@unpublished{GugatHabermannHintermuelleretal.2021, author = {Gugat, Martin and Habermann, Jens and Hinterm{\"u}ller, Michael and Huber, Olivier}, title = {Constrained exact boundary controllability of a semilinear model for pipeline gas flow}, year = {2021}, abstract = {While the quasilinear isothermal Euler equations are an excellent model for gas pipeline flow, the operation of the pipeline flow with high pressure and small Mach numbers allows us to obtain approximate solutions by a simpler semilinear model. We provide a derivation of the semilinear model that shows that the semilinear model is valid for sufficiently low Mach numbers and sufficiently high pressures. We prove an existence result for continuous solutions of the semilinear model that takes into account lower and upper bounds for the pressure and an upper bound for the magnitude of the Mach number of the gas flow. These state constraints are important both in the operation of gas pipelines and to guarantee that the solution remains in the set where the model is physically valid. We show the constrained exact boundary controllability of the system with the same pressure and Mach number constraints.}, language = {en} } @unpublished{GruebelHuberHuembsetal.2021, author = {Gr{\"u}bel, Julia and Huber, Olivier and H{\"u}mbs, Lukas and Klimm, Max and Schmidt, Martin and Schwartz, Alexandra}, title = {Nonconvex Equilibrium Models for Energy Markets: Exploiting Price Information to Determine the Existence of an Equilibrium}, pages = {29}, year = {2021}, abstract = {Motivated by examples from the energy sector, we consider market equilibrium problems (MEPs) involving players with nonconvex strategy spaces or objective functions, where the latter are assumed to be linear in market prices. We propose an algorithm that determines if an equilibrium of such an MEP exists and that computes an equilibrium in case of existence. Three key prerequisites have to be met. First, appropriate bounds on market prices have to be derived from necessary optimality conditions of some players. Second, a technical assumption is required for those prices that are not uniquely determined by the derived bounds. Third, nonconvex optimization problems have to be solved to global optimality. We test the algorithm on well-known instances from the power and gas literature that meet these three prerequisites. There, nonconvexities arise from considering the transmission system operator as an additional player besides producers and consumers who, e.g., switches lines or faces nonlinear physical laws. Our numerical results indicate that equilibria often exist, especially for the case of continuous nonconvexities in the context of gas market problems.}, language = {en} } @article{BarcenaPetiscoCavalcanteCocliteetal.2021, author = {B{\´a}rcena-Petisco, J.A. and Cavalcante, M. and Coclite, G.M. and de Nitti, N. and Zuazua, E.}, title = {Control of Hyperbolic and Parabolic Equations on Networks and Singular limits}, year = {2021}, abstract = {We study the controllability properties of the transport equation and of parabolic equations posed on a tree. Using a control localized on the exterior nodes, we prove that the hyperbolic and the parabolic systems are null-controllable. The hyperbolic proof relies on the method of characteristics, the parabolic one on duality arguments and Carleman inequalities. We also show that the parabolic system may not be controllable if we do not act on all exterior vertices because of symmetries. Moreover, we estimate the cost of the null-controllability of transport-diffusion equations with diffusivity ε > 0ε>0 and study its asymptotic behavior when ε → 0^+ε→0 + . We prove that the cost of the controllability decays for a time sufficiently large and explodes for short times. This is done by duality arguments allowing to reduce the problem to obtain observability estimates which depend on the viscosity parameter. These are derived by using Agmon and Carleman inequalities.}, language = {en} } @phdthesis{Nowak2021, author = {Nowak, Daniel}, title = {Nonconvex Nash Games - Solution Concepts and Algorithms}, publisher = {E-Publishing-Service der TU Darmstadt}, address = {Darmstadt}, doi = {10.26083/tuprints-00017637}, year = {2021}, abstract = {Game theory is a mathematical approach to model competition between several parties, called players. The goal of each player is to choose a strategy, which solves his optimization problem, i.e. minimizes or maximizes his objective function. Due to the competitive setting, this strategy may influence the optimization problems of other players. In the non-cooperative setting each player acts selfish, meaning he does not care about the objective of his opponents. A solution concept for this problem is a Nash equilibrium, which was introduced by John Forbes Nash in his Ph.D. thesis in 1950. Convexity of the optimization problems is a crucial assumption for the existence of Nash equilibria. This work investigates settings, where this convexity assumption fails to hold. The first part of this thesis extends results of Jong-Shi Pang and Gesualdo Scutari from their paper ``Nonconvex Games with Side Constraints'' published in 2011. In this publication, a game with possibly nonconvex objective functions and nonconvex individual and shared inequality constraints was investigated. We extend these results twofold. Firstly, we generalize the individual and shared polyhedral constraints to general convex constraints and, secondly, we introduce convex and nonconvex, individual and shared equality constraints. After a detailed comparison of solution concepts for the generalized Nash game and a related Nash game, we show that so-called quasi-Nash equilibria exist under similar assumptions than in the original work, provided some additional constraint qualification holds. Subsequently, we prove that the existence of Nash equilibria needs additional assumptions on the gradients of the equality constraints. Furthermore, a special case of a multi-leader multi-follower game is investigated. We show the convergence of epsilon-quasi-Nash equilibria to C-stationary points and prove that these are also Clarke-stationary under reasonable assumptions. In the second part of this thesis, an application in computation offloading is investigated. We consider several mobile users that are able to offload parts of a computation task to a connected server. However, the server has limited computation capacities which leads to competition among the mobile users. If a user decides to offload a part of his computation, he needs to wait for the server to finish before he can assemble the results of his computation. This leads to a vanishing constraint in the optimization problem of the mobile users which is a nonconvex and nonsmooth condition. We show the existence of a unique Nash equilibrium for the computation offloading game and provide an efficient algorithm for its computation. Furthermore, we present two extensions to this game, which inherit similar properties and we also show the limitations of these formulations. The third part investigates a hierarchical constrained Cournot game. In the upper level, several firms decide on capacities which act as constraints for the production variables. In the lower level the same firms engage in a Cournot competition, where they choose production variables to maximize profit. The prior chosen capacities are upper bounds on these production variables. This hierarchical setting induces nonconvexity and nonsmoothness in the upper level objective functions. After a detailed sensitivity analysis of the lower level, we give necessary optimality conditions for the upper level, i.e. for the hierarchical Cournot game. Using these conditions, we construct an algorithm which provably finds all Nash equilibria of the game, provided some assumptions are satisfied. This algorithm is numerically tested on several examples which are motivated by the gas market.}, language = {en} } @article{KrugLeugeringMartinetal.2021, author = {Krug, Richard and Leugering, G{\"u}nter and Martin, Alexander and Schmidt, Martin and Weninger, Dieter}, title = {Time-Domain Decomposition for Optimal Control Problems Governed by Semilinear Hyperbolic Systems with Mixed Two-Point Boundary Conditions}, series = {Control and Cybernetics}, journal = {Control and Cybernetics}, pages = {20}, year = {2021}, abstract = {In this article, we continue our work (Krug et al., 2021) on time-domain decomposition of optimal control problems for systems of semilinear hyperbolic equations in that we now consider mixed two-point boundary value problems and provide an in-depth well-posedness analysis. The more general boundary conditions significantly enlarge the scope of applications, e.g., to hyperbolic problems on metric graphs with cycles. We design an iterative method based on the optimality systems that can be interpreted as a decomposition method for the original optimal control problem into virtual control problems on smaller time domains.}, language = {en} } @article{KleinertMannsSchmidtetal.2021, author = {Kleinert, Thomas and Manns, Julian and Schmidt, Martin and Weninger, Dieter}, title = {Presolving Linear Bilevel Optimization Problems}, series = {EURO Journal on Computational Optimization}, journal = {EURO Journal on Computational Optimization}, number = {9}, doi = {10.1016/j.ejco.2021.100020}, year = {2021}, abstract = {Linear bilevel optimization problems are known to be strongly NP-hard and the computational techniques to solve these problems are often motivated by techniques from single-level mixed-integer optimization. Thus, during the last years and decades many branch-and-bound methods, cutting planes, or heuristics have been proposed. On the other hand, there is almost no literature on presolving linear bilevel problems although presolve is a very important ingredient in state-of-the-art mixed-integer optimization solvers. In this paper, we carry over standard presolve techniques from single-level optimization to bilevel problems and show that this needs to be done with great caution since a naive application of well-known techniques does often not lead to correctly presolved bilevel models. Our numerical study shows that presolve can also be very beneficial for bilevel problems but also highlights that these methods have a more heterogeneous effect on the solution process compared to what is known from single-level optimization. As a side result, our numerical experiments reveal that there is an urgent need for better and more heterogeneous test instance libraries to further propel the field of computational bilevel optimization.}, language = {en} } @unpublished{EgererGrimmGruebeletal.2021, author = {Egerer, Jonas and Grimm, Veronika and Gr{\"u}bel, Julia and Z{\"o}ttl, Gregor}, title = {Long-run market equilibria in coupled energy sectors: A study of uniqueness}, pages = {32}, year = {2021}, abstract = {We propose an equilibrium model for coupled markets of multiple energy sectors. The agents in our model are operators of sector-specific production and sector-coupling technologies, as well as price-sensitive consumers with varying demand. We analyze long-run investment in production capacity in each sector and investment in coupling capacity between sectors, as well as production decisions determined at repeated spot markets. We show that in our multi-sector model, multiplicity of equilibria may occur, even if all assumptions hold that would be sufficient for uniqueness in a single-sector model. We then contribute to the literature by deriving sufficient conditions for the uniqueness of short- and long-run market equilibrium in coupled markets of multiple energy sectors. We illustrate via simple examples that these conditions are indeed required to guarantee uniqueness in general. The uniqueness result is an important step to be able to incorporate the proposed market equilibrium problem in more complex computational multilevel equilibrium models, in which uniqueness of lower levels is a prerequisite for obtaining meaningful solutions. Our analysis also paves the way to understand and analyze more complex sector coupling models in the future.}, language = {en} } @article{RuizBaletZuazua2021, author = {Ruiz-Balet, Domenec and Zuazua, Enrique}, title = {Neural ODE Control for Classification, Approximation and Transport}, year = {2021}, 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} } @phdthesis{Kleinert2021, author = {Kleinert, Thomas}, title = {Algorithms for Mixed-Integer Bilevel Problems with Convex Followers}, year = {2021}, abstract = {Bilevel problems are optimization problems for which a subset of variables is constrained to be an optimal solution of another optimization problem. As such, bilevel problems are capable of modeling hierarchical decision processes. This is required by many real-world problems from a broad spectrum of applications such as energy markets, traffic planning, or critical infrastructure defense, to name only a few. However, the hierarchy of decisions makes bilevel optimization problems also very challenging to solve—both in theory and practice. This cumulative PhD thesis is concerned with computational bilevel optimization. In the first part, we summarize several solution approaches that we developed over the last years and highlight the significant computational progress that these methods provide. For linear bilevel problems, we review branch-and-bound methods, critically discuss their practical use, and propose valid inequalities to extend the methods to branch-and-cut approaches. Further, we demonstrate on a large test set that it is no longer necessary to use the well-known but error-prone big-M reformulation to solve linear bilevel problems. We also present a bilevel-specific heuristic that is based on a penalty alternating direction method. This heuristic is applicable to a broad class of bilevel problems, e.g., linear or mixed-integer quadratic bilevel problems. In a computational study, we show that the method computes optimal or close-to-optimal feasible points in a very short time and that it outperforms a state-of-the-art local method from the literature. Finally, we review global approaches for mixed-integer quadratic bilevel problems. In addition to a Benders-like decomposition, we present a multi-tree and a single-tree outer-approximation approach. A computational evaluation demonstrates that both variants outperform known benchmark algorithms. The second part of this thesis consists of reprints of our original articles and preprints. These articles contain all details and are referenced throughout the first part of the thesis.}, language = {en} } @article{EsteveGeshkovskiPighinetal., author = {Esteve, Carlos and Geshkovski, Borjan and Pighin, Dario and Zuazua, Enrique}, title = {Turnpike in Lipschitz-nonlinear optimal control}, abstract = {We present a new proof of the turnpike property for nonlinear optimal control problems, when the running target is a steady control-state pair of the underlying dynamics. Our strategy combines the construction of suboptimal quasi-turnpike trajectories via controllability, and a bootstrap argument, and does not rely on analyzing the optimality system or linearization techniques. This in turn allows us to address several optimal control problems for finite-dimensional, control-affine systems with globally Lipschitz (possibly nonsmooth) nonlinearities, without any smallness conditions on the initial data or the running target. These results are motivated by the large-layer regime of residual neural networks, commonly used in deep learning applications. We show that our methodology is applicable to controlled PDEs as well, such as the semilinear wave and heat equation with a globally Lipschitz nonlinearity, once again without any smallness assumptions.}, language = {en} } @article{KrugLeugeringMartinetal.2020, author = {Krug, Richard and Leugering, G{\"u}nter and Martin, Alexander and Schmidt, Martin and Weninger, Dieter}, title = {Time-Domain Decomposition for Optimal Control Problems Governed by Semilinear Hyperbolic Systems}, series = {SIAM Journal on Control and Optimization}, journal = {SIAM Journal on Control and Optimization}, pages = {28}, year = {2020}, abstract = {In this article, we extend the time-domain decomposition method described by Lagnese and Leugering (2003) to semilinear optimal control problems for hyperbolic balance laws with spatio-temporal varying coefficients. We provide the design of the iterative method applied to the global first-order optimality system, prove its convergence, and derive an a posteriori error estimate. The analysis is done entirely on the continuous level. A distinguishing feature of the method is that the decomposed optimality system can be interpreted as an optimality system of a local "virtual" optimal control problem. Thus, the iterative time-domain decomposition of the optimality system can be interpreted as an iterative parallel scheme for virtual optimal control problems on the subintervals. A typical example and further comments are given to show the range of potential applications. Moreover, we provide some numerical experiments to give a first interpretation of the role of the parameters involved in the iterative process.}, language = {en} } @article{KleinertLabbeLjubićetal.2021, author = {Kleinert, Thomas and Labb{\´e}, Martine and Ljubić, Ivana and Schmidt, Martin}, title = {A Survey on Mixed-Integer Programming Techniques in Bilevel Optimization}, series = {EURO Journal on Computational Optimization}, journal = {EURO Journal on Computational Optimization}, pages = {47}, year = {2021}, abstract = {Bilevel optimization is a field of mathematical programming in which some variables are constrained to be the solution of another optimization problem. As a consequence, bilevel optimization is able to model hierarchical decision processes. This is appealing for modeling real-world problems, but it also makes the resulting optimization models hard to solve in theory and practice. The scientific interest in computational bilevel optimization increased a lot over the last decade and is still growing. Independent of whether the bilevel problem itself contains integer variables or not, many state-of-the-art solution approaches for bilevel optimization make use of techniques that originate from mixed-integer programming. These techniques include branch-and-bound methods, cutting planes and, thus, branch-and-cut approaches, or problem-specific decomposition methods. In this survey article, we review bilevel-tailored approaches that exploit these mixed-integer programming techniques to solve bilevel optimization problems. To this end, we first consider bilevel problems with convex or, in particular, linear lower-level problems. The discussed solution methods in this field stem from original works from the 1980's but, on the other hand, are still actively researched today. Second, we review modern algorithmic approaches to solve mixed-integer bilevel problems that contain integrality constraints in the lower level. Moreover, we also briefly discuss the area of mixed-integer nonlinear bilevel problems. Third, we devote some attention to more specific fields such as pricing or interdiction models that genuinely contain bilinear and thus nonconvex aspects. Finally, we sketch a list of open questions from the areas of algorithmic and computational bilevel optimization, which may lead to interesting future research that will further propel this fascinating and active field of research.}, language = {en} } @unpublished{Thuerauf, author = {Th{\"u}rauf, Johannes}, title = {Deciding the Feasibility of a Booking in the European Gas Market is coNP-hard}, abstract = {We show that deciding the feasibility of a booking (FB) in the European entry-exit gas market is coNP-hard if a nonlinear potential-based flow model is used. The feasibility of a booking can be characterized by polynomially many load flow scenarios with maximum potential-difference, which are computed by solving nonlinear potential-based flow models. We use this existing characterization of the literature to prove that FB is coNP-hard by reducing Partition to the infeasibility of a booking. We further prove that computing a potential-difference maximizing load flow scenario is NP-hard even if we can determine the flow direction a priori. From the literature, it is known that FB can be decided in polynomial time on trees and a single cycle. Thus, our hardness result draws the first line that separates the easy from the hard variants of FB and finally answers that FB is hard in general.}, language = {en} } @article{KleinertLabbeSchmidtetal.2020, author = {Kleinert, Thomas and Labb{\´e}, Martine and Schmidt, Martin and Plein, Fr{\"a}nk}, title = {Closing the Gap in Linear Bilevel Optimization: A New Valid Primal-Dual Inequality}, series = {Optimization Letters}, journal = {Optimization Letters}, number = {15}, pages = {1027 -- 1040}, year = {2020}, abstract = {Linear bilevel optimization problems are often tackled by replacing the linear lower-level problem with its Karush-Kuhn-Tucker (KKT) conditions. The resulting single-level problem can be solved in a branch-and-bound fashion by branching on the complementarity constraints of the lower-level problem's optimality conditions. While in mixed-integer single-level optimization branch-and-cut has proven to be a powerful extension of branch-and-bound, in linear bilevel optimization not too many bilevel-tailored valid inequalities exist. In this paper, we briefly review existing cuts for linear bilevel problems and introduce a new valid inequality that exploits the strong duality condition of the lower level. We further discuss strengthened variants of the inequality that can be derived from McCormick envelopes. In a computational study, we show that the new valid inequalities can help to close the optimality gap very effectively on a large test set of linear bilevel instances.}, language = {de} } @article{BoettgerGrimmKleinertetal.2020, author = {B{\"o}ttger, Tom and Grimm, Veronika and Kleinert, Thomas and Schmidt, Martin}, title = {The Cost of Decoupling Trade and Transport in the European Entry-Exit Gas Market with Linear Physics Modeling}, series = {European Journal of Operational Research}, journal = {European Journal of Operational Research}, pages = {27}, year = {2020}, abstract = {Liberalized gas markets in Europe are organized as entry-exit regimes so that gas trade and transport are decoupled. The decoupling is achieved via the announcement of technical capacities by the transmission system operator (TSO) at all entry and exit points of the network. These capacities can be booked by gas suppliers and customers in long-term contracts. Only traders who have booked capacities up-front can "nominate" quantities for injection or withdrawal of gas via a day-ahead market. To ensure feasibility of the nominations for the physical network, the TSO must only announce technical capacities for which all possibly nominated quantities are transportable. In this paper, we use a four-level model of the entry-exit gas market to analyze possible welfare losses associated with the decoupling of gas trade and transport. In addition to the multilevel structure, the model contains robust aspects to cover the conservative nature of the European entry-exit system. We provide several reformulations to obtain a single-level mixed-integer quadratic problem. The overall model of the considered market regime is extremely challenging and we thus have to make the main assumption that gas flows are modeled as potential-based linear flows. Using the derived single-level reformulation of the problem, we show that the feasibility requirements for technical capacities imply significant welfare losses due to unused network capacity. Furthermore, we find that the specific structure of the network has a considerable influence on the optimal choice of technical capacities. Our results thus show that trade and transport are not decoupled in the long term. As a further source of welfare losses and discrimination against individual actors, we identify the minimum prices for booking capacity at the individual nodes.}, language = {en} } @unpublished{GrimmNowakScheweetal.2020, author = {Grimm, Veronika and Nowak, Daniel and Schewe, Lars and Schmidt, Martin and Schwartz, Alexandra and Z{\"o}ttl, Gregor}, title = {A Tractable Multi-Leader Multi-Follower Peak-Load-Pricing Model with Strategic Interaction}, doi = {10.1007/s10107-021-01708-0}, pages = {35}, year = {2020}, abstract = {While single-level Nash equilibrium problems are quite well understood nowadays, less is known about multi-leader multi-follower games. However, these have important applications, e.g., in the analysis of electricity and gas markets, where often a limited number of firms interacts on various subsequent markets. In this paper, we consider a special class of two-level multi-leader multi-follower games that can be applied, e.g., to model strategic booking decisions in the European entry-exit gas market. For this nontrivial class of games, we develop a solution algorithm that is able to compute the complete set of Nash equilibria instead of just individual solutions or a bigger set of stationary points. Additionally, we prove that for this class of games, the solution set is finite and provide examples for instances without any Nash equilibria in pure strategies. We apply the algorithm to a case study in which we compute strategic booking and nomination decisions in a model of the European entry-exit gas market system. Finally, we use our algorithm to provide a publicly available test library for the considered class of multi-leader multi-follower games. This library contains problem instances with different economic and mathematical properties so that other researchers in the field can test and benchmark newly developed methods for this challenging class of problems.}, language = {en} } @article{GugatHanteJin2020, author = {Gugat, Martin and Hante, Falk and Jin, Li}, title = {Closed loop control of gas flow in a pipe: Stability for a transient model}, series = {at - Automatisierungstechnik}, journal = {at - Automatisierungstechnik}, year = {2020}, abstract = {This contribution focuses on the analysis and control of friction-dominated flow of gas in pipes. The pressure in the gas flow is governed by a partial differential equation that is a doubly nonlinear parabolic equation of p-Laplace type, where p=2/3. Such equations exhibit positive solutions, finite speed of propagation and satisfy a maximum principle. The pressure is fixed on one end (upstream), and the flow is specified on the other end (downstream). These boundary conditions determine a unique steady equilibrium flow. We present a boundary feedback flow control scheme, that ensures local exponential stability of the equilibrium in an L2-sense. The analysis is done both for the pde system and an ode system that is obtained by a suitable spatial semi-discretization. The proofs are based upon suitably chosen Lyapunov functions.}, language = {en} } @unpublished{BiefelLiersRolfesetal.2020, author = {Biefel, Christian and Liers, Frauke and Rolfes, Jan and Schmidt, Martin}, title = {Affinely Adjustable Robust Linear Complementarity Problems}, pages = {20}, year = {2020}, abstract = {Linear complementarity problems are a powerful tool for modeling many practically relevant situations such as market equilibria. They also connect many sub-areas of mathematics like game theory, optimization, and matrix theory. Despite their close relation to optimization, the protection of LCPs against uncertainties - especially in the sense of robust optimization - is still in its infancy. During the last years, robust LCPs have only been studied using the notions of strict and Γ-robustness. Unfortunately, both concepts lead to the problem that the existence of robust solutions cannot be guaranteed. In this paper, we consider affinely adjustable robust LCPs. In the latter, a part of the LCP solution is allowed to adjust via a function that is affine in the uncertainty. We show that this notion of robustness allows to establish strong characterizations of solutions for the cases of uncertain matrix and vector, separately, from which existence results can be derived. Our main results are valid for the case of an uncertain LCP vector. Here, we additionally provide sufficient conditions on the LCP matrix for the uniqueness of a solution. Moreover, based on characterizations of the affinely adjustable robust solutions, we derive a mixed-integer programming formulation that allows to solve the corresponding robust counterpart. If, in addition, the certain LCP matrix is positive semidefinite, we prove polynomial-time solvability and uniqueness of robust solutions. If the LCP matrix is uncertain, characterizations of solutions are developed for every nominal matrix, i.e., these characterizations are, in particular, independent of the definiteness of the nominal matrix. Robust solutions are also shown to be unique for positive definite LCP matrix but both uniqueness and mixed-integer programming formulations still remain open problems if the nominal LCP matrix is not positive definite.}, language = {en} } @misc{Manns2020, type = {Master Thesis}, author = {Manns, Julian}, title = {Presolve of Linear Bilevel Programs}, year = {2020}, abstract = {Bilevel programs are complex optimization problems that can be used to model hierarchical decision processes, which occur e.g. in energy markets, critical infrastructure defense or pricing models. Even the most simple bilevel programs, where only linear objective functions and constraints appear, are non-convex optimization problems and equivalent single level formulations replace the lower level problem by its non-convex optimality constraints. This makes linear bilevel programs inherently difficult so solve. The simplification of mixed-integer linear programs before solving them, called presolve, significantly accelerated the solving of these problems. However, there is only very few literature on the topic of presolve of bilevel programs. In this thesis we review said literature on presolve of bilevel programs in the context of linear bilevel programming, derive new theoretical foundations for presolve of linear bilevel programs and then apply these results to analyze how common presolve techniques for linear and mixed integer programs can be used to presolve linear bilevel programs.}, language = {en} } @article{ScheweSchmidtThuerauf, author = {Schewe, Lars and Schmidt, Martin and Th{\"u}rauf, Johannes}, title = {Global Optimization for the Multilevel European Gas Market System with Nonlinear Flow Models on Trees}, series = {Journal of Global Optimization}, journal = {Journal of Global Optimization}, doi = {10.1007/s10898-021-01099-8}, abstract = {The European gas market is implemented as an entry-exit system, which aims to decouple transport and trading of gas. It has been modeled in the literature as a multilevel problem, which contains a nonlinear flow model of gas physics. Besides the multilevel structure and the nonlinear flow model, the computation of so-called technical capacities is another major challenge. These lead to nonlinear adjustable robust constraints that are computationally intractable in general. We provide techniques to equivalently reformulate these nonlinear adjustable constraints as finitely many convex constraints including integer variables in the case that the underlying network is tree-shaped. We further derive additional combinatorial constraints that significantly speed up the solution process. Using our results, we can recast the multilevel model as a single-level nonconvex mixed-integer nonlinear problem, which we then solve on a real-world network, namely the Greek gas network, to global optimality. Overall, this is the first time that the considered multilevel entry-exit system can be solved for a real-world sized network and a nonlinear flow model.}, language = {en} } @unpublished{AignerBurlacuLiersetal., author = {Aigner, Kevin-Martin and Burlacu, Robert and Liers, Frauke and Martin, Alexander}, title = {Solving AC Optimal Power Flow with Discrete Decisions to Global Optimality}, abstract = {We present a solution framework for general alternating current optimal power flow (AC OPF) problems that include discrete decisions. The latter occur, for instance, in the context of the curtailment of renewables or the switching of power generation units and transmission lines. Our approach delivers globally optimal solutions and is provably convergent. We model AC OPF problems with discrete decisions as mixed-integer nonlinear programs. The solution method starts from a known framework that uses piecewise linear relaxations. These relaxations are modeled as as mixed-integer linear programs and adaptively refined until some termination criterion is fulfilled. In this work, we extend and complement this approach by problem-specific as well as very general algorithmic enhancements. In particular, these are mixed-integer second-order cone programs as well as primal and dual cutting planes. For example objective cuts and no-good-cuts help to compute good feasible solutions as where outer approximation constraints tighten the relaxations. We present extensive numerical results for various AC OPF problems where discrete decisions play a major role. Even for hard instances with a large proportion of discrete decisions, the method is able to generate high quality solutions efficiently. Furthermore, we compare our approach with state-of-the-art MINLP. Our method outperforms all other algorithms.}, 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{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{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{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{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{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{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{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{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{GugatHerty2020, author = {Gugat, Martin and Herty, Michael}, title = {Modeling, Control and Numerics of Gas Networks}, year = {2020}, abstract = {In this article we survey recent progress on mathematical results on gas flow in pipe networks with a special focus on questions of control and stabilization. We briefly present the modeling of gas flow and coupling conditions for flow through vertices of a network. Our main focus is on gas models for spatially one-dimensional flow governed by hyperbolic balance laws. We survey results on classical solutions as well as weak solutions. We present results on well-posedness, controllability, feedback stabilization, the inclusion of uncertainty in the models and numerical methods.}, language = {en} } @unpublished{KleinertSchmidt2020, author = {Kleinert, Thomas and Schmidt, Martin}, title = {Why there is no need to use a big-M in linear bilevel optimization: A computational study of two ready-to-use approaches}, pages = {8}, year = {2020}, abstract = {Linear bilevel optimization problems have gained increasing attention both in theory as well as in practical applications of Operations Research (OR) during the last years and decades. The latter is mainly due to the ability of this class of problems to model hierarchical decision processes. However, this ability makes bilevel problems also very hard to solve. Since no general-purpose solvers are available, a "best-practice" has developed in the applied OR community, in which not all people want to develop tailored algorithms but "just use" bilevel optimization as a modeling tool for practice. This best-practice is the big-M reformulation of the Karush-Kuhn-Tucker (KKT) conditions of the lower-level problem - an approach that has been shown to be highly problematic by Pineda and Morales (2019). Choosing invalid values for M yields solutions that may be arbitrarily bad. Checking the validity of the big-Ms is however shown to be as hard as solving the original bilevel problem in Kleinert et al. (2019). Nevertheless, due to its appealing simplicity, especially w.r.t. the required implementation effort, this ready-to-use approach still is the most popular method. Until now, there has been a lack of approaches that are competitive both in terms of implementation effort and computational cost. In this note we demonstrate that there is indeed another competitive ready-to-use approach: If the SOS-1 technique is applied to the KKT complementarity conditions, adding the simple additional root-node inequality developed by Kleinert et al. (2020) leads to a competitive performance - without having all the possible theoretical disadvantages of the big-M approach.}, language = {en} } @article{BarcenaZuazua2020, author = {B{\´a}rcena, J.A. and Zuazua, Enrique}, title = {Averaged dynamics and control for heat equations with random diffusion}, year = {2020}, abstract = {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} } @article{SchusterStrauchGugatetal.2020, author = {Schuster, Michael and Strauch, Elisa and Gugat, Martin and Lang, Jens}, title = {Probabilistic Constrained Optimization on Flow Networks}, volume = {Optimization and Engineering}, doi = {https://doi.org/10.1007/s11081-021-09619-x}, pages = {50}, year = {2020}, abstract = {Uncertainty often plays an important role in dynamic flow problems. In this paper, we consider both, a stationary and a dynamic flow model with uncertain boundary data on networks. We introduce two different ways how to compute the probability for random boundary data to be feasible, discussing their advantages and disadvantages. In this context, feasible means, that the flow corresponding to the random boundary data meets some box constraints at the network junctions. The first method is the spheric radial decomposition and the second method is a kernel density estimation. In both settings, we consider certain optimization problems and we compute derivatives of the probabilistic constraint using the kernel density estimator. Moreover, we derive necessary optimality conditions for the stationary and the dynamic case. Throughout the paper, we use numerical examples to illustrate our results by comparing them with a classical Monte Carlo approach to compute the desired probability.}, language = {en} } @article{GugatGiesselmann2020, author = {Gugat, Martin and Giesselmann, Jan}, title = {Boundary feedback stabilization of a semilinear model for the flow in star-shaped gas networks}, address = {ESAIM:COCV}, doi = {10.1051/cocv/2021061}, year = {2020}, abstract = {The flow of gas through a pipeline network can be modelled by a coupled system of 1-d quasilinear hyperbolic equations. In this system, the influence of certain source terms that model friction effects is essential. Often for the solution of control problems it is convenient to replace the quasilinear model by a simpler semilinear model. In this paper, we analyze the behavior of such a semilinear model on a star-shaped network. The model is derived from the diagonal form of the quasilinear model by replacing the eigenvalues by the sound speed multiplied by 1 or -1 respectively. Thus in the corresponding eigenvalues the influence of the gas velocity is neglected, which is justified in the applications since it is much smaller than the sound speed in the gas. For a star-shaped network of horizontal pipes for suitable coupling conditions we present boundary feedback laws that stabilize the system state exponentially fast to a position of rest for sufficiently small initial data. We show the exponential decay of the \$H^1\$-norm for arbitrarily long pipes. This is remarkable since in general even for linear systems, for certain source terms the system can become exponentially unstable if the space interval is too long. Our proofs are based upon observability inequalities for the \$L^2\$ and the \$H^1\$-norm.}, language = {en} } @article{Burger2020, author = {Burger, Martin}, title = {Network structured kinetic models of social interactions}, edition = {Vietnam Journal of Mathematics}, doi = {10.1007/s10013-021-00505-8}, pages = {1-2ß0}, year = {2020}, abstract = {The aim of this paper is to study the derivation of appropriate meso- and macroscopic models for interactions as appearing in social processes. There are two main characteristics the models take into account, namely a network structure of interactions, which we treat by an appropriate mesoscopic description, and a different role of interacting agents. The latter differs from interactions treated in classical statistical mechanics in the sense that the agents do not have symmetric roles, but there is rather an active and a passive agent. We will demonstrate how a certain form of kinetic equations can be obtained to describe such interactions at a mesoscopic level and moreover obtain macroscopic models from monokinetics solutions of those. The derivation naturally leads to systems of nonlocal reaction-diffusion equations (or in a suitable limit local versions thereof), which can explain spatial phase separation phenomena found to emerge from the microscopic interactions. We will highlight the approach in three examples, namely the evolution and coarsening of dialects in human language, the construction of social norms, and the spread of an epidemic.}, language = {en} } @unpublished{RungeSoelchAlbertetal.2020, author = {Runge, Philipp and S{\"o}lch, Christian and Albert, Jakob and Wasserscheid, Peter and Z{\"o}ttl, Gregor and Grimm, Veronika}, title = {Economic comparison of electric fuels produced at excellent locations for renewable energies: A Scenario for 2035}, year = {2020}, abstract = {The use of electric fuels (e-fuels) enables CO2-neutral mobility and opens therefore an alternative to fossil-fuel-fired engines or battery-powered electric motors. This paper compares the cost-effectiveness of Fischer-Tropsch diesel, methanol, and hydrogen stored as cryogenic liquid (LH2) or in form of liquid organic hydrogen carriers (LOHCs). The production cost of those fuels are to a large extent driven by the energy-intensive electrolytic water splitting. The option of producing e-fuels in Germany competes with international locations with excellent conditions for renewable energy harvesting and thus very low levelized cost of electricity. We developed a mathematical model that covers the entire process chain. Starting with the production of the required resources such as fresh water, hydrogen, carbon dioxide, carbon monoxide, electrical and thermal energy, the subsequent chemical synthesis, the transport to filling stations in Germany and finally the energetic utilization of the fuels in the vehicle. We found that the choice of production site can have a major impact on the mobility cost using the respective fuels. Especially in case of diesel production, the levelized cost of electricity driven by the full load hours of the applied renewable energy source have a huge impact. An LOHC-based system is shown to be less dependent on the kind of electricity source compared to other technologies due to its comparatively low electricity consumption and the low cost for the hydrogenation units. The length of the transportation route and the price of the filling station infrastructure, on the other hand, clearly increase mobility cost for LOHC and LH2.}, language = {en} } @unpublished{BohlayerBuergerFleschutzetal.2020, author = {Bohlayer, Markus and B{\"u}rger, Adrian and Fleschutz, Markus and Braun, Marco and Z{\"o}ttl, Gregor}, title = {Multi-period investment pathways - Modeling approaches to design distributed energy systems under uncertainty}, year = {2020}, abstract = {Multi-modal distributed energy system planning is applied in the context of smart grids, industrial energy supply,and in the building energy sector. In real-world applications, these systems are commonly characterized by existing system structures of different age where monitoring and investment are conducted in a closed-loop, with the iterative possibility to invest. The literature contains two main approaches to approximate this computationally intensive multiperiod investment problem. The first approach simplifies the temporal decision-making process collapsing the multistage decision to a two-stage decision, considering uncertainty in the second stage decision variables. The second approach considers multi-period investments under the assumption of perfect foresight. In this work, we propose a multi-stage stochastic optimization problem that captures multi-period investment decisions under uncertainty and solves the problem to global optimality, serving as a first-best benchmark to the problem. To evaluate the performance of conventional approaches applied in a multi-year setup and to solve the multi-period problem at lower computational effort, we propose a rolling horizon heuristic that on the one hand reveals the performance of conventional approaches applied in a multi-period set-up and on the other hand enables planners to identify approximate solutions to the original multi-stage stochastic problem. Additionally, we consider an open-loop version of the rolling horizon algorithm to evaluate how single-period investments perform with respect to the entire scenario tree and compared to multi-period investments. We conduct a real-world case study and investigate solution quality as well as the computational performance of the proposed approaches. Our findings indicate that the approximation of multi-period investments by two-stage stochastic approaches yield the best results regarding constraint satisfaction, while deterministic multi-period approximations yield better economic and computational performance.}, language = {en} } @unpublished{GugatHerty2020, author = {Gugat, Martin and Herty, Michael}, title = {Limits of stabilizabilizy for a semilinear model for gas pipeline flow}, year = {2020}, abstract = {We present a positive and a negative stabilization result for a semilinear model of gas flow in pipelines. For feedback boundary conditions we obtain an unconditional stabilization result in the absence and conditional instability in the presence of the source term. We also obtain unconditional instability for the corresponding quasilinear model given by the isothermal Euler equations}, language = {en} } @article{SaracZuazua2021, author = {Sarac, Yesim and Zuazua, Enrique}, title = {Sidewise control of 1-d waves}, year = {2021}, abstract = {We analyze the sidewise controllability for the variable coefficients one-dimensional wave equation. The control is acting on one extreme of the string with the aim that the solution tracks a given path at the otherfree end. This sidewise control problem is also often referred to as nodal profile or tracking control. First, the problem is reformulated as a dual observability property for the corresponding adjoint system. Using sidewiseenergy propagation arguments the sidewise observability is shown to hold, ina sufficiently large time, in the class of BV-coefficients. We also present a number of open problems and perspectives for further research.}, language = {en} } @article{PleinThueraufLabbeetal.2021, author = {Plein, Fr{\"a}nk and Th{\"u}rauf, Johannes and Labb{\´e}, Martine and Schmidt, Martin}, title = {A Bilevel Optimization Approach to Decide the Feasibility of Bookings in the European Gas Market}, series = {Mathematical Methods of Operations Research}, journal = {Mathematical Methods of Operations Research}, doi = {10.1007/s00186-021-00752-y}, pages = {37}, year = {2021}, abstract = {The European gas market is organized as a so-called entry-exit system with the main goal to decouple transport and trading. To this end, gas traders and the transmission system operator (TSO) sign so-called booking contracts that grant capacity rights to traders to inject or withdraw gas at certain nodes up to this capacity. On a day-ahead basis, traders then nominate the actual amount of gas within the previously booked capacities. By signing a booking contract, the TSO guarantees that all nominations within the booking bounds can be transported through the network. This results in a highly challenging mathematical problem. Using potential-based flows to model stationary gas physics, feasible bookings on passive networks, i.e., networks without controllable elements, have been characterized in the recent literature. In this paper, we consider networks with linearly modeled active elements such as compressors or control valves. Since these active elements allow the TSO to control the gas flow, the single-level approaches for passive networks from the literature are no longer applicable. We thus present a bilevel model to decide the feasibility of bookings in networks with active elements. While this model is well-defined for general active networks, we focus on the class of networks for which active elements do not lie on cycles. This assumption allows us to reformulate the original bilevel model such that the lower-level problem is linear for every given upper-level decision. Consequently, we derive several single-level reformulations for this case. Besides the classic Karush-Kuhn-Tucker reformulation, we obtain three problem-specific optimal-value-function reformulations. The latter also lead to novel characterizations of feasible bookings in networks with active elements that do not lie on cycles. We compare the performance of our methods by a case study based on data from the GasLib.}, language = {en} } @unpublished{GugatSokolowski, author = {Gugat, Martin and Sokolowski, Jan}, title = {On Problems of Dynamic Optimal Nodal control for Gas Networks}, abstract = {We consider a dynamic ptimal control problem for gas pipeline systems. The flow is governed by a quasilinear hyperbolic model. Since in the operation of the gas networks regular solutions without shocks are desirable, we impose appropriate state and control constraint in order to guarantee that a classical solution is generated. Due to a W^{2;inf}-regularization term in the objective function, we can show the existence of an optimal control. Moreover, we give conditions that guarantee that the control becomes constant a the end of the control time interval if the weight of the regularization term is suffciently large.}, language = {en} } @article{Gugat2021, author = {Gugat, Martin}, title = {On the turnpike property with interior decay for optimal control problems}, series = {Mathematics of Control, Signals, and Systems}, journal = {Mathematics of Control, Signals, and Systems}, doi = {https://doi.org/10.1007/s00498-021-00280-4}, year = {2021}, abstract = {In this paper the turnpike phenomenon is studied for problems of optimal control where both pointwise-in-time state and control constraints can appear. We assume that in the objective function, a tracking term appears that is given as an integral over the time-interval [0, T] and measures the distance to a desired stationary state. In the optimal control problem, both the initial and the desired terminal state are prescribed. We assume that the system is exactly controllable in an abstract sense if the time horizon is long enough. We show that that the corresponding optimal control problems on the time intervals [0, T] give rise to a turnpike structure in the sense that for natural numbers n if T is su� ciently large, the contribution of the objective function from subintervals of [0, T] of the form [t - t/2^n, t + (T-t)/2^n] is of the order 1/min{t^n, (T-t)^n}. We also show that a similar result holds for epsilon-optimal solutions of the optimal control problems if epsilon > 0 is chosen suffciently small. At the end of the paper we present both systems that are governed by ordinary differential equations and systems governed by partial differential equations where the results can be applied.}, language = {en} } @unpublished{HeitschHenrionKleinertetal.2021, author = {Heitsch, Holger and Henrion, Ren{\´e} and Kleinert, Thomas and Schmidt, Martin}, title = {On Convex Lower-Level Black-Box Constraints in Bilevel Optimization with an Application to Gas Market Models with Chance Constraints}, pages = {34}, year = {2021}, abstract = {Bilevel optimization is an increasingly important tool to model hierarchical decision making. However, the ability of modeling such settings makes bilevel problems hard to solve in theory and practice. In this paper, we add on the general difficulty of this class of problems by further incorporating convex black-box constraints in the lower level. For this setup, we develop a cutting-plane algorithm that computes approximate bilevel-feasible points. We apply this method to a bilevel model of the European gas market in which we use a joint chance constraint to model uncertain loads. Since the chance constraint is not available in closed form, this fits into the black-box setting studied before. For the applied model, we use further problem-specific insights to derive bounds on the objective value of the bilevel problem. By doing so, we are able to show that we solve the application problem to approximate global optimality. In our numerical case study we are thus able to evaluate the welfare sensitivity in dependence of the achieved safety level of uncertain load coverage.}, language = {en} } @unpublished{Leugering2021, author = {Leugering, G{\"u}nter}, title = {Space-Time-Domain Decomposition for Optimal Control Problems Governed by Linear Hyperbolic Systems}, pages = {18}, 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{Leugering, author = {Leugering, G{\"u}nter}, title = {Industrial Applications of Optimal Control for Partial Differential Equations on Networks: Reduction and Decomposition Methods Applied to the Discrete-Continuous Control of Gas Flow in Complex Pipe Systems}, volume = {Computational Science and Its Applications}, number = {1st Edition}, pages = {25 -- 40}, abstract = {This chapter provides an exemplary road map—in a nutshell—from a given industrial application, the control of gas networks, which is far too complex for a direct approach, to a problem that can be actually handled using well-known methods in control theory. It also provides an iterative non-overlapping domain decomposition that can be interpreted as an Uzawa method. The chapter outline two strategies. The first one can be seen as a Jacobi-type approach. In the second approach, fix the integer controls s and decompose the corresponding optimality system for the entire graph into the subgraphs Gk by a another, but very similar, non-overlapping domain decomposition. The problem is the intrinsic coupling of integer controls, continuous controls, and nonlinear dynamics on a metric graph. The idea is to introduce a virtual control that aims at controlling classical in homogeneous Neumann condition including the iteration history at the interface as inhomogeneity to the Robin-type condition that appears in the decomposition.}, language = {en} } @unpublished{BiefelLiersRolfesetal., author = {Biefel, Christian and Liers, Frauke and Rolfes, Jan and Schewe, Lars and Z{\"o}ttl, Gregor}, title = {Robust Market Equilibria under Uncertain Cost}, pages = {24}, abstract = {We consider equilibrium problems under uncertainty where firms maximize their profits in a robust way when selling their output. Robust optimization plays an increasingly important role when best guaranteed objective values are to be determined, independently of the specific distributional assumptions regarding uncertainty. In particular, solutions are to be determined that are feasible regardless of how the uncertainty manifests itself within some predefined uncertainty set. Our analysis adopts the robust optimization perspective in the context of equilibrium problems. First, we consider a singlestage, nonadjustable robust setting. We then go one step further and study the more complex two-stage or adjustable case where a part of the variables can adjust to the realization of the uncertainty. We compare equilibrium outcomes with the corresponding centralized robust optimization problem where the sum of all profits are maximized. As we find, the market equilibrium for the perfectly competitive firms differs from the solution of the robust central planner, which is in stark contrast to classical results regarding the efficiency of market equilibria with perfectly competitive firms. For the different scenarios considered, we furthermore are able to determine the resulting price of anarchy. In the case of non-adjustable robustness, for fixed demand in every time step the price of anarchy is bounded whereas it is unbounded if the buyers are modeled by elastic demand functions. For the two-stage adjustable setting, we show how to compute subsidies for the firms that lead to robust welfare optimal equilibria.}, language = {en} } @unpublished{GugatKrugMartin2021, author = {Gugat, Martin and Krug, Richard and Martin, Alexander}, title = {Transient gas pipeline flow: Analytical examples, numerical simulation and a comparison to the quasi-static approach}, doi = {10.1007/s11081-021-09690-4}, year = {2021}, abstract = {The operation of gas pipeline flow with high pressure and small Mach numbers allows to model the flow by a semilinear hyperbolic system of partial differential equations. In this paper we present a number of transient and stationary analytical solutions of this model. They are used to discuss and clarify why a pde model is necessary to handle certain dynamic situations in the operation of gas transportation networks. We show that adequate numerical discretizations can capture the dynamical behavior sufficiently accurate. We also present examples that show that in certain cases an optimization approach that is based upon multi-period optimization of steady states does not lead to approximations that converge to the optimal state.}, language = {en} } @unpublished{AdelhuetteBiefelKuchlbaueretal., author = {Adelh{\"u}tte, Dennis and Biefel, Christitan and Kuchlbauer, Martina and Rolfes, Jan}, title = {Pareto Robust optimization on Euclidean vector spaces}, abstract = {Pareto efficiency for robust linear programs was introduced by Iancu and Trichakis in [9]. We generalize their approach and theoretical results to robust optimization problems in Euclidean spaces with linear uncertainty. Additionally, we demonstrate the value of this approach in an exemplary manner in the area of robust semidefinite programming (SDP). In particular, we prove that computing a Pareto robustly optimal solution for a robust SDP is tractable and illustrate the benefit of such solutions at the example of the maximal eigenvalue problem. Furthermore, we modify the famous algorithm of Goemans and Williamson [8] in order to compute cuts for the robust max cut problem that yield an improved approximation guarantee in non-worst-case scenarios.}, language = {en} }