@phdthesis{Turner2024, author = {Turner, Mark}, title = {Cutting Plane Selection for Mixed-Integer Linear Programming}, year = {2024}, abstract = {Mixed-Integer Linear Programming (MILP) is a ubiquitous and practical modelling paradigm that is essential for optimising a broad range of real-world systems. The backbone of all modern MILP solvers is the branch-and-cut algorithm, which is a hybrid of the branch-and-bound and cutting planes algorithms. Cutting planes (cuts) are linear inequalities that tighten the relaxation of a MILP. While a lot of research has gone into deriving valid cuts for MILPs, less emphasis has been put on determining which cuts to select. Cuts in general are generated in rounds, and a subset of the generated cuts must be added to the relaxation. The decision on which subset of cuts to add is called cut selection. This is a crucial task since adding too many cuts makes the relaxation large and slow to optimise over. Conversely, adding too few cuts results in an insufficiently tightened relaxation, and more relaxations need to be enumerated. To further emphasise the difficulty, the effectiveness of an applied cut is both dependent on the other applied cuts, and the state of the MILP solver. In this thesis, we present theoretical results on the importance and difficulty of cut selection, as well as practical results that use cut selection to improve general MILP solver performance. Improving general MILP solver performance is of great importance for practitioners and has many runoff effects. Reducing the solve time of currently solved systems can directly improve efficiency within the application area. In addition, improved performance enables larger systems to be modelled and optimised, and MILP to be used in areas where it was previously impractical due to time restrictions. Each chapter of this thesis corresponds to a publication on cut selection, where the contributions of this thesis can naturally be divided into four components. The first two components are motivated by instance-dependent performance. In practice, for each subroutine, including cut selection, MILP solvers have adjustable parameters with hard-coded default values. It is ultimately unrealistic to expect these default values to perform well for every instance. Rather, it would be ideal if the parameters were dependent on the given instance. To show this motivation is well founded, we first introduce a family of parametric MILP instances and cuts to showcase worst-case performance of cut selection for any fixed parameter value. We then introduce a graph neural network architecture and reinforcement learning framework for learning instance-dependent cut scoring parameters. In the following component, we formalise language for determining if a cut has theoretical usefulness from a polyhedral point of view in relation to other cuts. In addition, to overcome issues of infeasible projections and dual degeneracy, we introduce analytic center based distance measures. We then construct a lightweight multi-output regression model that predicts relative solver performance of an instance for a set of distance measures. The final two components are motivated by general MILP solver improvement via cut selection. Such improvement was shown to be possible, albeit difficult to achieve, by the first half of this thesis. We relate branch-and-bound and cuts through their underlying disjunctions. Using a history of previously computed Gomory mixed-integer cuts, we reduce the solve time of SCIP over the 67\% of affected MIPLIB 2017 instances by 4\%. In the final component, we introduce new cut scoring measures and filtering methods based on information from other MILP solving processes. The new cut selection techniques reduce the solve time of SCIP over the 97\% of affected MIPLIB 2017 instances by 5\%.}, language = {en} } @phdthesis{Hennings2023, author = {Hennings, Felix}, title = {Modeling and solving real-world transient gas network transport problems using mathematical programming}, year = {2023}, abstract = {This thesis considers the transient gas network control optimization problem for on-shore pipeline-based transmission networks with numerous gas routing options. As input, the problem is given the network's topology, its initial state, and future demands at the boundaries of the network, which prescribe the gas flow exchange and potentially the pressure values. The task is to find a set of future control measures for all the active, i.e., controllable, elements in the network that minimizes a combination of different penalty functions. The problem is examined in the context of a decision support tool for gas network dispatchers. This results in detailed models featuring a diverse set of constraints, large and challenging real-world instances, and demanding time limit requirements. All these factors further complicate the problem, which is already difficult to solve in theory due to the inherent combination of non-linear and combinatorial aspects. Our contributions concern different steps of the process of solving the problem. Regarding the model formulation, we investigate the validity of two common approximations of the gas flow description in transport pipes: neglecting the inertia term and assuming a friction term that linearly depends on the gas flow and the pressure. For both, we examine if they can be applied under real-world conditions by evaluating a large amount of historical state data of the network of our project partner, the gas network operator Open Grid Europe. While we can confirm that it is reasonable to ignore the influence of the inertia term, the friction term linearization leads to significant errors and, as a consequence, cannot be used for describing the general gas flow behavior in transport pipes. As another topic of this thesis, we introduce the target value concept as a more realistic approach to express control actions of dispatchers regarding regulators and compressor stations. Here, we derive the mechanisms defined for target values based on the gas flow principles in pipes and develop a mixed-integer programming model capturing their behavior. The accuracy of this model is demonstrated in comparison to a target-value-based industry-standard simulator. Furthermore, we present two heuristics for the transient gas network control optimization problem featuring target values that are based on approximative models for the target-value-based control and determine the final decisions in a post-processing step. To compare the performance of the two heuristics with the approach of directly solving the corresponding model, we evaluate them on a set of artificially created test instances. Finally, we develop problem-specific algorithms for two variants of the described problem. One considers the control optimization for a single network station, which represents a local operation site featuring a large number of active elements. The used transient model is very detailed and includes a sophisticated representation of the compressor stations. Based on the shortness of the pipes in the station, the corresponding algorithm finds valid solutions by solving a series of stationary model variants as well as a transient rolling horizon approach. As the second variant, we consider the problem on the entire network but assume an approximative model representing the control capabilities of network stations. Aside from a new description of the compression capabilities, we introduce an algorithm that uses a combination of sequential mixed-integer programming, two heuristics based on reduced time horizons, and a specialized dynamic branch-and-bound node limit to determine promising values for the binary variables of the model. Complete solutions for the problem are obtained by fixing the binary values and solving the remaining non-linear program. Both algorithms are investigated in extensive empirical studies based on real-world instances of the corresponding model variants.}, language = {en} } @misc{PrauseHoppmannBaumDefournyetal.2020, author = {Prause, Felix and Hoppmann-Baum, Kai and Defourny, Boris and Koch, Thorsten}, title = {The Maximum Diversity Assortment Selection Problem}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-81039}, year = {2020}, abstract = {In this paper, we introduce the Maximum Diversity Assortment Selection Problem (MADASS), which is a generalization of the 2-dimensional Cutting Stock Problem (2CSP). Given a set of rectangles and a rectangular container, the goal of 2CSP is to determine a subset of rectangles that can be placed in the container without overlapping, i.e., a feasible assortment, such that a maximum area is covered. In MADASS, we need to determine a set of feasible assortments, each of them covering a certain minimum threshold of the container, such that the diversity among them is maximized. Thereby, diversity is defined as minimum or average normalized Hamming-Distance of all assortment pairs. The MADASS Problem was used in the 11th AIMMS-MOPTA Competition in 2019. The methods we describe in this article and the computational results won the contest. In the following, we give a definition of the problem, introduce a mathematical model and solution approaches, determine upper bounds on the diversity, and conclude with computational experiments conducted on test instances derived from the 2CSP literature.}, language = {en} } @misc{AndersonTurnerKoch2020, author = {Anderson, Lovis and Turner, Mark and Koch, Thorsten}, title = {Generative deep learning for decision making in gas networks}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-81103}, year = {2020}, abstract = {A decision support system relies on frequent re-solving of similar problem instances. While the general structure remains the same in corresponding applications, the input parameters are updated on a regular basis. We propose a generative neural network design for learning integer decision variables of mixed-integer linear programming (MILP) formulations of these problems. We utilise a deep neural network discriminator and a MILP solver as our oracle to train our generative neural network. In this article, we present the results of our design applied to the transient gas optimisation problem. With the trained network we produce a feasible solution in 2.5s, use it as a warm-start solution, and thereby decrease global optimal solution solve time by 60.5\%.}, language = {en} } @article{StreubelTischendorfGriewank2020, author = {Streubel, Tom and Tischendorf, Caren and Griewank, Andreas}, title = {Piecewise Polynomial Taylor Expansions - The Generalization of Fa{\`a} di Bruno's Formula}, journal = {Modeling, Simulation and Optimization of Complex Processes HPSC 2018}, number = {Modeling, Simulation and Optimization of Complex Processes HPSC 2018}, publisher = {Springer International Publishing}, doi = {10.1007/978-3-030-55240-4_3}, pages = {63 -- 82}, year = {2020}, abstract = {We present an extension of Taylor's Theorem for the piecewise polynomial expansion of non-smooth evaluation procedures involving absolute value operations. Evaluation procedures are computer programs of mathematical functions in closed form expression and allow a different treatment of smooth operations or calls to the absolute value function. The well known classical Theorem of Taylor defines polynomial approximations of sufficiently smooth functions and is widely used for the derivation and analysis of numerical integrators for systems of ordinary differential- or differential-algebraic equations, for the construction of solvers for continuous non-linear optimization of finite dimensional objective functions and for root solving of non-linear systems of equations. The long term goal is the stabilization and acceleration of already known methods and the derivation of new methods by incorporating piecewise polynomial Taylor expansions. The herein provided proof of the higher order approximation quality of the new generalized expansions is constructive and allows efficiently designed algorithms for the execution and computation of the piecewise polynomial expansions. As a demonstration towards the ultimate goal we will derive a prototype of a {\\$}{\\$}k{\\$}{\\$}k-step method on the basis of polynomial interpolation and the proposed generalized expansions.}, language = {en} } @article{HoppmannBaumHenningsZitteletal.2021, author = {Hoppmann-Baum, Kai and Hennings, Felix and Zittel, Janina and Gotzes, Uwe and Spreckelsen, Eva-Maria and Spreckelsen, Klaus and Koch, Thorsten}, title = {An Optimization Approach for the Transient Control of Hydrogen Transport Networks}, journal = {Mathematical Methods of Operations Research}, number = {Special Issue on Energy Networks}, year = {2021}, language = {en} } @misc{HoppmannBaumMexiBurdakovetal.2020, author = {Hoppmann-Baum, Kai and Mexi, Gioni and Burdakov, Oleg and Casselgren, Carl Johan and Koch, Thorsten}, title = {Length-Constrained Cycle Partition with an Application to UAV Routing}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-80489}, year = {2020}, abstract = {In this article, we discuss the Length-Constrained Cycle Partition Problem (LCCP). Besides edge weights, the undirected graph in LCCP features an individual critical weight value for each vertex. A cycle partition, i.e., a vertex disjoint cycle cover, is a feasible solution if the length of each cycle is not greater than the critical weight of each of the vertices in the cycle. The goal is to find a feasible partition with the minimum number of cycles. In this article, we discuss theoretical properties, preprocessing techniques, and two mixed-integer programming models (MIP) for LCCP both inspired by formulations for the closely related Travelling Salesperson Problem (TSP). Further, we introduce conflict hypergraphs, whose cliques yield valid constraints for the MIP models. We conclude with a report on computational experiments conducted on (A)TSPLIB-based instances. As an example, we use a routing problem in which a fleet of uncrewed aerial vehicles (UAVs) patrols a set of areas.}, language = {en} } @article{GotzesHoppmannBaum2020, author = {Gotzes, Uwe and Hoppmann-Baum, Kai}, title = {Bounding the final rank during a round robin tournament with integer programming}, volume = {1866-1505}, journal = {Operational Research}, doi = {https://doi.org/10.1007/s12351-020-00546-w}, year = {2020}, abstract = {This article is mainly motivated by the urge to answer two kinds of questions regarding the Bundesliga, which is Germany's primary football (soccer) division having the highest average stadium attendance worldwide: "At any point in the season, what is the lowest final rank a certain team can achieve?" and "At any point in the season, what is the highest final rank a certain team can achieve?". Although we focus on the Bundesliga in particular, the integer programming formulations we introduce to answer these questions can easily be adapted to a variety of other league systems and tournaments.}, language = {en} } @misc{RehfeldtKoch2020, author = {Rehfeldt, Daniel and Koch, Thorsten}, title = {Implications, conflicts, and reductions for Steiner trees}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-80039}, year = {2020}, language = {en} } @misc{HoppmannBaumHenningsZitteletal.2020, author = {Hoppmann-Baum, Kai and Hennings, Felix and Zittel, Janina and Gotzes, Uwe and Spreckelsen, Eva-Maria and Spreckelsen, Klaus and Koch, Thorsten}, title = {From Natural Gas towards Hydrogen - A Feasibility Study on Current Transport Network Infrastructure and its Technical Control}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-79901}, year = {2020}, abstract = {This study examines the usability of a real-world, large-scale natural gas transport infrastructure for hydrogen transport. We investigate whether a converted network can transport the amounts of hydrogen necessary to satisfy current energy demands. After introducing an optimization model for the robust transient control of hydrogen networks, we conduct computational experiments based on real-world demand scenarios. Using a representative network, we demonstrate that replacing each turbo compressor unit by four parallel hydrogen compressors, each of them comprising multiple serial compression stages, and imposing stricter rules regarding the balancing of in- and outflow suffices to realize transport in a majority of scenarios. However, due to the reduced linepack there is an increased need for technical and non-technical measures leading to a more dynamic network control. Furthermore, the amount of energy needed for compression increases by 364\% on average.}, language = {en} } @article{HoppmannBaumHenningsLenzetal.2021, author = {Hoppmann-Baum, Kai and Hennings, Felix and Lenz, Ralf and Gotzes, Uwe and Heinecke, Nina and Spreckelsen, Klaus and Koch, Thorsten}, title = {Optimal Operation of Transient Gas Transport Networks}, volume = {22}, journal = {Optimization and Engineering}, doi = {10.1007/s11081-020-09584-x}, pages = {735 -- 781}, year = {2021}, abstract = {In this paper, we describe an algorithmic framework for the optimal operation of transient gas transport networks consisting of a hierarchical MILP formulation together with a sequential linear programming inspired post-processing routine. Its implementation is part of the KOMPASS decision support system, which is currently used in an industrial setting. Real-world gas transport networks are controlled by operating complex pipeline intersection areas, which comprise multiple compressor units, regulators, and valves. In the following, we introduce the concept of network stations to model them. Thereby, we represent the technical capabilities of a station by hand-tailored artificial arcs and add them to network. Furthermore, we choose from a predefined set of flow directions for each network station and time step, which determines where the gas enters and leaves the station. Additionally, we have to select a supported simple state, which consists of two subsets of artificial arcs: Arcs that must and arcs that cannot be used. The goal is to determine a stable control of the network satisfying all supplies and demands. The pipeline intersections, that are represented by the network stations, were initially built centuries ago. Subsequently, due to updates, changes, and extensions, they evolved into highly complex and involved topologies. To extract their basic properties and to model them using computer-readable and optimizable descriptions took several years of effort. To support the dispatchers in controlling the network, we need to compute a continuously updated list of recommended measures. Our motivation for the model presented here is to make fast decisions on important transient global control parameters, i.e., how to route the flow and where to compress the gas. Detailed continuous and discrete technical control measures realizing them, which take all hardware details into account, are determined in a subsequent step. In this paper, we present computational results from the KOMPASS project using detailed real-world data.}, language = {en} } @article{GriewankStreubelTischendorf2020, author = {Griewank, Andreas and Streubel, Tom and Tischendorf, Caren}, title = {On the abs-polynomial expansion of piecewise smooth functions}, journal = {Optimization Methods and Software}, publisher = {Taylor \& Francis}, doi = {10.1080/10556788.2020.1817448}, year = {2020}, abstract = {Tom Streubel has observed that for functions in abs-normal form, generalized Taylor expansions of arbitrary order \$\bar d-1\$ can be generated by algorithmic piecewise differentiation. Abs-normal form means that the real or vector valued function is defined by an evaluation procedure that involves the absolute value function \$|...|\$ apart from arithmetic operations and \$\bar d\$ times continuously differentiable univariate intrinsic functions. The additive terms in Streubel's expansion are abs-polynomial, i.e. involve neither divisions nor intrinsics. When and where no absolute values occur, Moore's recurrences can be used to propagate univariate Taylor polynomials through the evaluation procedure with a computational effort of \$\mathcal O({\bar d}^2)\$, provided all univariate intrinsics are defined as solutions of linear ODEs. This regularity assumption holds for all standard intrinsics, but for irregular elementaries one has to resort to Faa di Bruno's formula, which has exponential complexity in \$\bar d\$. As already conjectured we show that the Moore recurrences can be adapted for regular intrinsics to the abs-normal case. Finally, we observe that where the intrinsics are real analytic the expansions can be extended to infinite series that converge absolutely on spherical domains.}, language = {en} } @incollection{HoppmannBaumMexiBurdakovetal.2020, author = {Hoppmann-Baum, Kai and Mexi, Gioni and Burdakov, Oleg and Casselgren, Carl Johan and Koch, Thorsten}, title = {Minimum Cycle Partition with Length Requirements}, volume = {12296}, booktitle = {Integration of Constraint Programming, Artificial Intelligence, and Operations Research}, editor = {Hebrard, Emmanuel and Musliu, Nysret}, publisher = {Springer International Publishing}, address = {Cham}, isbn = {978-3-030-58941-7}, doi = {10.1007/978-3-030-58942-4_18}, pages = {273 -- 282}, year = {2020}, abstract = {In this article we introduce a Minimum Cycle Partition Problem with Length Requirements (CPLR). This generalization of the Travelling Salesman Problem (TSP) originates from routing Unmanned Aerial Vehicles (UAVs). Apart from nonnegative edge weights, CPLR has an individual critical weight value associated with each vertex. A cycle partition, i.e., a vertex disjoint cycle cover, is regarded as a feasible solution if the length of each cycle, which is the sum of the weights of its edges, is not greater than the critical weight of each of its vertices. The goal is to find a feasible partition, which minimizes the number of cycles. In this article, a heuristic algorithm is presented together with a Mixed Integer Programming (MIP) formulation of CPLR. We furthermore introduce a conflict graph, whose cliques yield valid constraints for the MIP model. Finally, we report on computational experiments conducted on TSPLIB-based test instances.}, language = {en} } @article{PrauseHoppmannBaumDefournyetal.2021, author = {Prause, Felix and Hoppmann-Baum, Kai and Defourny, Boris and Koch, Thorsten}, title = {The maximum diversity assortment selection problem}, volume = {93}, journal = {Mathematical Methods of Operations Research}, publisher = {Mathematical Methods of Operations Research}, doi = {https://doi.org/10.1007/s00186-021-00740-2}, pages = {521 -- 554}, year = {2021}, abstract = {In this article, we introduce the Maximum Diversity Assortment Selection Problem (MDASP), which is a generalization of the two-dimensional Knapsack Problem (2D-KP). Given a set of rectangles and a rectangular container, the goal of 2D-KP is to determine a subset of rectangles that can be placed in the container without overlapping, i.e., a feasible assortment, such that a maximum area is covered. MDASP is to determine a set of feasible assortments, each of them covering a certain minimum threshold of the container, such that the diversity among them is maximized. Thereby, diversity is defined as the minimum or average normalized Hamming distance of all assortment pairs. MDASP was the topic of the 11th AIMMS-MOPTA Competition in 2019. The methods described in this article and the resulting computational results won the contest. In the following, we give a definition of the problem, introduce a mathematical model and solution approaches, determine upper bounds on the diversity, and conclude with computational experiments conducted on test instances derived from the 2D-KP literature.}, language = {en} } @misc{YuekselErguenZittelWangetal.2020, author = {Yueksel-Erguen, Inci and Zittel, Janina and Wang, Ying and Hennings, Felix and Koch, Thorsten}, title = {Lessons learned from gas network data preprocessing}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-78262}, year = {2020}, abstract = {The German high-pressure natural gas transport network consists of thousands of interconnected elements spread over more than 120,000 km of pipelines built during the last 100 years. During the last decade, we have spent many person-years to extract consistent data out of the available sources, both public and private. Based on two case studies, we present some of the challenges we encountered. Preparing consistent, high-quality data is surprisingly hard, and the effort necessary can hardly be overestimated. Thus, it is particularly important to decide which strategy regarding data curation to adopt. Which precision of the data is necessary? When is it more efficient to work with data that is just sufficiently correct on average? In the case studies we describe our experiences and the strategies we adopted to deal with the obstacles and to minimize future effort. Finally, we would like to emphasize that well-compiled data sets, publicly available for research purposes, provide the grounds for building innovative algorithmic solutions to the challenges of the future.}, language = {en} } @misc{GriewankStreubelTischendorf2020, author = {Griewank, Andreas and Streubel, Tom and Tischendorf, Caren}, title = {On the abs-polynomial expansion of piecewise smooth functions}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-78448}, year = {2020}, abstract = {Tom Streubel has observed that for functions in abs-normal form, generalized Taylor expansions of arbitrary order \$\bar d-1\$ can be generated by algorithmic piecewise differentiation. Abs-normal form means that the real or vector valued function is defined by an evaluation procedure that involves the absolute value function \$|...|\$ apart from arithmetic operations and \$\bar d\$ times continuously differentiable univariate intrinsic functions. The additive terms in Streubel's expansion are abs-polynomial, i.e. involve neither divisions nor intrinsics. When and where no absolute values occur, Moore's recurrences can be used to propagate univariate Taylor polynomials through the evaluation procedure with a computational effort of \$\mathcal O({\bar d}^2)\$, provided all univariate intrinsics are defined as solutions of linear ODEs. This regularity assumption holds for all standard intrinsics, but for irregular elementaries one has to resort to Faa di Bruno's formula, which has exponential complexity in \$\bar d\$. As already conjectured we show that the Moore recurrences can be adapted for regular intrinsics to the abs-normal case. Finally, we observe that where the intrinsics are real analytic the expansions can be extended to infinite series that converge absolutely on spherical domains.}, language = {en} } @misc{BeckerHiller2019, author = {Becker, Kai-Helge and Hiller, Benjamin}, title = {Improved optimization models for potential-driven network flow problems via ASTS orientations}, issn = {1438-0064}, doi = {10.12752/7534}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-75347}, year = {2019}, abstract = {The class of potential-driven network flow problems provides important models for a range of infrastructure networks that lead to hard-to-solve MINLPs in real-world applications. On large-scale meshed networks the relaxations usually employed are rather weak due to cycles in the network. To address this situation, we introduce the concept of ASTS orientations, a generalization of bipolar orientations, as a combinatorial relaxation of feasible solutions of potential-driven flow problems, study their structure, and show how they can be used to strengthen existing relaxations and thus provide improved optimization models. Our computational results indicate that ASTS orientations can be used to derive much stronger bounds on the flow variables than existing bound tightening methods and to yield significant performance improvements for an existing state-of-the-art MILP model for large-scale gas networks.}, language = {en} } @masterthesis{Mexi2020, type = {Bachelor Thesis}, author = {Mexi, Gioni}, title = {Node Partitioning and Subtours Creation Problem (NPSC)}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-77094}, year = {2020}, language = {en} } @incollection{BennerGrundelHimpeetal.2019, author = {Benner, Peter and Grundel, Sara and Himpe, Christian and Huck, Christoph and Streubel, Tom and Tischendorf, Caren}, title = {Gas Network Benchmark Models}, booktitle = {Applications of Differential-Algebraic Equations: Examples and Benchmarks}, publisher = {Springer International Publishing}, isbn = {978-3-030-03718-5}, doi = {10.1007/11221_2018_5}, pages = {171 -- 197}, year = {2019}, abstract = {The simulation of gas transportation networks becomes increasingly more important as its use-cases broaden to more complex applications. Classically, the purpose of the gas network was the transportation of predominantly natural gas from a supplier to the consumer for long-term scheduled volumes. With the rise of renewable energy sources, gas-fired power plants are often chosen to compensate for the fluctuating nature of the renewables, due to their on-demand power generation capability. Such an only short-term plannable supply and demand setting requires sophisticated simulations of the gas network prior to the dispatch to ensure the supply of all customers for a range of possible scenarios and to prevent damages to the gas network. In this work we describe the modeling of gas networks and present benchmark systems to test implementations and compare new or extended models.}, language = {en} } @misc{GamrathKochMartinetal.2013, author = {Gamrath, Gerald and Koch, Thorsten and Martin, Alexander and Miltenberger, Matthias and Weninger, Dieter}, title = {Progress in Presolving for Mixed Integer Programming}, issn = {1438-0064}, doi = {10.1007/s12532-015-0083-5}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-42530}, year = {2013}, abstract = {Presolving attempts to eliminate redundant information from the problem formulation and simultaneously tries to strengthen the formulation. It can be very effective and is often essential for solving instances. Especially for mixed integer programming problems, fast and effective presolving algorithms are very important. In this paper, we report on three new presolving techniques. The first method searches for singleton continuous columns and tries to fix the corresponding variables. Then we present a presolving technique which exploits a partial order of the variables to induce fixings. Finally, we show an approach based on connected components in graphs. Our computational results confirm the profitable use of the algorithms in practice.}, language = {en} } @article{HoppmannBaumBurdakovMexietal.2022, author = {Hoppmann-Baum, Kai and Burdakov, Oleg and Mexi, Gioni and Casselgren, Carl Johan and Koch, Thorsten}, title = {Length-Constrained Cycle Partition with an Application to UAV Routing}, journal = {Optimization Methods and Software}, doi = {10.1080/10556788.2022.2053972}, year = {2022}, abstract = {This article discusses the Length-Constrained Cycle Partition Problem (LCCP), which constitutes a new generalization of the Travelling Salesperson Problem (TSP). Apart from nonnegative edge weights, the undirected graph in LCCP features a nonnegative critical length parameter for each vertex. A cycle partition, i.e., a vertex-disjoint cycle cover, is a feasible solution for LCCP if the length of each cycle is not greater than the critical length of each vertex contained in it. The goal is to find a feasible partition having a minimum number of cycles. Besides analyzing theoretical properties and developing preprocessing techniques, we propose an elaborate heuristic algorithm that produces solutions of good quality even for large-size instances. Moreover, we present two exact mixed-integer programming formulations (MIPs) for LCCP, which are inspired by well-known modeling approaches for TSP. Further, we introduce the concept of conflict hypergraphs, whose cliques yield valid constraints for the MIP models. We conclude with a discussion on computational experiments that we conducted using (A)TSPLIB-based problem instances. As a motivating example application, we describe a routing problem where a fleet of uncrewed aerial vehicles (UAVs) must patrol a given set of areas.}, language = {en} } @article{Dell’AmicoHadjidimitriouKochetal.2018, author = {Dell'Amico, M. and Hadjidimitriou, Natalia Selini and Koch, Thorsten and Petkovic, Milena}, title = {Forecasting Natural Gas Flows in Large Networks}, volume = {Lecture Notes in Computer Science}, journal = {Machine Learning, Optimization, and Big Data. MOD 2017.}, number = {vol 10710}, doi = {https://doi.org/10.1007/978-3-319-72926-8_14}, pages = {158 -- 171}, year = {2018}, abstract = {Natural gas is the cleanest fossil fuel since it emits the lowest amount of other remains after being burned. Over the years, natural gas usage has increased significantly. Accurate forecasting is crucial for maintaining gas supplies, transportation and network stability. This paper presents two methodologies to identify the optimal configuration o parameters of a Neural Network (NN) to forecast the next 24 h of gas flow for each node of a large gas network. In particular the first one applies a Design Of Experiments (DOE) to obtain a quick initial solution. An orthogonal design, consisting of 18 experiments selected among a total of 4.374 combinations of seven parameters (training algorithm, transfer function, regularization, learning rate, lags, and epochs), is used. The best result is selected as initial solution of an extended experiment for which the Simulated Annealing is run to find the optimal design among 89.100 possible combinations of parameters. The second technique is based on the application of Genetic Algorithm for the selection of the optimal parameters of a recurrent neural network for time series forecast. GA was applied with binary representation of potential solutions, where subsets of bits in the bit string represent different values for several parameters of the recurrent neural network. We tested these methods on three municipal nodes, using one year and half of hourly gas flow to train the network and 60 days for testing. Our results clearly show that the presented methodologies bring promising results in terms of optimal configuration of parameters and forecast error.}, language = {en} } @inproceedings{AndersonHiller2019, author = {Anderson, Lovis and Hiller, Benjamin}, title = {A Sweep-Plane Algorithm for the Computation of the Volume of a Union of Polytopes}, volume = {Operations Research Proceedings}, booktitle = {Operations Research Proceedings 2018}, doi = {10.1007/978-3-030-18500-8_12}, year = {2019}, abstract = {Optimization models often feature disjunctions of polytopes as submodels. Such a disjunctive set is initially at best) relaxed to its convex hull, which is then refined by branching. To measure the error of the convex relaxation, the (relative) difference between the volume of the convex hull and the volume of the disjunctive set may be used. This requires a method to compute the volume of the disjunctive set. We propose a revised variant of an old algorithm by Bieri and Nef (1983) for this purpose. The algorithm uses a sweep-plane to incrementally calculate the volume of the disjunctive set as a function of the offset parameter of the sweep-plane.}, language = {en} } @misc{HasenfelderLehmannRadonsetal.2018, author = {Hasenfelder, Richard and Lehmann, Lutz and Radons, Manuel and Streubel, Tom and Strohm, Christian and Griewank, Andreas}, title = {Computational aspects of the Generalized Trapezoidal Rule}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-68615}, year = {2018}, abstract = {In this article we analyze a generalized trapezoidal rule for initial value problems with piecewise smooth right hand side F:IR^n -> IR^n. When applied to such a problem, the classical trapezoidal rule suffers from a loss of accuracy if the solution trajectory intersects a non-differentiability of F. In such a situation the investigated generalized trapezoidal rule achieves a higher convergence order than the classical method. While the asymptotic behavior of the generalized method was investigated in a previous work, in the present article we develop the algorithmic structure for efficient implementation strategies and estimate the actual computational cost of the latter. Moreover, energy preservation of the generalized trapezoidal rule is proved for Hamiltonian systems with piecewise linear right hand side.}, language = {en} } @misc{StreubelTischendorfGriewank2018, author = {Streubel, Tom and Tischendorf, Caren and Griewank, Andreas}, title = {Piecewise Polynomial Taylor Expansions - The Generalization of Fa{\`a} di Bruno's Formula}, issn = {1438-0064}, doi = {10.1007/978-3-030-55240-4_3}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-68859}, year = {2018}, abstract = {We present an extension of Taylor's theorem towards nonsmooth evalua- tion procedures incorporating absolute value operaions. Evaluations procedures are computer programs of mathematical functions in closed form expression and al- low a different treatment of smooth operations and calls to the absolute value value function. The well known classical Theorem of Taylor defines polynomial approx- imation of sufficiently smooth functions and is widely used for the derivation and analysis of numerical integrators for systems of ordinary differential or differential algebraic equations, for the construction of solvers for the continuous nonlinear op- timization of finite dimensional objective functions and for root solving of nonlinear systems of equations. The herein provided proof is construtive and allow efficiently designed algorithms for the execution and computation of generalized piecewise polynomial expansions. As a demonstration we will derive a k-step method on the basis of polynomial interpolation and the proposed generalized expansions.}, language = {en} } @misc{LenzBecker2018, author = {Lenz, Ralf and Becker, Kai-Helge}, title = {Optimization of Capacity Expansion in Potential-driven Networks including Multiple Looping - A comparison of modelling approaches}, issn = {1438-0064}, doi = {10.1007/s00291-021-00648-7}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-69726}, year = {2018}, abstract = {In commodity transport networks such as natural gas, hydrogen and water networks, flows arise from nonlinear potential differences between the nodes, which can be represented by so-called "potential-driven" network models. When operators of these networks face increasing demand or the need to handle more diverse transport situations, they regularly seek to expand the capacity of their network by building new pipelines parallel to existing ones ("looping"). The paper introduces a new mixed-integer non-linear programming (MINLP) model and a new non-linear programming (NLP) model and compares these with existing models for the looping problem and related problems in the literature, both theoretically and experimentally. On this basis, we give recommendations about the circumstances under which a certain model should be used. In particular, it turns out that one of our novel models outperforms the existing models. Moreover, the paper is the first to include the practically relevant option that a particular pipeline may be looped several times.}, language = {en} } @misc{RadonsLehmannStreubeletal.2018, author = {Radons, Manuel and Lehmann, Lutz and Streubel, Tom and Griewank, Andreas}, title = {An Open Newton Method for Piecewise Smooth Systems}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-70418}, year = {2018}, abstract = {Recent research has shown that piecewise smooth (PS) functions can be approximated by piecewise linear functions with second order error in the distance to a given reference point. A semismooth Newton type algorithm based on successive application of these piecewise linearizations was subsequently developed for the solution of PS equation systems. For local bijectivity of the linearization at a root, a radius of quadratic convergence was explicitly calculated in terms of local Lipschitz constants of the underlying PS function. In the present work we relax the criterium of local bijectivity of the linearization to local openness. For this purpose a weak implicit function theorem is proved via local mapping degree theory. It is shown that there exist PS functions f:IR^2 --> IR^2 satisfying the weaker criterium where every neighborhood of the root of f contains a point x such that all elements of the Clarke Jacobian at x are singular. In such neighborhoods the steps of classical semismooth Newton are not defined, which establishes the new method as an independent algorithm. To further clarify the relation between a PS function and its piecewise linearization, several statements about structure correspondences between the two are proved. Moreover, the influence of the specific representation of the local piecewise linear models on the robustness of our method is studied. An example application from cardiovascular mathematics is given.}, language = {en} } @misc{BeckerHiller2018, author = {Becker, Kai-Helge and Hiller, Benjamin}, title = {ASTS Orientations on Undirected Graphs: Structural analysis and enumeration}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-69632}, year = {2018}, abstract = {All feasible flows in potential-driven networks induce an orientation on the undirected graph underlying the network. Clearly, these orientations must satisfy two conditions: they are acyclic and there are no "dead ends" in the network, i.e. each source requires outgoing flows, each sink requires incoming flows, and each transhipment vertex requires both an incoming and an outgoing flow. In this paper we will call orientations that satisfy these conditions acyclic source-transhipment-sink orientations (ASTS-orientation) and study their structure. In particular, we characterize graphs that allow for such an orientation, describe a way to enumerate all possible ASTS-orientations of a given graph, present an algorithm to simplify and decompose a graph before such an enumeration and shed light on the role of zero flows in the context of ASTS-orientations.}, language = {en} } @misc{HillerBecker2018, author = {Hiller, Benjamin and Becker, Kai-Helge}, title = {Improving relaxations for potential-driven network flow problems via acyclic flow orientations}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-69622}, year = {2018}, abstract = {The class of potential-driven network flow problems provides important models for a range of infrastructure networks. For real-world applications, they need to be combined with integer models for switching certain network elements, giving rise to hard-to-solve MINLPs. We observe that on large-scale real-world meshed networks the usually employed relaxations are rather weak due to cycles in the network. We propose acyclic flow orientations as a combinatorial relaxation of feasible solutions of potential-driven flow problems and show how they can be used to strengthen existing relaxations. First computational results indicate that the strengthend model is much tighter than the original relaxation, thus promising a computational advantage.}, language = {en} } @misc{Schweiger2018, author = {Schweiger, Jonas}, title = {Exploiting structure in non-convex quadratic optimization}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-69476}, year = {2018}, abstract = {The amazing success of computational mathematical optimization over the last decades has been driven more by insights into mathematical structures than by the advance of computing technology. In this vein, we address applications, where nonconvexity in the model poses principal difficulties. This paper summarizes the dissertation of Jonas Schweiger for the occasion of the GOR dissertation award 2018. We focus on the work on non-convex quadratic programs and show how problem specific structure can be used to obtain tight relaxations and speed up Branch\&Bound methods. Both a classic general QP and the Pooling Problem as an important practical application serve as showcases.}, language = {en} } @misc{AndersonHiller2018, author = {Anderson, Lovis and Hiller, Benjamin}, title = {A Sweep-Plane Algorithm for the Computation of the Volume of a Union of Polytopes}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-69489}, year = {2018}, abstract = {Optimization models often feature disjunctions of polytopes as submodels. Such a disjunctive set is initially (at best) relaxed to its convex hull, which is then refined by branching. To measure the error of the convex relaxation, the (relative) difference between the volume of the convex hull and the volume of the disjunctive set may be used. This requires a method to compute the volume of the disjunctive set. Naively, this can be done via inclusion/exclusion and leveraging the existing code for the volume of polytopes. However, this is often inefficient. We propose a revised variant of an old algorithm by Bieri and Nef (1983) for this purpose. The algorithm uses a sweep-plane to incrementally calculate the volume of the disjunctive set as a function of the offset parameter of the sweep-plane.}, language = {en} } @misc{LuedtkeD'AmbrosioLinderothetal.2018, author = {Luedtke, James and D'Ambrosio, Claudia and Linderoth, Jeff and Schweiger, Jonas}, title = {Strong Convex Nonlinear Relaxations of the Pooling Problem}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-67824}, year = {2018}, abstract = {We investigate new convex relaxations for the pooling problem, a classic nonconvex production planning problem in which input materials are mixed in intermediate pools, with the outputs of these pools further mixed to make output products meeting given attribute percentage requirements. Our relaxations are derived by considering a set which arises from the formulation by considering a single product, a single attibute, and a single pool. The convex hull of the resulting nonconvex set is not polyhedral. We derive valid linear and convex nonlinear inequalities for the convex hull, and demonstrate that different subsets of these inequalities define the convex hull of the nonconvex set in three cases determined by the parameters of the set. Computational results on literature instances and newly created larger test instances demonstrate that the inequalities can significantly strengthen the convex relaxation of the pq-formulation of the pooling problem, which is the relaxation known to have the strongest bound.}, language = {en} } @misc{HillerSaitenmacherWalther2018, author = {Hiller, Benjamin and Saitenmacher, Ren{\´e} and Walther, Tom}, title = {Analysis of operating modes of complex compressor stations}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-68179}, year = {2018}, abstract = {We consider the modeling of operation modes for complex compressor stations (i.e., ones with several in- or outlets) in gas networks. In particular, we propose a refined model that allows to precompute tighter relaxations for each operation mode. These relaxations may be used to strengthen the compressor station submodels in gas network optimization problems. We provide a procedure to obtain the refined model from the input data for the original model. This procedure is based on a nontrivial reduction of the graph representing the gas flow through the compressor station in an operation mode.}, language = {en} } @misc{LuedtkeD'AmbrosioLinderothetal.2018, author = {Luedtke, James and D'Ambrosio, Claudia and Linderoth, Jeff and Schweiger, Jonas}, title = {Strong Convex Nonlinear Relaxations of the Pooling Problem: Extreme Points}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-67801}, year = {2018}, abstract = {We investigate new convex relaxations for the pooling problem, a classic nonconvex production planning problem in which products are mixed in intermediate pools in order to meet quality targets at their destinations. In this technical report, we characterize the extreme points of the convex hull of our non-convex set, and show that they are not finite, i.e., the convex hull is not polyhedral. This analysis was used to derive valid nonlinear convex inequalities and show that, for a specific case, they characterize the convex hull of our set. The new valid inequalities and computational results are presented in ZIB Report 18-12.}, language = {en} } @article{GotzesGotzes2018, author = {Gotzes, Uwe and Gotzes, Claudia}, title = {The Jeep Problem: A Flexible MIP Formulation}, volume = {1}, journal = {Journal of Nepal Mathematical Society}, issn = {2616-0161}, pages = {11 -- 17}, year = {2018}, abstract = {This article deals with the Jeep Problem (also known as Desert Crossing Problem), which reads as follows: An unlimited supply of fuel is available at one edge of a desert, but there is no source on the desert itself. A vehicle can carry enough fuel to go a certain distance, and it can built up its own refuelling stations. What is the minimum amount of fuel the vehicle will require in order to cross the desert? Under these mild conditions this question is answered since the 1940s. But what is the answer if the caches are restricted to certain areas or if the fuel consumption does not depend linearly on the distance travelled? To answer these and similar questions we develop and solve a flexible mixed-integer programming (MIP) model for the classical problem and enhance it with new further aspects of practical relevance.}, language = {en} } @misc{LenzSerrano2021, author = {Lenz, Ralf and Serrano, Felipe}, title = {Tight Convex Relaxations for the Expansion Planning Problem}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-81459}, year = {2021}, abstract = {Secure energy transport is considered as highly relevant for the basic infrastructure of nowadays society and economy. To satisfy increasing demands and to handle more diverse transport situations, operators of energy networks regularly expand the capacity of their network by building new network elements, known as the expansion planning problem. A key constraint function in expansion planning problems is a nonlinear and nonconvex potential loss function. In order to improve the algorithmic performance of state-of-the-art MINLP solvers, this paper presents an algebraic description for the convex envelope of this function. Through a thorough computational study, we show that this tighter relaxation tremendously improve the performance of the MINLP solver SCIP on a large test set of practically relevant instances for the expansion planning problem. In particular, the results show that our achievements lead to an improvement of the solver performance for a development version by up to 58\%.}, language = {en} } @misc{FuegenschuhHillerHumpolaetal.2011, author = {F{\"u}genschuh, Armin and Hiller, Benjamin and Humpola, Jesco and Koch, Thorsten and Lehmann, Thomas and Schwarz, Robert and Schweiger, Jonas and Szab{\´o}, J{\´a}cint}, title = {Gas Network Topology Optimization for Upcoming Market Requirements}, doi = {10.1109/EEM.2011.5953035}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-12348}, number = {11-09}, year = {2011}, abstract = {Gas distribution networks are complex structures that consist of passive pipes, and active, controllable elements such as valves and compressors. Controlling such network means to find a suitable setting for all active components such that a nominated amount of gas can be transmitted from entries to exits through the network, without violating physical or operational constraints. The control of a large-scale gas network is a challenging task from a practical point of view. In most companies the actual controlling process is supported by means of computer software that is able to simulate the flow of the gas. However, the active settings have to be set manually within such simulation software. The solution quality thus depends on the experience of a human planner. When the gas network is insufficient for the transport then topology extensions come into play. Here a set of new pipes or active elements is determined such that the extended network admits a feasible control again. The question again is how to select these extensions and where to place them such that the total extension costs are minimal. Industrial practice is again to use the same simulation software, determine extensions by experience, add them to the virtual network, and then try to find a feasible control of the active elements. The validity of this approach now depends even more on the human planner. Another weakness of this manual simulation-based approach is that it cannot establish infeasibility of a certain gas nomination, unless all settings of the active elements are tried. Moreover, it is impossible to find a cost-optimal network extension in this way. In order to overcome these shortcomings of the manual planning approach we present a new approach, rigorously based on mathematical optimization. Hereto we describe a model for finding feasible controls and then extend this model such that topology extensions can additionally and simultaneously be covered. Numerical results for real-world instances are presented and discussed.}, language = {en} } @inproceedings{FuegenschuhHillerHumpolaetal.2011, author = {F{\"u}genschuh, Armin and Hiller, Benjamin and Humpola, Jesco and Koch, Thorsten and Lehmann, Thomas and Schwarz, Robert and Schweiger, Jonas and Szabo, Jacint}, title = {Gas Network Topology Optimization for Upcoming Market Requirements}, booktitle = {International Conference on the European Energy Market (EEM)}, doi = {10.1109/EEM.2011.5953035}, pages = {346 -- 351}, year = {2011}, abstract = {Gas distribution networks are complex structures that consist of passive pipes, and active, controllable elements such as valves and compressors. Controlling such network means to find a suitable setting for all active components such that a nominated amount of gas can be transmitted from entries to exits through the network, without violating physical or operational constraints. The control of a large-scale gas network is a challenging task from a practical point of view. In most companies the actual controlling process is supported by means of computer software that is able to simulate the flow of the gas. However, the active settings have to be set manually within such simulation software. The solution quality thus depends on the experience of a human planner. When the gas network is insufficient for the transport then topology extensions come into play. Here a set of new pipes or active elements is determined such that the extended network admits a feasible control again. The question again is how to select these extensions and where to place them such that the total extension costs are minimal. Industrial practice is again to use the same simulation software, determine extensions by experience, add them to the virtual network, and then try to find a feasible control of the active elements. The validity of this approach now depends even more on the human planner. Another weakness of this manual simulation-based approach is that it cannot establish infeasibility of a certain gas nomination, unless all settings of the active elements are tried. Moreover, it is impossible to find a cost-optimal network extension in this way. In order to overcome these shortcomings of the manual planning approach we present a new approach, rigorously based on mathematical optimization. Hereto we describe a model for finding feasible controls and then extend this model such that topology extensions can additionally and simultaneously be covered. Numerical results for real-world instances are presented and discussed.}, language = {en} } @article{FuegenschuhGeisslerGollmeretal.2013, author = {F{\"u}genschuh, Armin and Geißler, Bj{\"o}rn and Gollmer, Ralf and Hayn, Christine and Henrion, Ren{\´e} and Hiller, Benjamin and Humpola, Jesco and Koch, Thorsten and Lehmann, Thomas and Martin, Alexander and Mirkov, Radoslava and Morsi, Antonio and R{\"o}misch, Werner and R{\"o}vekamp, Jessica and Schewe, Lars and Schmidt, Martin and Schultz, R{\"u}diger and Schwarz, Robert and Schweiger, Jonas and Stangl, Claudia and Steinbach, Marc and Willert, Bernhard}, title = {Mathematical optimization for challenging network planning problems in unbundled liberalized gas markets}, volume = {5}, journal = {Energy Systems}, number = {3}, publisher = {Springer Berlin Heidelberg}, address = {Berlin}, doi = {10.1007/s12667-013-0099-8}, pages = {449 -- 473}, year = {2013}, abstract = {The recently imposed new gas market liberalization rules in Germany lead to a change of business of gas network operators. While previously network operator and gas vendor were united, they were forced to split up into independent companies. The network has to be open to any other gas trader at the same conditions, and free network capacities have to be identified and publicly offered in a non-discriminatory way. We discuss how these changing paradigms lead to new and challenging mathematical optimization problems. This includes the validation of nominations, that asks for the decision if the network's capacity is sufficient to transport a specific amount of flow, the verification of booked capacities and the detection of available freely allocable capacities, and the topological extension of the network with new pipelines or compressors in order to increase its capacity. In order to solve each of these problems and to provide meaningful results for the practice, a mixture of different mathematical aspects have to be addressed, such as combinatorics, stochasticity, uncertainty, and nonlinearity. Currently, no numerical solver is available that can deal with such blended problems out-of-the-box. The main goal of our research is to develop such a solver, that moreover is able to solve instances of realistic size. In this article, we describe the main ingredients of our prototypical software implementations.}, language = {en} } @misc{HumpolaFuegenschuhLehmann2013, author = {Humpola, Jesco and F{\"u}genschuh, Armin and Lehmann, Thomas}, title = {A Primal Heuristic for MINLP based on Dual Information}, issn = {1438-0064}, doi = {10.1007/s13675-014-0029-0}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-43110}, year = {2013}, abstract = {We present a novel heuristic algorithm to identify feasible solutions of a mixed-integer nonlinear programming problem arising in natural gas transportation: the selection of new pipelines to enhance the network's capacity to a desired level in a cost-efficient way. We solve this problem in a linear programming based branch-and-cut approach, where we deal with the nonlinearities by linear outer approximation and spatial branching. At certain nodes of the branching tree, we compute a KKT point for a nonlinear relaxation. Based on the information from the KKT point we alter some of the integer variables in a locally promising way. We describe this heuristic for general MINLPs and then show how to tailor the heuristic to exploit our problem-specific structure. On a test set of real-world instances, we are able to increase the chance of identifying feasible solutions by some order of magnitude compared to standard MINLP heuristics that are already built in the general-purpose MINLP solver SCIP.}, language = {en} } @article{HumpolaFuegenschuhKoch2016, author = {Humpola, Jesco and F{\"u}genschuh, Armin and Koch, Thorsten}, title = {Valid inequalities for the topology optimization problem in gas network design}, volume = {38}, journal = {OR Spectrum}, number = {3}, publisher = {Springer}, address = {Berlin Heidelberg}, doi = {https://doi.org/10.1007/s00291-015-0390-2}, pages = {597 -- 631}, year = {2016}, abstract = {One quarter of Europe's energy demand is provided by natural gas distributed through a vast pipeline network covering the whole of Europe. At a cost of 1 million Euro per km extending the European pipeline network is already a multi-billion Euro business. Therefore, automatic planning tools that support the decision process are desired. Unfortunately, current mathematical methods are not capable of solving the arising network design problems due to their size and complexity. In this article, we will show how to apply optimization methods that can converge to a proven global optimal solution. By introducing a new class of valid inequalities that improve the relaxation of our mixed-integer nonlinear programming model, we are able to speed up the necessary computations substantially.}, language = {en} } @article{HumpolaJoormannOucherifetal.2015, author = {Humpola, Jesco and Joormann, Imke and Oucherif, Djamal and Pfetsch, Marc and Schewe, Lars and Schmidt, Martin and Schwarz, Robert}, title = {GasLib - A Library of Gas Network Instances}, journal = {Optimization Online}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-57950}, year = {2015}, abstract = {The development of mathematical simulation and optimization models and algorithms for solving gas transport problems is an active field of research. In order to test and compare these models and algorithms, gas network instances together with demand data are needed. The goal of GasLib is to provide a set of publicly available gas network instances that can be used by researchers in the field of gas transport. The advantages are that researchers save time by using these instances and that different models and algorithms can be compared on the same specified test sets. The library instances are encoded in an XML format. In this paper, we explain this format and present the instances that are available in the library.}, language = {en} } @article{DomschkeGrossHanteetal.2015, author = {Domschke, Pia and Groß, Martin and Hante, Falk M. and Hiller, Benjamin and Schewe, Lars and Schmidt, Martin}, title = {Mathematische Modellierung, Simulation und Optimierung von Gastransportnetzwerken}, volume = {156}, journal = {gwf - Gas+Energie}, number = {11}, pages = {880 -- 885}, year = {2015}, language = {de} } @incollection{KochPfetschRoevekamp2015, author = {Koch, Thorsten and Pfetsch, Marc and R{\"o}vekamp, Jessica}, title = {Introduction}, booktitle = {Evaluating Gas Network Capacities}, publisher = {Society for Industrial and Applied Mathematics}, isbn = {9781611973686}, pages = {3 -- 16}, year = {2015}, language = {en} } @inproceedings{MartinGeisslerHeynetal.2011, author = {Martin, Alexander and Geißler, Bj{\"o}rn and Heyn, Christine and Hiller, Benjamin and Humpola, Jesco and Koch, Thorsten and Lehmann, Thomas and Morsi, Antonio and Pfetsch, Marc and Schewe, Lars and Schmidt, Martin and Schultz, R{\"u}diger and Schwarz, Robert and Schweiger, Jonas and Steinbach, Marc and Willert, Bernhard}, title = {Optimierung Technischer Kapazit{\"a}ten in Gasnetzen}, booktitle = {Optimierung in der Energiewirtschaft}, publisher = {VDI-Verlag, D{\"u}sseldorf}, pages = {105 -- 114}, year = {2011}, language = {en} } @misc{HillerSaitenmacherWalther2016, author = {Hiller, Benjamin and Saitenmacher, Ren{\´e} and Walther, Tom}, title = {Analysis of operating modes of complex compressor stations}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-61092}, year = {2016}, abstract = {We consider the modeling of operation modes for complex compressor stations (i.e., ones with several in- or outlets) in gas networks. In particular, we propose a refined model that allows to precompute tighter relaxations for each operation mode. These relaxations may be used to strengthen the compressor station submodels in gas network optimization problems. We provide a procedure to obtain the refined model from the input data for the original model.}, language = {en} } @article{GamrathKochMaheretal.2017, author = {Gamrath, Gerald and Koch, Thorsten and Maher, Stephen J. and Rehfeldt, Daniel and Shinano, Yuji}, title = {SCIP-Jack - A solver for STP and variants with parallelization extensions}, volume = {9}, journal = {Mathematical Programming Computation}, number = {2}, doi = {10.1007/s12532-016-0114-x}, pages = {231 -- 296}, year = {2017}, abstract = {The Steiner tree problem in graphs is a classical problem that commonly arises in practical applications as one of many variants. While often a strong relationship between different Steiner tree problem variants can be observed, solution approaches employed so far have been prevalently problem-specific. In contrast, this paper introduces a general-purpose solver that can be used to solve both the classical Steiner tree problem and many of its variants without modification. This versatility is achieved by transforming various problem variants into a general form and solving them by using a state-of-the-art MIP-framework. The result is a high-performance solver that can be employed in massively parallel environments and is capable of solving previously unsolved instances.}, language = {en} } @misc{HillerWalther2018, author = {Hiller, Benjamin and Walther, Tom}, title = {Improving branching for disjunctive polyhedral models using approximate convex decompositions}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-67462}, year = {2018}, abstract = {Disjunctive sets arise in a variety of optimization models and much esearch has been devoted to obtain strong relaxations for them. This paper focuses on the evaluation of the relaxation during the branch-and-bound search process. We argue that the branching possibilities (\ie binary variables) of the usual formulations are unsuitable to obtain strong bounds early in the search process as they do not capture the overall shape of the the entire disjunctive set. To analyze and exploit the shape of the disjunctive set we propose to compute a hierarchy of approximate convex decompositions and show how to extend the known formulations to obtain improved branching behavior.}, language = {en} } @article{GriewankHasenfelderRadonsetal.2018, author = {Griewank, Andreas and Hasenfelder, Richard and Radons, Manuel and Lehmann, Lutz and Streubel, Tom}, title = {Integrating Lipschitzian dynamical systems using piecewise algorithmic differentiation}, volume = {33}, journal = {Optimization Methods and Software}, publisher = {Taylor \& Francis}, doi = {10.1080/10556788.2017.1378653}, pages = {1089 -- 1107}, year = {2018}, abstract = {In this article we analyse a generalized trapezoidal rule for initial value problems with piecewise smooth right-hand side F : IR^n -> IR^n based on a generalization of algorithmic differentiation. When applied to such a problem, the classical trapezoidal rule suffers from a loss of accuracy if the solution trajectory intersects a nondifferentiability of F. The advantage of the proposed generalized trapezoidal rule is threefold: Firstly, we can achieve a higher convergence order than with the classical method. Moreover, the method is energy preserving for piecewise linear Hamiltonian systems. Finally, in analogy to the classical case we derive a third-order interpolation polynomial for the numerical trajectory. In the smooth case, the generalized rule reduces to the classical one. Hence, it is a proper extension of the classical theory. An error estimator is given and numerical results are presented.}, language = {en} } @article{GriewankStreubelLehmannetal.2018, author = {Griewank, Andreas and Streubel, Tom and Lehmann, Lutz and Radons, Manuel and Hasenfelder, Richard}, title = {Piecewise linear secant approximation via algorithmic piecewise differentiation}, volume = {33}, journal = {Optimization Methods and Software}, number = {4-6}, publisher = {Taylor \& Francis}, doi = {10.1080/10556788.2017.1387256}, pages = {1108 -- 1126}, year = {2018}, abstract = {It is shown how piecewise differentiable functions F : IR^n -> IR^m that are defined by evaluation programmes can be approximated locally by a piecewise linear model based on a pair of sample points \check x and \hat x. We show that the discrepancy between function and model at any point x is of the bilinear order O(||x - \check x||*||x - \hat x||). As an application of the piecewise linearization procedure we devise a generalized Newton's method based on successive piecewise linearization and prove for it sufficient conditions for convergence and convergence rates equalling those of semismooth Newton. We conclude with the derivation of formulas for the numerically stable implementation of the aforedeveloped piecewise linearization methods.}, language = {en} } @article{SchweigerLiers2018, author = {Schweiger, Jonas and Liers, Frauke}, title = {A Decomposition Approach for Optimal Gas Network Extension with a Finite Set of Demand Scenarios}, volume = {19}, journal = {Optimization and Engineering}, number = {2}, publisher = {Springer}, pages = {297 -- 326}, year = {2018}, abstract = {Today's gas markets demand more flexibility from the network operators which in turn have to invest into their network infrastructure. As these investments are very cost-intensive and long-living, network extensions should not only focus on a single bottleneck scenario, but should increase the flexibility to fulfill different demand scenarios. In this work, we formulate a model for the network extension problem for multiple demand scenarios and propose a scenario decomposition in order to solve the arising challenging optimization tasks. In fact, each subproblem consists of a mixed-integer nonlinear optimization problem (MINLP). Valid bounds on the objective value are derived even without solving the subproblems to optimality. Furthermore, we develop heuristics that prove capable of improving the initial solutions substantially. Results of computational experiments on realistic network topologies are presented. It turns out that our method is able to solve these challenging instances to optimality within a reasonable amount of time.}, language = {en} }