TY - JOUR A1 - Gugat, Martin A1 - Schuster, Michael A1 - Zuazua, Enrique T1 - M. Gugat, M. Schuster, E. Zuazua. The Finite-Time Turnpike Phenomenon for Optimal Control Problems: Stabilization by Non-Smooth Tracking Terms, in “Stabilization of Distributed Parameter Systems: Design Methods and Applications”. Grigory Sklyar Alexander Zuyev Eds., ICIAM 2019 SEMA SIMAI Springer Series 2, p. 17-42. ISSN 2199-3041 N2 - In this paper, problems of optimal control are considered where in the objective function, in addition to the control cost, there is a tracking term that measures the distance to a desired stationary state. The tracking term is given by some norm, and therefore it is in general not differentiable. In the optimal control problem, the initial state is prescribed. We assume that the system is either exactly controllable in the classical sense or nodal profile controllable. We show that both for systems that are governed by ordinary differential equations and for infinite-dimensional systems, for example, for boundary control systems governed by the wave equation, under certain assumptions, the optimal system state is steered exactly to the desired state after finite time. Y1 - 2020 VL - SEMA SIMAI Springer Series 2 SP - 17 EP - 42 PB - Springer International Publishing ET - Grigory Sklyar Alexander Zuyev Eds., ICIAM 2019 ER - TY - INPR A1 - Sarna, Neeraj A1 - Giesselmann, Jan A1 - Benner, Peter T1 - Data-Driven Snapshot Calibration via Monotonic Feature Matching N2 - Snapshot matrices of hyperbolic equations have a slow singular value decay, resulting in inefficient reduced-order models. We develop on the idea of inducing a faster singular value decay by computing snapshots on a transformed spatial domain, or the so-called snapshot calibration/transformation. We are particularly interested in problems involving shock collision, shock rarefaction-fan collision, shock formation, etc. For such problems, we propose a realizable algorithm to compute the spatial transform using monotonic feature matching. We consider discontinuities and kinks as features, and by carefully partitioning the parameter domain, we ensure that the spatial transform has properties that are desirable both from a theoretical and an implementation standpoint. We use these properties to prove that our method results in a fast $m$-width decay of a so-called calibrated manifold. A crucial observation we make is that due to calibration, the $m$-width does not only depend on $m$ but also on the accuracy of the full order model, which is in contrast to elliptic and parabolic problems that do not need calibration. The method we propose only requires the solution snapshots and not the underlying partial differential equation (PDE) and is therefore, data-driven. We perform several numerical experiments to demonstrate the effectiveness of our method. Y1 - 2020 ER - TY - JOUR A1 - Schuster, Michael A1 - Strauch, Elisa A1 - Gugat, Martin A1 - Lang, Jens T1 - Probabilistic Constrained Optimization on Flow Networks N2 - 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. KW - Probabilistic Constraints KW - Flow Networks KW - Gas Networks KW - Spheric Radial Decomposition KW - Kernel Density Estimator Y1 - 2020 U6 - https://doi.org/https://doi.org/10.1007/s11081-021-09619-x VL - Optimization and Engineering ER - TY - INPR A1 - Aigner, Kevin-Martin A1 - Burlacu, Robert A1 - Liers, Frauke A1 - Martin, Alexander T1 - Solving AC Optimal Power Flow with Discrete Decisions to Global Optimality N2 - 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. KW - Mixed-Integer Nonlinear Programming KW - Second-Order Cone Programming KW - AC Optimal Power Flow KW - Discrete Decisions KW - Piecewise Linear Relaxation Y1 - 2020 ER - TY - JOUR A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Thürauf, Johannes T1 - Global Optimization for the Multilevel European Gas Market System with Nonlinear Flow Models on Trees JF - Journal of Global Optimization N2 - 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. Y1 - 2020 U6 - https://doi.org/10.1007/s10898-021-01099-8 ER - TY - THES A1 - Manns, Julian T1 - Presolve of Linear Bilevel Programs N2 - 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. Y1 - 2020 ER - TY - INPR A1 - Biefel, Christian A1 - Liers, Frauke A1 - Rolfes, Jan A1 - Schmidt, Martin T1 - Affinely Adjustable Robust Linear Complementarity Problems N2 - 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. KW - Linear Complementarity Problems KW - Adjustable Robustness KW - Robust Optimization KW - Existence KW - Uniqueness Y1 - 2020 ER - TY - JOUR A1 - Gugat, Martin A1 - Hante, Falk A1 - Jin, Li T1 - Closed loop control of gas flow in a pipe: Stability for a transient model JF - at - Automatisierungstechnik N2 - 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. Y1 - 2020 ER - TY - INPR A1 - Grimm, Veronika A1 - Nowak, Daniel A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Schwartz, Alexandra A1 - Zöttl, Gregor T1 - A Tractable Multi-Leader Multi-Follower Peak-Load-Pricing Model with Strategic Interaction N2 - 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. KW - Game theory KW - Nash-Cournot equilibria KW - Multi-leader multi-follower game KW - Peak-load pricing Y1 - 2020 U6 - https://doi.org/10.1007/s10107-021-01708-0 ER - TY - JOUR A1 - Böttger, Tom A1 - Grimm, Veronika A1 - Kleinert, Thomas A1 - Schmidt, Martin T1 - The Cost of Decoupling Trade and Transport in the European Entry-Exit Gas Market with Linear Physics Modeling JF - European Journal of Operational Research N2 - 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. KW - Entry-Exit Gas Market KW - Gas Market Design KW - Multilevel Optimization KW - Robust Optimization Y1 - 2020 ER - TY - JOUR A1 - Kleinert, Thomas A1 - Labbé, Martine A1 - Schmidt, Martin A1 - Plein, Fränk T1 - Closing the Gap in Linear Bilevel Optimization: A New Valid Primal-Dual Inequality JF - Optimization Letters N2 - 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. Y1 - 2020 IS - 15 SP - 1027 EP - 1040 ER - TY - INPR A1 - Thürauf, Johannes T1 - Deciding the Feasibility of a Booking in the European Gas Market is coNP-hard N2 - 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. KW - Potential-based flows, Gas networks, Computational complexity, European entry-exit market, Bookings Y1 - 2020 ER - TY - INPR A1 - Aigner, Kevin-Martin A1 - Clarner, Jan-Patrick A1 - Liers, Frauke A1 - Martin, Alexander T1 - Robust Approximation of Chance Constrained DC Optimal Power Flow under Decision-Dependent Uncertainty N2 - We propose a mathematical optimization model and its solution for joint chance constrained DC Optimal Power Flow. In this application, it is particularly important that there is a high probability of transmission limits being satisfied, even in the case of uncertain or fluctuating feed-in from renewable energy sources. In critical network situations where the network risks overload, renewable energy feed-in has to be curtailed by the transmission system operator (TSO). The TSO can reduce the feed-in in discrete steps at each network node. The proposed optimization model minimizes curtailment while ensuring that there is a high probability of transmission limits being maintained. The latter is modeled via (joint) chance constraints that are computationally challenging. Thus, we propose a solution approach based on the robust safe approximation of these constraints. Hereby, probabilistic constraints are replaced by robust constraints with suitably defined uncertainty sets constructed from historical data. The uncertainty sets are calculated by encompassing randomly drawn scenarios using the scenario approach proposed by Margellos et al. (IEEE Transactions on Automatic Control, 59 (2014)). The ability to discretely control the power feed-in then leads to a robust optimization problem with decision-dependent uncertainties, i.e. the uncertainty sets depend on decision variables. We propose an equivalent mixed-integer linear reformulation for box uncertainties with the exact linearization of bilinear terms. Finally, we present numerical results for different test cases from the Nesta archive, as well as for a real network. We consider the discrete curtailment of solar feed-in, for which we use real-world weather and network data. The experimental tests demonstrate the effectiveness of this method and run times are very fast. Moreover, on average the calculated robust solutions lead only to a small increase in curtailment, when compared to nominal solutions. KW - OR in energy KW - optimal power flow KW - chance constrained programming KW - robust optimization KW - decision-dependent uncertainty Y1 - 2020 ER - TY - INPR A1 - Habeck, Oliver A1 - Pfetsch, Marc E. T1 - Combinatorial Acyclicity Models for Potential-based Flows N2 - Potential-based flows constitute a basic model to represent physical behavior in networks. Under natural assumptions, the flow in such networks must be acyclic. The goal of this paper is to exploit this property for the solution of corresponding optimization problems. To this end, we introduce several combinatorial models for acyclic flows, based on binary variables for flow directions. We compare these models and introduce a particular model that tries to capture acyclicity together with the supply/demand behavior. We analyze properties of this model, including variable fixing rules. Our computational results show that the usage of the corresponding constraints speeds up solution times by about a factor of 3 on average and a speed-up of a factor of almost 5 for the time to prove optimality. KW - Network Optimization KW - Potential networks KW - Potential-based flows KW - acyclic flows Y1 - 2020 ER - TY - JOUR A1 - Roland, Marius A1 - Schmidt, Martin T1 - Mixed-Integer Nonlinear Optimization for District Heating Network Expansion JF - at - Automatisierungstechnik N2 - We present a mixed-integer nonlinear optimization model for computing the optimal expansion of an existing tree-shaped district heating network given a number of potential new consumers. To this end, we state a stationary and nonlinear model of all hydraulic and thermal effects in the pipeline network as well as nonlinear models for consumers and the network's depot. For the former, we consider the Euler momentum and the thermal energy equation. The thermal aspects are especially challenging. Here, we develop a novel polynomial approximation that we use in the optimization model. The expansion decisions are modeled by binary variables for which we derive additional valid inequalities that greatly help to solve the highly challenging problem. Finally, we present a case study in which we identify three major aspects that strongly influence investment decisions: the estimated average power demand of potentially new consumers, the distance between the existing network and the new consumers, and thermal losses in the network. KW - District heating networks KW - Network expansion KW - Mixed-integer nonlinear optimization Y1 - 2020 ER - TY - INPR A1 - Klimm, Max A1 - Pfetsch, Marc A1 - Raber, Rico A1 - Skutella, Martin T1 - On the Robustness of Potential-Based Flow Networks N2 - Potential-based flows provide a simple yet realistic mathematical model of transport in many real-world infrastructure networks such as, e.g., electricity, gas, or water networks, where the flow along each edge is controlled via the (difference of) potentials at its end nodes. A potential-based flow network is robust if the maximal difference of node potentials needed to satisfy a set of demands cannot increase if demands are decreased. This notion of robustness is motivated by infrastructure networks where users first make reservations for certain demands that may be larger than the actual amounts sent later on. Here node potentials correspond to physical quantities such as the pressures or the voltages and must be guaranteed to lie within a fixed range, even if the actual amounts are smaller than the previously reserved demands. Our main results are a precise characterization of such robust networks for the case of point-to-point demands via forbidden node-labeled graph minors, as well as an efficient algorithm for testing robustness. Y1 - 2019 ER - TY - JOUR A1 - Gabriel, Steven A. A1 - Leal, Marina A1 - Schmidt, Martin T1 - Solving Binary-Constrained Mixed Complementarity Problems Using Continuous Reformulations JF - Computers & Operations Research N2 - Mixed complementarity problems are of great importance in practice since they appear in various fields of applications like energy markets, optimal stopping, or traffic equilibrium problems. However, they are also very challenging due to their inherent, nonconvex structure. In addition, recent applications require the incorporation of integrality constraints. Since complementarity problems often model some kind of equilibrium, these recent applications ask for equilibrium points that additionally satisfy certain integer conditions. Obviously, this makes the problem even harder to solve. The solution approach used most frequently in the literature is to recast the complementarity conditions as disjunctive constraints using additional binary variables and big-M constraints. However, both latter aspects create issues regarding the tractability and correctness of the reformulation. In this paper, we follow the opposite route and restate the integrality conditions as complementarity constraints, leading to purely continuous reformulations that can be tackled by local solvers. We study these reformulations theoretically and provide a numerical study that shows that continuous reformulations are useful in practice both in terms of solution times and solution quality. KW - Binary-constrained mixed complementarity problems KW - Mixed-integer optimization KW - Continuous reformulations KW - Spatial price equilibrium problems Y1 - 2020 ER - TY - INPR A1 - Kuchlbauer, Martina A1 - Liers, Frauke A1 - Stingl, Michael T1 - Adaptive bundle methods for nonlinear robust optimization N2 - Currently, there are few theoretical or practical approaches available for general nonlinear robust optimization. Moreover, the approaches that do exist impose restrictive assumptions on the problem structure. We present an adaptive bundle method for nonlinear and non-convex robust optimization problems with a suitable notion of inexactness in function values and subgradients. As the worst case evaluation requires a global solution to the adversarial problem, it is a main challenge in a general non-convex nonlinear setting. Moreover, computing elements of an epsilon-perturbation of the Clarke subdifferential in the l2-norm sense is in general prohibitive for this class of problems. In this article, instead of developing an entirely new bundle concept, we demonstrate how existing approaches, such as Noll's bundle method for non-convex minimization with inexact information (Computational and analytical mathematics 50: 555-592, 2013) can be modified to be able to cope with this situation. Extending the non-convex bundle concept to the case of robust optimization in this way, we prove convergence under two assumptions: Firstly, that the objective function is lower C1 and secondly, that approximately optimal solutions to the adversarial maximization problem are available. The proposed method is hence applicable to a rather general setting of nonlinear robust optimization problems. In particular, we do not rely on a specific structure of the adversary's constraints. The considered class of robust optimization problems covers the case that the worst-case adversary only needs to be evaluated up to a certain precision. One possibility to evaluate the worst case with the desired degree of precision is the use of techniques from mixed-integer linear programming (MIP). We investigate the procedure on some analytic examples. As applications, we study the gas transport problem under uncertainties in demand and in physical parameters that affect pressure losses in the pipes. Computational results for examples in large realistic gas network instances demonstrate the applicability as well as the efficiency of the method. Y1 - 2020 ER - TY - THES A1 - Philippi, Nora T1 - Analysis and Numerical Approximation of Transport Equations on Networks N2 - This work deals with the analysis and numerical approximation of transport problems on networks. Appropriate coupling conditions are proposed that allow to establish well-posedness of the continuous problem by semigroup theory. A discontinuous Galerkin method is proposed for the space discretization and its well-posedness and order optimal convergence rates are proven. In addition, the time discretization by the implicit Euler method is investigated. Y1 - 2020 ER - TY - JOUR A1 - Çelebi, Emre A1 - Krebs, Vanessa A1 - Schmidt, Martin T1 - Γ-Robust Electricity Market Equilibrium Models with Transmission and Generation Investments JF - Energy Systems N2 - We consider uncertain robust electricity market equilibrium problems including transmission and generation investments. Electricity market equilibrium modeling has a long tradition but is, in most of the cases, applied in a deterministic setting in which all data of the model are known. Whereas there exist some literature on stochastic equilibrium problems, the field of robust equilibrium models is still in its infancy. We contribute to this new field of research by considering Γ-robust electricity market equilibrium models on lossless DC networks with transmission and generation investments. We state the nominal market equilibrium problem as a mixed complementarity problem as well as its variational inequality and welfare optimization counterparts. For the latter, we then derive a Γ-robust formulation and show that it is indeed the counterpart of a market equilibrium problem with robustified player problems. Finally, we present two case studies to gain insights into the general effects of robustification on electricity market models. In particular, our case studies reveal that the transmission system operator tends to act more risk-neutral in the robust setting, whereas generating firms clearly behave more risk-averse. KW - Robust optimization KW - Robust market equilibria KW - Electricity market equilibrium models KW - Transmission and generation investment KW - Perfect competition Y1 - 2020 ER -