renewable energy sources, hydrogen will potentially play an important role for energy storage. The efficient transport of this gas is possible via pipelines. An understanding of the possibilities to control the gas flow in pipelines is one of the main building blocks towards the optimal use of gas. For the operation of gas transport networks it is important to take into account the randomness of the consumers' demand, where often information on the probability distribution is available. Hence in an efficient optimal control model the corresponding probability should be included and the optimal control should be such that the state that is generated by the optimal control satisfies given state constraints with large probability. We comment on the modelling of gas pipeline flow and the problems of optimal nodal control with random demand, where the aim of the optimization is to determine controls that generate states that satisfy given pressure bounds with large probability. We include the H2 norm of the control as control cost, since this avoids large pressure fluctuations which are harmful in the transport of hydrogen since they can cause embrittlement of the pipeline metal.}, language = {en} } @article{GugatLazar2023, author = {Gugat, Martin and Lazar, Martin}, title = {Turnpike Properties for Partially Uncontrollable Systems}, volume = {Automatica}, number = {149}, year = {2023}, abstract = {We analyse the turnpike properties for a general, infinite dimensional, linear-quadratic (LQ) optimal control problem, both in the deterministic and in the stochastic case. The novelty of the paper is twofold. Firstly, it obtains positive turnpike results for systems that are (partially) uncontrollable. Secondly, it provides turnpike results for averaged control associated to a family of problems that depend on a random parameter, which is the first turnpike type result in the averaged controllability framework.}, language = {en} } @unpublished{SchusterSakamoto2023, author = {Schuster, Michael and Sakamoto, Noboru}, title = {A Turnpike Result for Optimal Boundary Control Problems with the Transport Equation under Uncertainty}, year = {2023}, abstract = {In this paper we analyze the turnpike phenomenon for optimal boundary control problems with a linear transport equation with source term. The convex objective function depends on the boundary traces of the transport equation and is strictly convex with respect to the boundary control. We show an integral turnpike result for an optimal Dirichlet boundary control problem in the sense that if the time horizon goes to infinity, then the dynamic optimal control converges to the corresponding steady state optimal control. The novelty of this work is two-sided. On the one hand, even if turnpike results for this kind of optimal boundary control problem already exist, we present a new direct proof without using adjoint calculus that leads to sharper estimates. On the other hand we consider uncertainty in the initial data and/or in the source term. We show that the integral turnpike result also holds considering uncertainty. Throughout the paper we use numerical examples to illustrate the results.}, language = {en} } @unpublished{GiesselmannGugatKunkel2023, author = {Giesselmann, Jan and Gugat, Martin and Kunkel, Teresa}, title = {Observer-based data assimilation for barotropic gas transport using distributed measurements}, year = {2023}, abstract = {We consider a state estimation problem for gas pipeline flow modeled by the one-dimensional barotropic Euler equations. In order to reconstruct the system state, we construct an observer system of Luenberger type based on distributed measurements of one state variable. First, we show the existence of Lipschitz-continuous semi-global solutions of the observer system and of the original system for initial and boundary data satisfying smallness and compatibility conditions for a single pipe and for general networks. Second, based on an extension of the relative energy method we prove that the state of the observer system converges exponentially in the long time limit towards the original system state. We show this for a single pipe and for star-shaped networks.}, language = {en} } @article{EgererGrimmNiazmandetal.2023, author = {Egerer, Jonas and Grimm, Veronika and Niazmand, Kiana and Runge, Philipp}, title = {The economics of global green ammonia trade - "Shipping Australian wind and sunshine to Germany"}, series = {Applied Energy}, volume = {334}, journal = {Applied Energy}, doi = {https://doi.org/10.1016/j.apenergy.2023.120662}, pages = {15}, year = {2023}, abstract = {This paper contributes to understanding the transformation of global energy trade to green energy carriers, focusing on green ammonia as the foreseeable first green hydrogen carrier. We provide a comprehensive overview of today's ammonia trade and assess scaling options for the trade of green ammonia. To that aim, we develop an optimization model for the integrated assessment of the green ammonia value chain that covers all steps from green ammonia production in an exporting country, up to delivery to a harbor in an importing country. The model endogenously chooses among different technology options and determines cost minimal operation. In a case study, we apply the model to the large-scale import of ammonia from Australia to Germany in a scenario for 2030. The results show that green ammonia can reach cost parity with gray ammonia even for moderate gas prices (but not necessarily with blue ammonia) if CO2 prices are high enough. We also provide a sensitivity analysis with respect to the interest rate and other key technical and economic parameters and show that cracking ammonia to provide pure hydrogen comes at a 45 \% cost markup per MWh at the destination.}, language = {en} } @unpublished{HanteSchmidt2023, author = {Hante, Falk M. and Schmidt, Martin}, title = {Gas Transport Network Optimization: Mixed-Integer Nonlinear Models}, pages = {8}, year = {2023}, abstract = {Although modern societies strive towards energy systems that are entirely based on renewable energy carriers, natural gas is still one of the most important energy sources. This became even more obvious in Europe with Russia's 2022 war against the Ukraine and the resulting stop of gas supplies from Russia. Besides that it is very important to use this scarce resource efficiently. To this end, it is also of significant relevance that its transport is organized in the most efficient, i.e., cost- or energy-efficient, way. The corresponding mathematical optimization models have gained a lot of attention in the last decades in different optimization communities. These models are highly nonlinear mixed-integer problems that are constrained by algebraic constraints and partial differential equations (PDEs), which usually leads to models that are not tractable. Hence, simplifications have to be made and in this chapter, we present a commonly accepted finite-dimensional stationary model, i.e., a model in which the steady-state solutions of the PDEs are approximated with algebraic constraints. For more details about the involved PDEs and the treatment of transient descriptions we refer to Hante and Schmidt (2023). The presented finite-dimensional as well as mixed-integer nonlinear and nonconvex model is still highly challenging if it needs to be solved for real-world gas transport networks. Hence, we also review some classic solution approaches from the literature.}, language = {en} } @unpublished{HanteSchmidt2023, author = {Hante, Falk M. and Schmidt, Martin}, title = {Gas Transport Network Optimization: PDE-Constrained Models}, pages = {8}, year = {2023}, abstract = {The optimal control of gas transport networks was and still is a very important topic for modern economies and societies. Accordingly, a lot of research has been carried out on this topic during the last years and decades. Besides mixed-integer aspects in gas transport network optimization, one of the main challenges is that a physically and technically detailed modeling of transient gas dynamics leads to theoretically and computationally highly demanding models involving nonlinear partial differential equations (PDEs). For further background on the application, historical notes and a detailed discussion of mixed-integer aspects for stationary descriptions we refer to Hante and Schmidt (2023). In this chapter, we focus on the most common modeling approaches concerning transient descriptions, point out the challenges, and summarize important contributions concerning the optimization of the most relevant control parameters for this particular class of problems.}, language = {en} } @unpublished{KrugLeugeringMartinetal.2022, author = {Krug, Richard and Leugering, G{\"u}nter and Martin, Alexander and Schmidt, Martin and Weninger, Dieter}, title = {A Consensus-Based Alternating Direction Method for Mixed-Integer and PDE-Constrained Gas Transport Problems}, pages = {26}, year = {2022}, abstract = {We consider dynamic gas transport optimization problems, which lead to large-scale and nonconvex mixed-integer nonlinear optimization problems (MINLPs) on graphs. Usually, the resulting instances are too challenging to be solved by state-of-the-art MINLP solvers. In this paper, we use graph decompositions to obtain multiple optimization problems on smaller blocks, which can be solved in parallel and which may result in simpler classes of optimization problems since not every block necessarily contains mixed-integer or nonlinear aspects. For achieving feasibility at the interfaces of the several blocks, we employ a tailored consensus-based penalty alternating direction method. Our numerical results show that such decomposition techniques can outperform the baseline approach of just solving the overall MINLP from scratch. However, a complete answer to the question of how to decompose MINLPs on graphs in dependence of the given model is still an open topic for future research.}, language = {en} } @unpublished{SchmidtThuerauf2022, author = {Schmidt, Martin and Th{\"u}rauf, Johannes}, title = {An Exact Method for Nonlinear Network Flow Interdiction Problems}, pages = {28}, year = {2022}, abstract = {We study network flow interdiction problems with nonlinear and nonconvex flow models. The resulting model is a max-min bilevel optimization problem in which the follower's problem is nonlinear and nonconvex. In this game, the leader attacks a limited number of arcs with the goal to maximize the load shed and the follower aims at minimizing the load shed by solving a transport problem in the interdicted network. We develop an exact algorithm consisting of lower and upper bounding schemes that computes an optimal interdiction under the assumption that the interdicted network remains weakly connected. The main challenge consists of computing valid upper bounds for the maximal load shed, whereas lower bounds can directly be derived from the follower's problem. To compute an upper bound, we propose solving a specific bilevel problem, which is derived from restricting the flexibility of the follower when adjusting the load flow. This bilevel problem still has a nonlinear and nonconvex follower's problem, for which we then prove necessary and sufficient optimality conditions. Consequently, we obtain equivalent single-level reformulations of the specific bilevel model to compute upper bounds. Our numerical results show the applicability of this exact approach using the example of gas networks.}, language = {en} } @unpublished{ShyshkanovaWalther2022, author = {Shyshkanova, Ganna and Walther, Andrea}, title = {Contact Pressure over Doubly Connected Rectangular Domains and Punch Shape Optimization}, year = {2022}, abstract = {Contact problems arise in a variety of industrial processes, engineering and biomechanical systems. 3-D contact problem for a rigid punch with a doubly connected base bounded by the lines close to rectangles is in consideration. An analytic-numerical technique is developed for its solving. The problem contains Fredholm integral equations of the first kind, which are transformed into the second kind by means of regularization. Using the simple layer potential expansion, the kernels of the integrals are presented in the form of expansions in the powers of the polar radius. The difference between the values of the desired function at different points and the subsequent interpolation of the terms are proposed to smooth the kernels and eliminate singularities. The integral equations are reduced to one-dimension and then solved using quadrature formulas. Subsequently a punch shape is taken as a desired function, and as a minimizing functional is considered the root-mean-square deviation of the pressure distribution arising under the punch from some optimal distribution. In this case, the values of the total forces and moments applied to the punch are assumed to be given, which leads to restrictions imposed on the distributions by the equilibrium conditions. The normal displacements are determined which arising under the action of the found contact pressure on the elastic half-space. The desired punch shape is found using the simple layer potential. A solution to the problem is obtained for the punch with the doubly connected base bounded by lines close to rectangles.}, language = {en} } @unpublished{KreimeierPokuttaWaltheretal.2022, author = {Kreimeier, Timo and Pokutta, Sebastian and Walther, Andrea and Woodstock, Zev}, title = {On a Frank-Wolfe Approach for Abs-smooth Functions}, year = {2022}, abstract = {We propose an algorithm which appears to be the first bridge between the fields of conditional gradient methods and abs-smooth optimization. Our nonsmooth nonconvex problem setting is motivated by machine learning, since the broad class of abs-smooth functions includes, for instance, the squared \$\ell_2\$-error of a neural network with ReLU or hinge Loss activation. To overcome the nonsmoothness in our problem, we propose a generalization to the traditional Frank-Wolfe gap and prove that first-order minimality is achieved when it vanishes. We derive a convergence rate for our algorithm which is identical to the smooth case. Although our algorithm necessitates the solution of a subproblem which is more challenging than the smooth case, we provide an efficient numerical method for its partial solution, and we identify several applications where our approach fully solves the subproblem. Numerical and theoretical convergence is demonstrated, yielding several conjectures.}, language = {en} } @unpublished{GruebelKrugSchmidtetal.2022, author = {Gr{\"u}bel, Julia and Krug, Richard and Schmidt, Martin and Wollner, Winnifried}, title = {A Successive Linear Relaxation Method for MINLPs with Multivariate Lipschitz Continuous Nonlinearities}, pages = {34}, year = {2022}, abstract = {We present a novel method for mixed-integer optimization problems with multivariate and Lipschitz continuous nonlinearities. In particular, we do not assume that the nonlinear constraints are explicitly given but that we can only evaluate them and that we know their global Lipschitz constants. The algorithm is a successive linear relaxation method in which we alternate between solving a master problem, which is a mixed-integer linear relaxation of the original problem, and a subproblem, which is designed to tighten the linear relaxation of the next master problem by using the Lipschitz information about the respective functions. By doing so, we follow the ideas of Schmidt et al. (2018, 2021) and improve the tackling of multivariate constraints. Although multivariate nonlinearities obviously increase modeling capabilities, their incorporation also significantly increases the computational burden of the proposed algorithm. We prove the correctness of our method and also derive a worst-case iteration bound. Finally, we show the generality of the addressed problem class and the proposed method by illustrating that both bilevel optimization problems with nonconvex and quadratic lower levels as well as nonlinear and mixed-integer models of gas transport can be tackled by our method. We provide the necessary theory for both applications and briefly illustrate the outcomes of the new method when applied to these two problems.}, language = {en} } @unpublished{HorlaenderSchmidt2022, author = {Horl{\"a}nder, Andreas and Schmidt, Martin}, title = {A Penalty Branch-and-Bound Method for Mixed-Integer Quadratic Bilevel Problems}, pages = {9}, year = {2022}, abstract = {We propose an algorithm for solving bilevel problems with mixed-integer convex-quadratic upper level as well as convex-quadratic and continuous lower level. The method is based on a classic branch-and-bound procedure, where branching is performed on the integer constraints and on the complementarity constraints resulting from the KKT reformulation of the lower-level problem. However, instead of branching on constraints as usual, suitably chosen penalty terms are added to the objective function in order to create new subproblems in the tree. We prove the correctness of the method and present its applicability by some first numerical results.}, language = {en} } @unpublished{AignerBaermannBraunetal.2023, author = {Aigner, Kevin-Martin and B{\"a}rmann, Andreas and Braun, Kristin and Liers, Frauke and Pokutta, Sebastian and Schneider, Oskar and Sharma, Kartikey and Tschuppik, Sebastian}, title = {Data-driven Distributionally Robust Optimization over Time}, year = {2023}, abstract = {Stochastic Optimization (SO) is a classical approach for optimization under uncertainty that typically requires knowledge about the probability distribution of uncertain parameters. As the latter is often unknown, Distributionally Robust Optimization (DRO) provides a strong alternative that determines the best guaranteed solution over a set of distributions (ambiguity set). In this work, we present an approach for DRO over time that uses online learning and scenario observations arriving as a data stream to learn more about the uncertainty. Our robust solutions adapt over time and reduce the cost of protection with shrinking ambiguity. For various kinds of ambiguity sets, the robust solutions converge to the SO solution. Our algorithm achieves the optimization and learning goals without solving the DRO problem exactly at any step. We also provide a regret bound for the quality of the online strategy which converges at a rate of \$ O(\log T / \sqrt{T})\$, where \$T\$ is the number of iterations. Furthermore, we illustrate the effectiveness of our procedure by numerical experiments on mixed-integer optimization instances from popular benchmark libraries and give practical examples stemming from telecommunications and routing. Our algorithm is able to solve the DRO over time problem significantly faster than standard reformulations.}, language = {en} } @unpublished{MolanSchmidt2022, author = {Molan, Ioana and Schmidt, Martin}, title = {Using Neural Networks to Solve Linear Bilevel Problems with Unknown Lower Level}, pages = {16}, year = {2022}, abstract = {Bilevel problems are used to model the interaction between two decision makers in which the lower-level problem, the so-called follower's problem, appears as a constraint in the upper-level problem of the so-called leader. One issue in many practical situations is that the follower's problem is not explicitly known by the leader. For such bilevel problems with unknown lower-level model we propose the use of neural networks to learn the follower's optimal response for given decisions of the leader based on available historical data of pairs of leader and follower decisions. Integrating the resulting neural network in a single-level reformulation of the bilevel problem leads to a challenging model with a black-box constraint. We exploit Lipschitz optimization techniques from the literature to solve this reformulation and illustrate the applicability of the proposed method with some preliminary case studies using academic and linear bilevel instances.}, language = {en} } @unpublished{GutinaBaermannRoederetal.2022, author = {Gutina, Daria and B{\"a}rmann, Andreas and Roeder, Georg and Schellenberger, Martin and Liers, Frauke}, title = {Optimisation over Decision Trees - A Case Study for the Design of Stable Direct-Current Electricity Networks}, year = {2022}, abstract = {In many real-world mixed-integer optimisation problems from engineering, the side constraints can be subdivided into two categories: constraints which describe a certain logic to model a feasible allocation of resources (such as a maximal number of available assets, working time requirements, maintenance requirements, contractual obligations, etc.), and constraints which model physical processes and the related quantities (such as current, pressure, temperature, etc.). While the first type of constraints can often easily be stated in terms of a mixed-integer program (MIP), the second part may involve the incorporation of complex non-linearities, partial differential equations or even a black-box simulation of the involved physical process. In this work, we propose the integration of a trained tree-based classifier - a decision-tree or a random forest, into a mixed-integer optimization model as a possible remedy. We assume that the classifier has been trained on data points produced by a detailed simulation of a given complex process to represent the functional relationship between the involved physical quantities. We then derive MIP-representable reformulations of the trained classifier such that the resulting model can be solved using state-of-the-art solvers. At the hand of several use cases in terms of possible optimisation goals, we show the broad applicability of our framework that is easily extendable to other tasks beyond engineering. In a detailed real-world computational study for the design of stable direct- current power networks, we demonstrate that our approach yields high-quality solutions in reasonable computation times.}, language = {en} } @article{GugatSchusterSteffensen2022, author = {Gugat, Martin and Schuster, Michael and Steffensen, Sonja}, title = {A Dynamic Multilevel Model of the European Gas Market}, number = {Volume 2023}, publisher = {Communications in Optimization Theory}, pages = {1 -- 26}, year = {2022}, abstract = {The European gas market is governed by rules that are agreed on by the European Union. We present a mathematical market model that takes into account this structure, where the technical system operator (TSO) offers certain transportation capacities that can be booked and later nominated within the previously chosen bookings. The TSO also fixes booking fees and defines an operational control of the gas pipeline system in order to deliver the gas according to the nominations. Since the gas flow is governed by a system of partial differential equations, to realize this control structure partial differential equations (PDEs) should be involved in the model. While the four level gas market model has been discussed previously, in this paper we take into account the flow model by PDEs in the discussion of the model and in the reduction to a single level problem, where we also state the corresponding necessary optimality conditions.}, language = {en} } @phdthesis{Gruebel, author = {Gr{\"u}bel, Julia}, title = {Existence, Uniqueness, and Algorithms for Equilibria in Competitive Energy Markets}, doi = {10.25353/ubtr-xxxx-4c49-7f53}, abstract = {Due to the transition towards climate neutrality, energy markets are rapidly evolving. New technologies are developed that allow electricity from renewable energy sources to be stored or to be converted into other energy commodities. As a consequence, new players enter the markets and existing players gain more importance. Market equilibrium problems are capable of capturing these changes and therefore enable us to answer contemporary research questions with regard to energy market design and climate policy. This cumulative dissertation is devoted to the study of different market equilibrium problems that address such emerging aspects in liberalized energy markets. In the first part, we review a well-studied competitive equilibrium model for energy commodity markets and extend this model by sector coupling, by temporal coupling, and by a more detailed representation of physical laws and technical requirements. Moreover, we summarize our main contributions of the last years with respect to analyzing the market equilibria of the resulting equilibrium problems. For the extension regarding sector coupling, we derive sufficient conditions for ensuring uniqueness of the short-run equilibrium a priori and for verifying uniqueness of the long-run equilibrium a posteriori. Furthermore, we present illustrative examples that each of the derived conditions is indeed necessary to guarantee uniqueness in general. For the extension regarding temporal coupling, we provide sufficient conditions for ensuring uniqueness of demand and production a priori. These conditions also imply uniqueness of the short-run equilibrium in case of a single storage operator. However, in case of multiple storage operators, examples illustrate that charging and discharging decisions are not unique in general. We conclude the equilibrium analysis with an a posteriori criterion for verifying uniqueness of a given short-run equilibrium. Since the computation of equilibria is much more challenging due to the temporal coupling, we shortly review why a tailored parallel and distributed alternating direction method of multipliers enables to efficiently compute market equilibria. For the extension regarding physical laws and technical requirements, we show that, in nonconvex settings, existence of an equilibrium is not guaranteed and that the fundamental welfare theorems therefore fail to hold. In addition, we argue that the welfare theorems can be re-established in a market design in which the system operator is committed to a welfare objective. For the case of a profit-maximizing system operator, we propose an algorithm that indicates existence of an equilibrium and that computes an equilibrium in the case of existence. Based on well-known instances from the literature on the gas and electricity sector, we demonstrate the broad applicability of our algorithm. Our computational results suggest that an equilibrium often exists for an application involving nonconvex but continuous stationary gas physics. In turn, integralities introduced due to the switchability of DC lines in DC electricity networks lead to many instances without an equilibrium. Finally, we state sufficient conditions under which the gas application has a unique equilibrium and the line switching application has finitely many. In the second part, all preprints belonging to this cumulative dissertation are provided. These preprints, as well as two journal articles to which the author of this thesis contributed, are referenced within the extended summary in the first part and contain more details.}, language = {en} } @unpublished{BeckLjubicSchmidt2022, author = {Beck, Yasmine and Ljubic, Ivana and Schmidt, Martin}, title = {A Survey on Bilevel Optimization Under Uncertainty}, pages = {57}, year = {2022}, abstract = {Bilevel optimization is a very active field of applied mathematics. The main reason is that bilevel optimization problems can serve as a powerful tool for modeling hierarchical decision making processes. This ability, however, also makes the resulting problems challenging to solve - both in theory and practice. Fortunately, there have been significant algorithmic advances in the field of bilevel optimization so that we can solve much larger and also more complicated problems today compared to what was possible to solve two decades ago. This results in more and more challenging bilevel problems that researchers try to solve today. This survey gives a detailed overview of one of these more challenging classes of bilevel problems: bilevel optimization under uncertainty. We review the classic ways of addressing uncertainties in bilevel optimization using stochastic or robust techniques. Moreover, we highlight that the sources of uncertainty in bilevel optimization are much richer than for usual, i.e., single-level, problems since not only the problem's data can be uncertain but also the (observation of the) decisions of the two players can be subject to uncertainty. We thus also review the field of bilevel optimization under limited observability, the area of problems considering only near-optimal decisions, and discuss intermediate solution concepts between the optimistic and pessimistic cases. Finally, we also review the rich literature on applications studied using uncertain bilevel problems such as in energy, for interdiction games and security applications, in management sciences, and networks.}, language = {en} } @article{HernandezLecarosZamorano2022, author = {Hernandez, Martin and Lecaros, Rodrigo and Zamorano, Sebastian}, title = {Averaged turnpike property for differential equations with random constant coefficients}, year = {2022}, abstract = {This paper studies the integral turnpike and turnpike in average for a class of random or- dinary differential equations. We prove that, under suitable assumptions on the matrices that define the system, the optimal solutions for an optimal distributed control tracking problem remain, in an averaged sense, sufficiently close to the associated random stationary optimal solution for the majority of the time horizon}, language = {en} } @incollection{Leugering2022, author = {Leugering, G{\"u}nter}, title = {Nonoverlapping Domain Decomposition in Space and Time for Optimal Control Problems on Metric Graphs by the Example of Gas Flow in Pipe Networks}, year = {2022}, abstract = {We consider non-overlapping domain decomposition methods for ordinary and partial differential equations and corresponding optimal control problems on metric graphs. As an exemplary context, we chose a semilinear approximation of the Euler system and a doubly nonlinear parabolic model that has come to be known as friction dominated flow in gas pipe networks. By this choice, we encounter hyperbolic, parabolic and elliptic linear and nonlinear problems in a single highly motivating application. We depart from the classical domain decomposition methods described by P.L. Lions and J.L. Lions and O. Pironneau and extend those to problems on metric graphs. The choice of methods is determined by the desire to use a control concept that has come to be known as virtual controls which, in turn, possibly lead to a fully parallel decomposition of the corresponding optimality systems. In a second step, we extend the methods to p-Laplace problems on networks. The analysis, due to space limitations, will appear in a forthcoming publication. See however J.E. Lagnese and G. Leugering. Furthermore, we then describe methods for space and time domain decomposition or optimal control problems in the spirit of J.E. Lagnese and G. Leugering. We finally provide some comments on PINN-based approximations of the methods described before. We provide numerical evidence for all algorithms discussed.}, language = {en} } @unpublished{BeckSchmidtThueraufetal.2022, author = {Beck, Yasmine and Schmidt, Martin and Th{\"u}rauf, Johannes and Bienstock, Daniel}, title = {On a Computationally Ill-Behaved Bilevel Problem with a Continuous and Nonconvex Lower Level}, pages = {16}, year = {2022}, abstract = {It is well known that bilevel optimization problems are hard to solve both in theory and practice. In this paper, we highlight a further computational difficulty when it comes to solving bilevel problems with continuous but nonconvex lower levels. Even if the lower-level problem is solved to ɛ-feasibility regarding its nonlinear constraints for an arbitrarily small but positive ɛ, the obtained bilevel solution as well as its objective value may be arbitrarily far away from the actual bilevel solution and its actual objective value. This result even holds for bilevel problems for which the nonconvex lower level is uniquely solvable, for which the strict complementarity condition holds, for which the feasible set is convex, and for which Slater's constraint qualification is satisfied for all feasible upper-level decisions. Since the consideration of ɛ-feasibility cannot be avoided when solving nonconvex problems to global optimality, our result shows that computational bilevel optimization with continuous and nonconvex lower levels needs to be done with great care. Finally, we illustrate that the nonlinearities in the lower level are the key reason for the observed bad behavior by showing that linear bilevel problems behave much better at least on the level of feasible solutions.}, language = {en} } @unpublished{HannesVolkerRolandetal.2022, author = {Hannes, D{\"a}nschel and Volker, Mehrmann and Roland, Marius and Schmidt, Martin}, title = {Adaptive Nonlinear Optimization of District Heating Networks Based on Model and Discretization Catalogs}, pages = {29}, year = {2022}, abstract = {We propose an adaptive optimization algorithm for operating district heating networks in a stationary regime. The behavior of hot water flow in the pipe network is modeled using the incompressible Euler equations and a suitably chosen energy equation. By applying different simplifications to these equations, we derive a catalog of models. Our algorithm is based on this catalog and adaptively controls where in the network which model is used. Moreover, the granularity of the applied discretization is controlled in a similar adaptive manner. By doing so, we are able to obtain optimal solutions at low computational costs that satisfy a prescribed tolerance w.r.t. the most accurate modeling level. To adaptively control the switching between different levels and the adaptation of the discretization grids, we derive error measure formulas and a posteriori error measure estimators. Under reasonable assumptions we prove that the adaptive algorithm terminates after finitely many iterations. Our numerical results show that the algorithm is able to produce solutions for problem instances that have not been solvable before.}, language = {en} } @unpublished{Leugering2022, author = {Leugering, G{\"u}nter}, title = {Nonoverlapping Domain Decomposition for Instantaneous Optimal Control of Friction Dominated Flow in a Gas-Network}, pages = {15}, year = {2022}, abstract = {We consider a non-overlapping domain decomposition method for an optimal control problem related to the flow of gas in a pipe network. The equations of motions are taken to be represented by a friction dominated model derived from a semi-linear approximation of the fully nonlinear isothermal Euler gas equations. This involves a p-Laplace-type problem on the graph with p = 3/2. We continue the work by Leugering and Mophou where such a problem has been discussed in the context of an instantaneous control strategy. We provide a non-overlapping domain decomposition in the spirit of P.L. Lions for elliptic problems and extend the method to the first order optimality system.}, language = {en} } @unpublished{SchaeferAguilarUlbrich2021, author = {Sch{\"a}fer Aguilar, Paloma and Ulbrich, Stefan}, title = {Convergence of numerical adjoint schemes arising from optimal boundary control problems of hyperbolic conservation laws}, year = {2021}, abstract = {We study the convergence of discretization schemes for the adjoint equation arising in the adjoint-based derivative computation for optimal boundary control problems governed by entropy solutions of conservation laws. As boundary control we consider piecewise continuously differentiable controls with possible discontinuities at switching times, where the smooth parts as well as the switching times serve as controls. The derivative of tracking-type objective functionals with respect to the smooth controls and the switching times can then be represented by an adjoint-based formula. The main difficulties arise from the fact that the correct adjoint state is the reversible solution of a transport equation with discontinuous coefficient and boundary conditions that lead in general to discontinuous adjoints. Moreover, the solution of the adjoint equation is non-unique and the so-called reversible solution leads to the correct adjoint-based derivative representation. We study discrete adjoint schemes of monotone difference schemes in conservation form such as Engquist-Osher or Lax-Friedrichs scheme. We also allow that the state is computed by another numerical scheme satisfying certain convergence properties. We proof convergence results of the discrete adjoint to the reversible solution.}, language = {en} } @unpublished{AignerSchaumannvonLoeperetal., author = {Aigner, Kevin-Martin and Schaumann, Peter and von Loeper, Freimut and Martin, Alexander and Schmidt, Volker and Liers, Frauke}, title = {Robust DC Optimal Power Flow with Modeling of Solar Power Supply Uncertainty via R-Vine Copulas}, abstract = {We present a robust approximation of joint chance constrained DC Optimal Power Flow in combination with a model-based prediction of uncertain power supply via R-vine copulas. It is applied to optimize the discrete curtailment of solar feed-in in an electrical distribution network and guarantees network stability under fluctuating feed-in. This is modeled by a two-stage mixed-integer stochastic optimization problem proposed by Aigner et al. (European Journal of Operational Research, (2021)). The solution approach is based on the approximation of chance constraints via robust constraints using suitable uncertainty sets. The resulting robust optimization problem has a known equivalent tractable reformulation. To compute uncertainty sets that lead to an inner approximation of the stochastic problem, an R-vine copula model is fitted to the distribution of the multi-dimensional power forecast error, i.e., the difference between the forecasted solar power and the measured feed-in at several network nodes. The uncertainty sets are determined by encompassing a sufficient number of samples drawn from the R-vine copula model. Furthermore, an enhanced algorithm is proposed to fit R-vine copulas which can be used to draw conditional samples for given solar radiation forecasts. The experimental results obtained for real-world weather and network data demonstrate the effectiveness of the combination of stochastic programming and model-based prediction of uncertainty via copulas. We improve the outcomes of previous work by showing that the resulting uncertainty sets are much smaller and lead to less conservative solutions while maintaining the same probabilistic guarantees.}, language = {en} } @article{GugatWeiland2021, author = {Gugat, Martin and Weiland, Sven}, title = {Nodal Stabilization of the Flow in a Network with a Cycle}, series = {Journal of Optimization, Differential Equations and their Applications}, volume = {29}, journal = {Journal of Optimization, Differential Equations and their Applications}, number = {2}, pages = {1 -- 24}, year = {2021}, abstract = {In this paper we discuss an approach to the stability analysis for classical solutions of closed loop systems that is based upon the tracing of the evolution of the Riemann invariants along the characteristics. We consider a network where several edges are coupled through node conditions that govern the evolution of the Riemann invariants through the nodes of the network. The analysis of the decay of the Riemann invariants requires to follow backwards all the characteristics that enter such a node and contribute to the evolution. This means that with each nodal reflection/crossing the number of characteristics that contribute to the evolution increases. We show how for simple networks with a suffcient number of damping nodal controlers it is possible to keep track of this family of characteristics and use this approach to analyze the exponential stability of the system. The analysis is based on an adapted version of Gronwall's lemma that allows us to take into account the possible increase of the Riemann invariants when the characteristic curves cross a node of the network. Our example is motivated by applications in the control of gas pipeline flow, where the graphs of the networks often contain many cycles.}, language = {en} } @unpublished{SauterTischendorf2021, author = {Sauter, Henning and Tischendorf, Caren}, title = {Least squares collocation for the simulation of gas network DAEs}, year = {2021}, language = {en} } @unpublished{SauterTischendorf2021, author = {Sauter, Henning and Tischendorf, Caren}, title = {Optimal control of gas network DAEs}, year = {2021}, language = {en} } @unpublished{HalbigHuembsRoeseletal.2021, author = {Halbig, Katrin and H{\"u}mbs, Lukas and R{\"o}sel, Florian and Schewe, Lars and Weninger, Dieter}, title = {Computing optimality certificates for convex mixed-integer nonlinear problems}, year = {2021}, abstract = {Every optimization problem has a corresponding verification problem which verifies whether a given optimal solution is in fact optimal. In the literature there are a lot of such ways to verify optimality for a given solution, e.g., the branch-and-bound tree. To simplify this task, Baes et al. introduced optimality certificates for convex mixed-integer nonlinear programs and proved that these are bounded in the number of integer variables. We introduce an algorithm to compute the certificates and conduct computational experiments. Through the experiments we show that the optimality certificates can be surprisingly small.}, language = {en} } @unpublished{KreimeierWaltherGriewank2021, author = {Kreimeier, Timo and Walther, Andrea and Griewank, Andreas}, title = {An active signature method for constrained abs-linear minimization}, year = {2021}, abstract = {In this paper we consider the solution of optimization tasks with a piecewise linear objective function and piecewise linear constraints. First, we state optimality conditions for that class of problems using the abs-linearization approach and prove that they can be verified in polynomial time. Subsequently, we propose an algorithm called Constrained Active Signature Method that explicitly exploits the piecewise linear structure to solve such optimization problems. Convergence of the algorithm within a finite number of iterations is proven. Numerical results for various testcases including linear complementarity conditions and a bi-level problem illustrate the performance of the new algorithm.}, language = {en} } @unpublished{GrimmHintermuellerHuberetal., author = {Grimm, Veronika and Hinterm{\"u}ller, Michael and Huber, Olivier and Schewe, Lars and Schmidt, Martin and Z{\"o}ttl, Gregor}, title = {A PDE-Constrained Generalized Nash Equilibrium Approach for Modeling Gas Markets with Transport}, abstract = {We investigate a class of generalized Nash equilibrium problems (GNEPs) in which the objectives of the individuals are interdependent and the shared constraint consists of a system of partial differential equations. This setup is motivated by the modeling of strategic interactions of competing firms, which explicitly take into account the dynamics of transporting a commodity, such as natural gas, through a network. We establish the existence of a variational equilibrium of the GNEP. In the case of symmetric firms, we identify an equivalent optimization problem. We use this model to numerically explore the impact of linepacking, that is the use of the network as a temporary storage device. In particular, we study the firms' decisions under various linepacking abilities and analyze which market participants benefit from it.}, language = {en} } @unpublished{KreimeierSauterStreubeletal.2021, author = {Kreimeier, Timo and Sauter, Henning and Streubel, Tom and Tischendorf, Caren and Walther, Andrea}, title = {Solving Least-Squares Collocated Differential Algebraic Equations by Successive Abs-Linear Minimization - A Case Study on Gas Network Simulation}, year = {2021}, abstract = {This paper studies the numerical simulation of gas networks with regulating elements using differential algebraic equations (DAEs) in combination with least-squares collocation. In contrast to classical collocation methods, more collocation points than degrees of freedom for the collocation polynomials are used. Recently, it has been shown that such a least-squares collocation has a regularizing effect for DAEs, in particular for DAEs with higher index. In each time step of the numerical integration, one has to solve a system of nonlinear equations that is nonsmooth due to the regulating elements in the gas networks. We consider four solvers one of which explicitly exploits the inherent nonsmooth nature. Numerical results are given for three different test cases with increasing complexity illustrating the feasibility of the proposed approach to approximate a solution of the DAE and the advantageous performance of the nonsmooth solver that is based on the concept of abs-linearization.}, language = {en} } @article{Henrion2021, author = {Henrion, Rene}, title = {On M-stationarity conditions for probabilistic MPECs}, year = {2021}, abstract = {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.}, language = {en} } @unpublished{BeckLjubicSchmidt2021, author = {Beck, Yasmine and Ljubic, Ivana and Schmidt, Martin}, title = {Exact Methods for Discrete Γ-Robust Interdiction Problems with an Application to the Bilevel Knapsack Problem}, pages = {39}, year = {2021}, abstract = {Developing solution methods for discrete bilevel problems is known to be a challenging task - even if all parameters of the problem are exactly known. Many real-world applications of bilevel optimization, however, involve data uncertainty. We study discrete min-max problems with a follower who faces uncertainties regarding the parameters of the lower-level problem. Adopting a Γ-robust approach, we present an extended formulation and a multi-follower formulation to model this type of problem. For both settings, we provide a generic branch-and-cut framework. Specifically, we investigate interdiction problems with a monotone Γ-robust follower and we derive problem-tailored cuts, which extend existing techniques that have been proposed for the deterministic case. For the Γ-robust knapsack interdiction problem, we computationally evaluate and compare the performance of the proposed algorithms for both modeling approaches.}, language = {en} } @unpublished{Hante, author = {Hante, Falk}, title = {Relaxation methods for optimal switching control of PDE-dynamical systems}, abstract = {The chapter reviews certain computational approaches to solve optimal control problems for evolution-type partial differential equations, where some control functions are limited to switching. The mechanism that enforces switching is modeled as integer restrictions. This brings a combinatorial aspect into the apart from switching already computationally very demanding optimization problems. Recently, great advances have been made to tackle such problems rigorously using relaxation and combinatorial integral approximation. An overview of these methods and the known theoretical results concerning convergence and error estimates are provided in a consistent manner. Further, we point to applications with benchmark character as well as to open problems.}, language = {en} } @unpublished{CattaruzzaLabbePetrisetal.2021, author = {Cattaruzza, Diego and Labb{\´e}, Martine and Petris, Matteo and Roland, Marius and Schmidt, Martin}, title = {Exact and Heuristic Solution Techniques for Mixed-Integer Quantile Minimization Problems}, pages = {35}, year = {2021}, abstract = {We consider mixed-integer linear quantile minimization problems that yield large-scale problems that are very hard to solve for real-world instances. We motivate the study of this problem class by two important real-world problems: a maintenance planning problem for electricity networks and a quantile-based variant of the classic portfolio optimization problem. For these problems, we develop valid inequalities and present an overlapping alternating direction method. Moreover, we discuss an adaptive scenario clustering method for which we prove that it terminates after a finite number of iterations with a global optimal solution. We study the computational impact of all presented techniques and finally show that their combination leads to an overall method that can solve the maintenance planning problem on large-scale real-world instances provided by the ROADEF challenge 2020 and that they also lead to significant improvements when solving a quantile-version of the classic portfolio optimization problem.}, language = {en} } @unpublished{BreitenSchulze2021, author = {Breiten, Tobias and Schulze, Philipp}, title = {Structure-Preserving Linear Quadratic Gaussian Balanced Truncation for Port-Hamiltonian Descriptor Systems}, year = {2021}, abstract = {We present a new balancing-based structure-preserving model reduc- tion technique for linear port-Hamiltonian descriptor systems. The pro- posed method relies on a modification of a set of two dual generalized algebraic Riccati equations that arise in the context of linear quadratic Gaussian balanced truncation for differential algebraic systems. We de- rive an a priori error bound with respect to a right coprime factorization of the underlying transfer function thereby allowing for an estimate with respect to the gap metric. We further theoretically and numerically ana- lyze the influence of the Hamiltonian and a change thereof, respectively. With regard to this change of the Hamiltonian, we provide a novel proce- dure that is based on a recently introduced Kalman-Yakubovich-Popov inequality for descriptor systems. Numerical examples demonstrate how the quality of reduced-order models can significantly be improved by first computing an extremal solution to this inequality.}, language = {en} } @unpublished{KlimmPfetschRaberetal.2021, author = {Klimm, Max and Pfetsch, Marc and Raber, Rico and Skutella, Martin}, title = {Reduction of Potential-Based Flow Networks}, year = {2021}, abstract = {We consider potential-based flow networks with terminal nodes at which flow can enter or leave the network and physical properties such as voltages or pressures are measured and controlled. We study conditions under which such a network can be reduced to a smaller, equivalent network with the same behavior at the terminal nodes. Potential-based flow networks are widely used to model infrastructure networks such as electricity, gas, or water networks. In contrast to Kron's reduction for electrical networks, we prove that, in general, potential-based flow networks with at least three terminals cannot be reduced to smaller networks whose size only depends on the number of terminals. On the other hand, we show that it is possible to represent a special class of potential-based flow networks by a complete graph on the terminals, and we establish a characterization of networks that can be reduced to a path network. Our results build on fundamental properties of effective resistances proved in this paper, including explicit formulae for their dependence on edge resistances of the network and their metric properties.}, language = {en} } @unpublished{KlimmPfetschRaberetal.2021, author = {Klimm, Max and Pfetsch, Marc and Raber, Rico and Skutella, Martin}, title = {Approximation of Binary Second Order Cone Programs of Packing Type}, year = {2021}, abstract = {This paper considers binary second order cone programs of packing type where a linear objective is optimized under m second order cone packing constraints and all decision variables are binary. We show that when m is part of the input, these problems cannot be approximated within a factor of 1/(m + 1)^(1-ε) for any ε > 0, unless P = NP. We then propose approximation algorithms based on different algorithmic principles that almost match this approximation factor: a pipage rounding technique that solves fractional relaxations of the problems and modifies the solutions so that few fractional variables remain, a greedy approach, and a randomized rounding technique. While all algorithms have similar theoretical approximation guarantees in the order of 1/m, we also test the algorithms on realistic instances that arise in the context of gas transportation networks. This empirical study reveals in particular that taking the best of the proposed algorithms produces highly competitive solutions that yield on average 96 \% of the value of an optimal solution.}, language = {en} } @article{KlimmWarode2021, author = {Klimm, Max and Warode, Philipp}, title = {Parametric computation of minimum cost flows with piecewise quadratic costs}, series = {Mathematics of Operations Research}, journal = {Mathematics of Operations Research}, doi = {10.1287/moor.2021.1151}, year = {2021}, abstract = {We develop algorithms solving parametric flow problems with separable, continuous, piecewise quadratic, and strictly convex cost functions. The parameter to be considered is a common multiplier on the demand of all nodes. Our algorithms compute a family of flows that are each feasible for the respective demand and minimize the costs among the feasible flows for that demand. For single commodity networks with homogenous cost functions, our algorithm requires one matrix multiplication for the initialization, a rank 1 update for each nondegenerate step and the solution of a convex quadratic program for each degenerate step. For nonhomogeneous cost functions, the initialization requires the solution of a convex quadratic program instead. For multi-commodity networks, both the initialization and every step of the algorithm require the solution of a convex program. As each step is mirrored by a breakpoint in the output this yields output-polynomial algorithms in every case.}, language = {en} } @unpublished{PflugGrieshammerUihleinetal.2021, author = {Pflug, Lukas and Grieshammer, Max and Uihlein, Andrian and Stingl, Michael}, title = {CSG: A stochastic gradient method for a wide class of optimization problems appearing in a machine learning or data-driven context}, pages = {22}, year = {2021}, abstract = {In a recent article the so called continuous stochastic gradient method (CSG) for the efficient solution of a class of stochastic optimization problems was introduced. While the applicability of known stochastic gradient type methods is typically limited to so called expected risk functions, no such limitation exists for CSG. The key to this lies in the computation of design dependent integration weights, which allows for an optimal usage of available information leading to stronger convergence properties. However, due to the nature of the formula for these integration weights, the practical applicability was essentially limited to problems, in which stochasticity enters via a low-dimensional and suficiently simple probability distribution. In this paper the scope of the CSG method is significantly extended presenting new ways of calculating the integration weights. A full convergence analysis for this new variant of the CSG method is presented and its efficiency is demonstrated in comparison to more classical stochastic gradient methods by means of a number of problem classes, relevant in stochastic optimization and machine learning.}, language = {en} } @unpublished{BurgerHumpertPietschmann2021, author = {Burger, Martin and Humpert, Ina and Pietschmann, Jan}, title = {Dynamic optimal transport on networks}, year = {2021}, abstract = {In this paper we study a dynamical optimal transport problem on a network that allows for transport of mass between different edges if a penalty κ is paid. We show existence of minimisers using duality and discuss the relationships of the distance-functional to other metrics such as the Fisher-Rao and the classical Wasserstein metric and analyse the resulting distance functional in the limiting case κ → ∞.}, language = {en} } @unpublished{RuizBaletAffiliZuazua, author = {Ruiz-Balet, Dom{\`e}nec and Affili, Elisa and Zuazua, Enrique}, title = {Interpolation and approximation via Momentum ResNets and Neural ODEs}, abstract = {In this article, we explore the effects of memory terms in continuous-layer Deep Residual Networks by studying Neural ODEs (NODEs). We investigate two types of models. On one side, we consider the case of Residual Neural Networks with dependence on multiple layers, more precisely Momentum ResNets. On the other side, we analyse a Neural ODE with auxiliary states playing the role of memory states. We examine the interpolation and universal approximation properties for both architectures through a simultaneous control perspective. We also prove the ability of the second model to represent sophisticated maps, such as parametrizations of time-dependent functions. Numerical simulations complement our study.}, language = {en} } @techreport{OggioniSchwartzZoettletal.2022, type = {Working Paper}, author = {Oggioni, Giorgia and Schwartz, Alexandra and Z{\"o}ttl, Gregor and Wiertz, Ann-Kathrin}, title = {Dynamic Pricing and Strategic Retailers in the Energy Sector: A Multi-Leader-Follower Approach}, pages = {41}, year = {2022}, abstract = {We consider strategic retail pricing in markets, where retail companies buy commodities at fluctuating wholesale prices and resell them to final consumers by applying dynamic retail tariffs. This is of especially large relevance in the context of energy markets where substantial wholesale price fluctuations are observed. Policy makers currently foster the introduction of such dynamic tariff schemes. From a modelling point of view, we propose a multi-leader-follower problem to investigate the implications of strategic retail pricing and we compare the impacts of implementing dynamic tariffs on retailers and final consumers. Our analysis tackles different aspects: first, we formulate the model and provide theoretical results. Second, we develop algorithms, which solve the multi-leader-follower problem and allow us to characterize the resulting market equilibria. Third, we calibrate and solve our framework based on data of the German retail electricity market for the years 2020 and 2021. This allows us to quantitatively assess the impact of introducing real time prices on retailers' profits and customers' benefits. As our results show, dynamic real-time pricing on the one hand typically increases market efficiency, which confirms previous results obtained without the explicit consideration of strategic behavior. On the other hand, however, as a novel aspect, dynamic real-time pricing turns out to significantly reduce equilibrium profits in case of strategic firms. This effect is especially large in environments with strongly fluctuating wholesale prices.}, language = {en} } @article{ReussWelderThueraufetal.2019, author = {Reuß, Markus and Welder, Lara and Th{\"u}rauf, Johannes and Linßen, Jochen and Grube, Thomas and Schewe, Lars and Schmidt, Martin and Stolten, Detlef and Robinius, Martin}, title = {Modeling Hydrogen Networks for Future Energy Systems: A Comparison of Linear and Nonlinear Approaches}, series = {International Journal of Hydrogen Energy}, journal = {International Journal of Hydrogen Energy}, doi = {10.1016/j.ijhydene.2019.10.080}, year = {2019}, abstract = {Common energy system models that integrate hydrogen transport in pipelines typically simplify fluid flow models and reduce the network size in order to achieve solutions quickly. This contribution analyzes two different types of pipeline network topologies (namely, star and tree networks) and two different fluid flow models (linear and nonlinear) for a given hydrogen capacity scenario of electrical reconversion in Germany to analyze the impact of these simplifications. For each network topology, robust demand and supply scenarios are generated. The results show that a simplified topology, as well as the consideration of detailed fluid flow, could heavily influence the total pipeline investment costs. For the given capacity scenario, an overall cost reduction of the pipeline costs of 37\% is observed for the star network with linear cost compared to the tree network with nonlinear fluid flow. The impact of these improvements regarding the total electricity reconversion costs has led to a cost reduction of 1.4\%, which is fairly small. Therefore, the integration of nonlinearities into energy system optimization models is not recommended due to their high computational burden. However, the applied method for generating robust demand and supply scenarios improved the credibility and robustness of the network topology, while the simplified fluid flow consideration can lead to infeasibilities. Thus, we suggest the utilization of the nonlinear model for post- processing to prove the feasibility of the results and strengthen their credibility, while retaining the computational performance of linear modeling.}, language = {en} } @article{GugatHenrionHeitsch2021, author = {Gugat, Martin and Henrion, Ren{\´e} and Heitsch, Holger}, title = {A turnpike property for optimal control problems with dynamic probabilistic constraints}, series = {Journal of Convex Analysis}, volume = {30}, journal = {Journal of Convex Analysis}, number = {3}, publisher = {Heldermann Verlag 2023}, pages = {1025 -- 1052}, year = {2021}, abstract = {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.}, language = {en} } @unpublished{GeiersbachHintermueller2021, author = {Geiersbach, Caroline and Hinterm{\"u}ller, Michael}, title = {Optimality conditions and Moreau-Yosida regularization for almost sure state constraints}, year = {2021}, abstract = {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.}, language = {en} } @unpublished{GahururuHintermuellerSurowiec2021, author = {Gahururu, Deborah and Hinterm{\"u}ller, Michael and Surowiec, Thomas}, title = {Risk-Neutral PDE-Constrained Generalized Nash Equilibrium Problems}, year = {2021}, abstract = {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.}, language = {en} } @article{KreimeierKuchlbauerLiersetal.2022, author = {Kreimeier, Timo and Kuchlbauer, Martina and Liers, Frauke and Stingl, Michael and Walther, Andrea}, title = {Towards the Solution of Robust Gas Network Optimization Problems Using the Constrained Active Signature Method}, year = {2022}, abstract = {This work studies robust gas network optimization under uncertainties in demand and in the physical parameters. The corresponding optimization problems are nonconvex in node pressures and flows along the pipes. They are thus very difficult to solve for realistic instance sizes. In recent approaches, an adaptive bundle method has been developed, where one solves the occurring adversarial problems via iteratively refined piecewise linear relaxations. These subproblems need to be solved always from scratch using mixed-integer linear programming (MIP). As alternative to the MIP solver, we employ here a nonsmooth optimization approach that allows a warm start strategy such that it can profit from the results obtained for coarser relaxations. We evaluate the approach for realistic gas network topologies and outline possibilities for future research.}, language = {en} } @phdthesis{Huck2018, author = {Huck, Christoph}, title = {Perturbation analysis and numerical discretisation of hyperbolic partial differential algebraic equations describing flow networks}, doi = {10.18452/19596}, year = {2018}, abstract = {This thesis addresses several aspects regarding modelling, analysis and numerical simulation of gas networks. Hereby, our focus lies on (partial) differential-algebraic equations, thus systems of partial and ordinary differential equations which are coupled by algebraic equations. These coupled systems allow an easy approach towards the modelling of dynamic structures on networks. Therefore, they are well suited for gas networks, which have gained a rise of attention in society, politics and science due to the focus towards renewable energies. We give an introduction towards gas network modelling that includes the most common elements that also appear in real gas networks and present two PDAE systems: One for pipe networks and one that includes additional elements like resistors and compressors. Furthermore, we investigate the impact of perturbations onto the pipe network PDAE, where we explicitly allow perturbations to affect the system in the differential as well as in the algebraic components. We conclude that the solution of the PDAE possesses stability properties. In addition, this thesis introduces a new spatial discretisation that is adapted to the net- work topology. This topology-adapted semi-discretisation results in a DAE which possesses the same perturbation behaviour as the space continuous PDAE. Furthermore, we present a topology based decoupling procedure that allows to reformulate the DAE as an ordinary differential equation (ODE), which represents the inherent dynamics of the DAE system. This ODE, together with a decoupled set of algebraic equations, can be derived from the topology and element information directly. We conclude by demonstrating the established results for several benchmark networks. This includes a comparison of numerical solutions for the decoupled ODE and the DAE system. In addition we present the advantages of the topology-adapted spatial discretisation over existing well established methods.}, language = {en} } @article{UlbrichChristianManns2021, author = {Ulbrich, Stefan and Christian, Kirches and Manns, Paul}, title = {Compactness and convergence rates in the combinatorial integral approximation decomposition}, year = {2021}, abstract = {The combinatorial integral approximation decomposition splits the optimization of a discrete-valued control into two steps: solving a continuous relaxation of the discrete control problem, and computing a discrete-valued approximation of the relaxed control. Different algorithms exist for the second step to construct piecewise constant discrete-valued approximants that are defined on given decompositions of the domain. It is known that the resulting discrete controls can be constructed such that they converge to a relaxed control in the weak^* topology of L^\infty if the grid constant of this decomposition is driven to zero. We exploit this insight to formulate a general approximation result for optimization problems, which feature discrete and distributed optimization variables, and which are governed by a compact control-to-state operator. We analyze the topology induced by the grid refinements and prove convergence rates of the control vectors for two problem classes. We use a reconstruction problem from signal processing to demonstrate both the applicability of the method outside the scope of differential equations, the predominant case in the literature, and the effectiveness of the approach.}, language = {en} } @article{UlbrichManns, author = {Ulbrich, Stefan and Manns, Paul}, title = {a simplified newton method to generate snapshots for POD models of semilinear optimal controlproblems}, abstract = {n PDE-constrained optimization, proper orthogonal decomposition (POD) provides a surrogate model of a (potentially expensive) PDE discretization, on which optimization iterations are executed. Because POD models usually provide good approximation quality only locally, they have to be updated during optimization. Updating the POD model is usually expensive, however,and therefore often impossible in a model-predictive control (MPC) context. Thus, reduced models of mediocre quality might be accepted. We take the view of a simplified Newton method for solving semilinear evolution equations to derive an algorithm that can serve as an offline phase to produce a POD model. Approaches that build the POD model with impulse response snapshots can be regarded as the first Newton step in this context.In particular, POD models that are based on impulse response snapshots are extended by adding a second simplified Newton step. This procedure improves the approximation quality of the POD model significantly by introducing a moderate amount of extra computational costs during optimization or the MPC loop. We illustrate our findings with an example satisfying our assumptions.}, language = {en} } @unpublished{VeldmanZuazua, author = {Veldman, D and Zuazua, E}, title = {A framework for randomized time-splitting in linear-quadratic optimal control}, abstract = {Inspired by the successes of stochastic algorithms in the training of deep neural networks and the simulation of interacting particle systems, we propose and analyze a framework for randomized time-splitting in linear-quadratic optimal control. In our proposed framework, the linear dynamics of the original problem is replaced by a randomized dynamics. To obtain the randomized dynamics, the system matrix is split into simpler submatrices and the time interval of interest is split into subintervals. The randomized dynamics is then found by selecting randomly one or more submatrices in each subinterval. We show that the dynamics, the minimal values of the cost functional, and the optimal control obtained with the proposed randomized time-splitting method converge in expectation to their analogues in the original problem when the time grid is refined. The derived convergence rates are validated in several numerical experiments. Our numerical results also indicate that the proposed method can lead to a reduction in computational cost for the simulation and optimal control of large-scale linear dynamical systems.}, language = {en} } @unpublished{GeshkovskiZuazua, author = {Geshkovski, B and Zuazua, E}, title = {Control and Deep Learning: Some connections}, abstract = {This note is an extended abstract for a talk given by the second author during the workshop "Challenges in Optimization with Complex PDE-Systems", at Oberwolfach, in February 2021. It is superfluous to state the impact that deep learning has had on modern technology, as it powers many tools of modern society, ranging from web search to content filtering on social networks. A key paradigm of deep learning is that of supervised learning, which may be seen as a compound and high-dimensional simultaneous control problem. This is the viewpoint adopted by our group. And here we present some of our main findings.}, language = {en} } @article{GeshkovskiZuazua2021, author = {Geshkovski, B and Zuazua, E}, title = {Controllability of one-dimensional viscous free boundary flows}, volume = {59}, number = {3}, doi = {https://doi.org/10.1137/19M1285354}, pages = {1830 -- 1850}, year = {2021}, abstract = {In this work, we address the local controllability of a one-dimensional free boundary problem for a fluid governed by the viscous Burgers equation. The free boundary manifests itself as one moving end of the interval, and its evolution is given by the value of the fluid velocity at this endpoint. We prove that, by means of a control actuating along the fixed boundary, we may steer the fluid to constant velocity in addition to prescribing the free boundary's position, provided the initial velocities and interface positions are close enough.}, language = {en} } @unpublished{EsteveKouhkouhPighinetal., author = {Esteve, C and Kouhkouh, H and Pighin, D and Zuazua, E}, title = {The Turnpike property and the long-time behavior of the Hamilton-Jacobi equation}, abstract = {In this work, we analyze the consequences that the so-called turnpike property has on the long-time behavior of the value function corresponding to a finite-dimensional linear-quadratic optimal control problem with general terminal cost and constrained controls. We prove that, when the time horizon TTT tends to infinity, the value function asymptotically behaves as W(x)+c T+λW(x) + c\, T + \lambda W(x)+cT+λ, and we provide a control interpretation of each of these three terms, making clear the link with the turnpike property. As a by-product, we obtain the long-time behavior of the solution to the associated Hamilton-Jacobi-Bellman equation in a case where the Hamiltonian is not coercive in the momentum variable. As a result of independent interest, we provide a new turnpike result for the linear-quadratic optimal control problem with constrained control. As a main feature, our turnpike result applies to the case when the steady optimum may saturate the control constraints. This prevented us from proving the turnpike property with an exponential rate, which is well-known to hold for the unconstrained case.}, language = {en} } @article{JoheacTrelatZuazua2021, author = {Joheac, J and Trelat, E and Zuazua, E}, title = {Nonnegative control of finite-dimensional linear systems}, volume = {38}, number = {2}, doi = {https://doi.org/10.1016/j.anihpc.2020.07.004}, pages = {301 -- 346}, year = {2021}, abstract = {We consider the controllability problem for finite-dimensional linear autonomous control systems with nonnegative controls. Despite the Kalman condition, the unilateral nonnegativity control constraint may cause a positive minimal controllability time. When this happens, we prove that, if the matrix of the system has a real eigenvalue, then there is a minimal time control in the space of Radon measures, which consists of a finite sum of Dirac impulses. When all eigenvalues are real, this control is unique and the number of impulses is less than half the dimension of the space. We also focus on the control system corresponding to a finite-difference spatial discretization of the one-dimensional heat equation with Dirichlet boundary controls, and we provide numerical simulations.}, language = {en} } @unpublished{BiccariWarmaZuazua, author = {Biccari, U and Warma, M and Zuazua, E}, title = {Control and Numerical approximation of Fractional Diffusion Equations}, abstract = {The aim of this work is to give a broad panorama of the control properties of fractional diffusive models from a numerical analysis and simulation perspective. We do this by surveying several research results we obtained in the last years, focusing in particular on the numerical computation of controls, though not forgetting to recall other relevant contributions which can be currently found in the literature of this prolific field. Our reference model will be a non-local diffusive dynamics driven by the fractional Laplacian on a bounded domain ΩΩΩ. The starting point of our analysis will be a Finite Element approximation for the associated elliptic model in one and two space-dimensions, for which we also present error estimates and convergence rates in the L2L^2L2 and energy norm. Secondly, we will address two specific control scenarios: firstly, we consider the standard interior control problem, in which the control is acting from a small subset ω⊂Ωω ⊂ Ωω⊂Ω. Secondly, we move our attention to the exterior control problem, in which the control region O⊂ΩcO ⊂ Ω cO⊂Ωc is located outside ΩΩΩ. This exterior control notion extends boundary control to the fractional framework, in which the non-local nature of the models does not allow for controls supported on ∂Ω∂Ω∂Ω. We will conclude by discussing the interesting problem of simultaneous control, in which we consider families of parameter-dependent fractional heat equations and we aim at designing a unique control function capable of steering all the different realizations of the model to the same target configuration. In this framework, we will see how the employment of stochastic optimization techniques may help in alleviating the computational burden for the approximation of simultaneous controls. Our discussion is complemented by several open problems related with fractional models which are currently unsolved and may be of interest for future investigation.}, language = {en} } @unpublished{HeilandZuazua2021, author = {Heiland, J and Zuazua, E}, title = {Classical system theory revisited for Turnpike in standard state space systems and impulse controllable descriptor systems}, year = {2021}, abstract = {The concept of turnpike connects the solution of long but finite time horizon optimal control problems with steady state optimal controls. A key ingredient of the analysis of turnpike phenomena is the linear quadratic regulator problem and the convergence of the solution of the associated differential Riccati equation as the terminal time approaches infinity. This convergence has been investigated in linear systems theory in the 1980s. We extend classical system theoretic results for the investigation of turnpike properties of standard state space systems and descriptor systems. We present conditions for turnpike phenomena in the non detectable case and for impulse controllable descriptor systems. For the latter, in line with the theory for standard linear systems,we establish existence and convergence of solutions to a generalized differential Riccati equation.}, language = {en} } @unpublished{GeshkovskiZuazua, author = {Geshkovski, B and Zuazua, E}, title = {Optimal actuator design via Brunovsky's normal form}, abstract = {In this paper, by using the Brunovsky normal form, we provide a reformulation of the problem consisting in finding the actuator design which minimizes the controllability cost for finite-dimensional linear systems with scalar controls. Such systems may be seen as spatially discretized linear partial differential equations with lumped controls. The change of coordinates induced by Brunovsky's normal form allows us to remove the restriction of having to work with diagonalizable system dynamics, and does not entail a randomization procedure as done in past literature on diffusion equations or waves. Instead, the optimization problem reduces to a minimization of the norm of the inverse of a change of basis matrix, and allows for an easy deduction of existence of solutions, and for a clearer picture of some of the problem's intrinsic symmetries. Numerical experiments help to visualize these artifacts, indicate further open problems, and also show a possible obstruction of using gradient-based algorithms - this is alleviated by using an evolutionary algorithm.}, language = {en} } @unpublished{BiccariZuazua2021, author = {Biccari, U and Zuazua, E}, title = {Multilevel control by duality}, year = {2021}, abstract = {We discuss the multilevel control problem for linear dynamical systems, consisting in designing a piece-wise constant control function taking values in a finite-dimensional set. In particular, we provide a complete characterization of multilevel controls through a duality approach, based on the minimization of a suitable cost functional. In this manner we build optimal multi-level controls and characterize the time needed for a given ensemble of levels to assure the controllability of the system. Moreover, this method leads to efficient numerical algorithms for computing multilevel controls.}, language = {en} } @article{KoZuazua2021, author = {Ko, D and Zuazua, E}, title = {Model predictive control with random batch methods for a guiding problem}, volume = {31}, number = {8}, doi = {https://doi.org/10.1142/S0218202521500329}, pages = {1569 -- 1592}, year = {2021}, abstract = {We model, simulate and control the guiding problem for a herd of evaders under the action of repulsive drivers. The problem is formulated in an optimal control framework, where the drivers (controls) aim to guide the evaders (states) to a desired region of the Euclidean space. The numerical simulation of such models quickly becomes unfeasible for a large number of interacting agents. To reduce the computational cost, we use the Random Batch Method (RBM), which provides a computationally feasible approximation of the dynamics. At each time step, the RBM randomly divides the set of particles into small subsets (batches), considering only the interactions inside each batch. Due to the averaging effect, the RBM approximation converges to the exact dynamics as the time discretization gets finer. We propose an algorithm that leads to the optimal control of a fixed RBM approximated trajectory using a classical gradient descent. The resulting control is not optimal for the original complete system, but rather for the reduced RBM model. We then adopt a Model Predictive Control (MPC) strategy to handle the error in the dynamics. While the system evolves in time, the MPC strategy consists in periodically updating the state and computing the optimal control over a long-time horizon, which is implemented recursively in a shorter time-horizon. This leads to a semi-feedback control strategy. Through numerical experiments we show that the combination of RBM and MPC leads to a significant reduction of the computational cost, preserving the capacity of controlling the overall dynamics.}, language = {en} } @unpublished{BarcenaPetiscoZuazua2021, author = {Barcena-Petisco, J.A. and Zuazua, E}, title = {Averaged dynamics and control for heat equations with random diffusion}, year = {2021}, abstract = {This paper deals with the averaged dynamics for heat equations in the degenerate case where the diffusivity coefficient, assumed to be constant, is allowed to take the null value. First we prove that the averaged dynamics is analytic. This allows to show that, most often, the averaged dynamics enjoys the property of unique continuation and is approximately controllable. We then determine if the averaged dynamics is actually null controllable or not depending on how the density of averaging behaves when the diffusivity vanishes. In the critical density threshold the dynamics of the average is similar to the \frac{1}{2}-fractional Laplacian, which is wellknown to be critical in the context of the controllability of fractional diffusion processes. Null controllability then fails (resp. holds) when the density weights more (resp. less) in the null diffusivity regime than in this critical regime.}, language = {en} } @unpublished{WangZhangZuazua2021, author = {Wang, G and Zhang, Y and Zuazua, E}, title = {Flow decomposition for heat equations with memory}, year = {2021}, abstract = {We build up a decomposition for the flow generated by the heat equation with a real analytic memory kernel. It consists of three components: The first one is of parabolic nature; the second one gathers the hyperbolic component of the dynamics, with null velocity of propagation; the last one exhibits a finite smoothing effect. This decomposition reveals the hybrid parabolic-hyperbolic nature of the flow and clearly illustrates the significant impact of the memory term on the parabolic behavior of the system in the absence of memory terms.}, language = {en} } @unpublished{ZhongJieZuazua, author = {Zhong-Jie, H and Zuazua, E}, title = {Slow decay and Turnpike for Infinite-horizon Hyperbolic LQ problems}, abstract = {This paper is devoted to analysing the explicit slow decay rate and turnpike in the infinite-horizon linear quadratic optimal control problems for hyperbolic systems. Assume that some weak observability or controllability are satisfied, by which, the lower and upper bounds of the corresponding algebraic Riccati operator are estimated, respectively. Then based on these two bounds, the explicit slow decay rate of the closed-loop system with Riccati-based optimal feedback control is obtained. The averaged turnpike property for this problem is also further discussed. We then apply these results to the LQ optimal control problems constraint to networks of onedimensional wave equations and also some multi-dimensional ones with local controls which lack of GCC (Geometric Control Condition).}, language = {en} } @unpublished{RuizBaletZuazua, author = {Ruiz-Balet, D and Zuazua, E}, title = {Neural ODE Control for Classification, Approximation and Transport}, abstract = {We analyze Neural Ordinary Differential Equations (NODEs) from a control theoretical perspective to address some of the main properties and paradigms of Deep Learning (DL), in particular, data classification and universal approximation. These objectives are tackled and achieved from the perspective of the simultaneous control of systems of NODEs. For instance, in the context of classification, each item to be classified corresponds to a different initial datum for the control problem of the NODE, to be classified, all of them by the same common control, to the location (a subdomain of the euclidean space) associated to each label. Our proofs are genuinely nonlinear and constructive, allowing us to estimate the complexity of the control strategies we develop. The nonlinear nature of the activation functions governing the dynamics of NODEs under consideration plays a key role in our proofs, since it allows deforming half of the phase space while the other half remains invariant, a property that classical models in mechanics do not fulfill. This very property allows to build elementary controls inducing specific dynamics and transformations whose concatenation, along with properly chosen hyperplanes, allows achieving our goals in finitely many steps. The nonlinearity of the dynamics is assumed to be Lipschitz. Therefore, our results apply also in the particular case of the ReLU activation function. We also present the counterparts in the context of the control of neural transport equations, establishing a link between optimal transport and deep neural networks.}, language = {en} } @article{LanceTrelatZuazua2020, author = {Lance, G and Trelat, E and Zuazua, E}, title = {Shape turnpike for linear parabolic PDE models}, volume = {142}, doi = {10.1016/j.sysconle.2020.104733}, pages = {104 -- 733}, year = {2020}, abstract = {We introduce and study the turnpike property for time-varying shapes, within the viewpoint of optimal control. We focus here on second-order linear parabolic equations where the shape acts as a source term and we seek the optimal time-varying shape that minimizes a quadratic criterion. We first establish existence of optimal solutions under some appropriate sufficient conditions. We then provide necessary conditions for optimality in terms of adjoint equations and, using the concept of strict dissipativity, we prove that state and adjoint satisfy the measure-turnpike property, meaning that the extremal time-varying solution remains essentially close to the optimal solution of an associated static problem. We show that the optimal shape enjoys the exponential turnpike property in term of Hausdorff distance for a Mayer quadratic cost. We illustrate the turnpike phenomenon in optimal shape design with several numerical simulations.}, language = {en} } @unpublished{Leugering2021, author = {Leugering, G{\"u}nter}, title = {Space-time-domain decomposition for optimal control problems governed by linear hyperbolic systems}, year = {2021}, abstract = {In this article, we combine a domain decomposition method in space and time for optimal control problems with PDE-constraints described by Lagnese and Leugering to a simultaneous space-time decomposition applied to optimal control problems for systems of linear hyperbolic equations with distributed control. We thereby extend the recent work by Krug et al. and answer a long standing open question as to whether the combination of time- and space domain decomposition for the method under consideration can be put into one single convergent iteration procedure. The algorithm is designed for a semi-elliptic system of equations obtained from the hyperbolic optimality system by the way of reduction to the adjoint state. The focus is on the relation to the classical procedure introduced by Lions for elliptic problems.}, language = {en} } @article{JoormannPfetsch2018, author = {Joormann, Imke and Pfetsch, Marc E.}, title = {Complexity of Minimum Irreducible Infeasible Subsystem Covers for Flow Networks}, series = {Discrete Applied Mathematics}, volume = {244}, journal = {Discrete Applied Mathematics}, doi = {10.1016/j.dam.2018.02.025}, pages = {124 -- 142}, year = {2018}, abstract = {For an infeasible network flow system with supplies and demands, we consider the problem of finding a minimum irreducible infeasible subsystem cover, i.e., a smallest set of constraints that must be dropped to obtain a feasible system. The special cases of covers which only contain flow balance constraints (node cover) or only flow bounds (arc cover) are investigated as well. We show strong NP-hardness of all three variants. Furthermore, we show that finding minimum arc covers for assignment problems is still hard and as hard to approximate as the set covering problem. However, the minimum arc cover problem is polynomially solvable for networks on cactus graphs. This leads to the development of two different fixed parameter algorithms with respect to the number of elementary cycles connected at arcs and the treewidth, respectively. The latter can be adapted for node covers and the general case.}, language = {en} } @unpublished{Gugat2021, author = {Gugat, Martin}, title = {Optimal boundary control of the wave equation: The finite-time turnpike phenomenon}, year = {2021}, abstract = {It is well-known that vibrating strings can be steered to a position of rest in finite time by suitably defined boundary control functions, if the time horizon is suffciently long. In optimal control problems, the desired terminal state is often enforced by terminal conditions, that add an additional diffculty to the optimal control problem. In this paper we present an optimal control problem for the wave equation with a time-dependent weight in the objective function such that for a suffciently long time horizon, the optimal state reaches a position of rest in finite time without prescribing a terminal constraint. This situation can be seen as a realization of the finite-time turnpike phenomenon that has been studied recently in [1].}, language = {en} }