TY - GEN A1 - Fügenschuh, Armin A1 - Humpola, Jesco T1 - A Unified View on Relaxations for a Nonlinear Network Flow Problem N2 - We consider a nonlinear nonconvex network flow problem that arises, for example, in natural gas or water transmission networks. Given is such network with active and passive components, that is, valves, compressors, pressure regulators (active) and pipelines (passive), and a desired amount of flow at certain specified entry and exit nodes of the network. Besides flow conservation constraints in the nodes the flow must fulfill nonlinear nonconvex pressure loss constraints on the arcs subject to potential values (i.e., pressure levels) in both end nodes of each arc. The problem is how to numerically compute this flow and pressures. We review an existing approach of Maugis (1977) and extend it to the case of networks with active elements (for example, compressors). We further examine different ways of relaxations for the nonlinear network flow model. We compare different approaches based on nonlinear optimization numerically on a set of test instances. T3 - ZIB-Report - 13-31 KW - Nonlinear Network Flow KW - Mixed-Integer Nonlinear Programming KW - Relaxations Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-18857 ER - TY - GEN A1 - Humpola, Jesco A1 - Fügenschuh, Armin T1 - A New Class of Valid Inequalities for Nonlinear Network Design Problems N2 - We consider a nonlinear nonconvex network design problem that arises in the extension of natural gas transmission networks. Given is such network with active and passive components, that is, valves, compressors, pressure regulators (active) and pipelines (passive), and a desired amount of flow at certain specified entry and exit nodes of the network. Besides flow conservation constraints in the nodes the flow must fulfill nonlinear nonconvex pressure loss constraints on the arcs subject to potential values (i.e., pressure levels) in both end nodes of each arc. Assume that there does not exist a feasible flow that fulfills all physical constraints and meets the desired entry and exit amounts. Then a natural question is where to extend the network by adding pipes in the most economic way such that this flow becomes feasible. Answering this question is computationally demanding because of the difficult problem structure. We use mixed-integer nonlinear programming techniques that rely on an outer approximation of the overall problem, and a branching on decision variables. We formulate a new class of valid inequalities (or cutting planes) which reduce the overall solution time when added to the formulation. We demonstrate the computational merits of our approach on test instances. T3 - ZIB-Report - 13-06 KW - Network Design KW - Mixed-Integer Nonlinear Programming KW - Cutting Planes Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-17771 SN - 1438-0064 ER - TY - GEN A1 - Fügenschuh, Armin A1 - Geißler, Björn A1 - Gollmer, Ralf A1 - Hayn, Christine A1 - Henrion, Rene A1 - Hiller, Benjamin A1 - Humpola, Jesco A1 - Koch, Thorsten A1 - Lehmann, Thomas A1 - Martin, Alexander A1 - Mirkov, Radoslava A1 - Morsi, Antonio A1 - Römisch, Werner A1 - Rövekamp, Jessica A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Schultz, Rüdiger A1 - Schwarz, Robert A1 - Schweiger, Jonas A1 - Stangl, Claudia A1 - Steinbach, Marc A1 - Willert, Bernhard T1 - Mathematical Optimization for Challenging Network Planning Problems in Unbundled Liberalized Gas Markets N2 - 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 where 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 show that these new paradigms lead to new and challenging mathematical optimization problems. In order to solve them and to provide meaningful results for practice, all aspects of the underlying problems, such as combinatorics, stochasticity, uncertainty, and nonlinearity, have to be addressed. With such special-tailored solvers, free network capacities and topological network extensions can, for instance, be determined. T3 - ZIB-Report - 13-13 KW - Gas Market Liberalization KW - Entry-Exit Model KW - Gas Network Access Regulation KW - Mixed-Integer Nonlinear Nonconvex Stochastic Optimization Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-17821 SN - 1438-0064 ER - TY - GEN A1 - Pfetsch, Marc A1 - Fügenschuh, Armin A1 - Geißler, Björn A1 - Geißler, Nina A1 - Gollmer, Ralf A1 - Hiller, Benjamin A1 - Humpola, Jesco A1 - Koch, Thorsten A1 - Lehmann, Thomas A1 - Martin, Alexander A1 - Morsi, Antonio A1 - Rövekamp, Jessica A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Schultz, Rüdiger A1 - Schwarz, Robert A1 - Schweiger, Jonas A1 - Stangl, Claudia A1 - Steinbach, Marc A1 - Vigerske, Stefan A1 - Willert, Bernhard T1 - Validation of Nominations in Gas Network Optimization: Models, Methods, and Solutions N2 - In this article we investigate methods to solve a fundamental task in gas transportation, namely the validation of nomination problem: Given a gas transmission network consisting of passive pipelines and active, controllable elements and given an amount of gas at every entry and exit point of the network, find operational settings for all active elements such that there exists a network state meeting all physical, technical, and legal constraints. We describe a two-stage approach to solve the resulting complex and numerically difficult mixed-integer non-convex nonlinear feasibility problem. The first phase consists of four distinct algorithms facilitating mixed-integer linear, mixed-integer nonlinear, reduced nonlinear, and complementarity constrained methods to compute possible settings for the discrete decisions. The second phase employs a precise continuous nonlinear programming model of the gas network. Using this setup, we are able to compute high quality solutions to real-world industrial instances whose size is significantly larger than networks that have appeared in the literature previously. T3 - ZIB-Report - 12-41 Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-16531 SN - 1438-0064 ER - TY - GEN A1 - Maher, Stephen J. A1 - Miltenberger, Matthias A1 - Pedroso, João Pedro A1 - Rehfeldt, Daniel A1 - Schwarz, Robert A1 - Serrano, Felipe T1 - PySCIPOpt: Mathematical Programming in Python with the SCIP Optimization Suite N2 - SCIP is a solver for a wide variety of mathematical optimization problems. It is written in C and extendable due to its plug-in based design. However, dealing with all C specifics when extending SCIP can be detrimental to development and testing of new ideas. This paper attempts to provide a remedy by introducing PySCIPOpt, a Python interface to SCIP that enables users to write new SCIP code entirely in Python. We demonstrate how to intuitively model mixed-integer linear and quadratic optimization problems and moreover provide examples on how new Python plug-ins can be added to SCIP. T3 - ZIB-Report - 16-64 KW - SCIP, Mathematical optimization, Python, Modeling Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-61348 SN - 1438-0064 ER - TY - GEN A1 - Gamrath, Gerald A1 - Koch, Thorsten A1 - Maher, Stephen J. A1 - Rehfeldt, Daniel A1 - Shinano, Yuji T1 - SCIP-Jack – A solver for STP and variants with parallelization extensions N2 - 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. T3 - ZIB-Report - 16-41 KW - Steiner tree problem KW - SCIP-Jack KW - Steiner tree variants KW - maximum-weight connected subgraph KW - prize-collecting Steiner tree Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-60170 SN - 1438-0064 ER - TY - JOUR A1 - Humpola, Jesco A1 - Serrano, Felipe T1 - Sufficient pruning conditions for MINLP in gas network design JF - EURO Journal on Computational Optimization N2 - 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 Euros per kilometer the extension of the European pipeline network is already a multi-billion Euro business. Therefore, automatic planning tools that support the decision process are desired. We model the topology optimization problem in gas networks by a mixed-integer nonlinear program (MINLP). This gives rise to a so-called active transmission problem, a continuous nonlinear non-convex feasibility problem which emerges from the MINLP model by fixing all integral variables. We offer novel sufficient conditions for proving the infeasibility of this active transmission problem. These conditions can be expressed in the form of a mixed-integer program (MILP), i.e., the infeasibility of a non-convex continuous nonlinear program (NLP) can be certified by solving an MILP. This result provides an efficient pruning procedure in a branch-and-bound algorithm. Our computational results demonstrate a substantial speedup for the necessary computations. Y1 - 2017 U6 - https://doi.org/10.1007/s13675-016-0077-8 VL - 5 IS - 1-2 SP - 239 EP - 261 PB - Springer Berlin Heidelberg ER - TY - CHAP A1 - Gotzes, Uwe A1 - Heinecke, Nina A1 - Hiller, Benjamin A1 - Rövekamp, Jessica A1 - Koch, Thorsten T1 - Regulatory rules for gas markets in Germany and other European countries T2 - Evaluating gas network capacities Y1 - 2015 SN - 978-1-611973-68-6 SP - 45 EP - 64 PB - Society for Industrial and Applied Mathematics ER - TY - GEN A1 - Walther, Tom A1 - Hiller, Benjamin A1 - Saitenmacher, René T1 - Polyhedral 3D Models for compressors in gas networks N2 - Compressor machines are crucial elements in a gas transmission network, required to compensate for the pressure loss caused by friction in the pipes. Modelling all physical and technical details of a compressor machine involves a large amount of nonlinearity, which makes it hard to use such models in the optimization of large-scale gas networks. In this paper, we are going to describe a modelling approach for the operating range of a compressor machine, starting from a physical reference model and resulting in a polyhedral representation in the 3D space of mass flow throughput as well as in- and outlet pressure. T3 - ZIB-Report - 17-66 KW - gas compressor modeling, polyhedral model, accuracy, z factor Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-65864 SN - 1438-0064 ER - TY - THES A1 - Schweiger, Jonas T1 - Exploiting structure in non-convex quadratic optimization and gas network planning under uncertainty N2 - 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 and uncertainty in the data pose principal difficulties. The first part of the thesis deals with non-convex quadratic programs. Branch&Bound methods for this problem class depend on tight relaxations. We contribute in several ways: First, we establish a new way to handle missing linearization variables in the well-known Reformulation-Linearization-Technique (RLT). This is implemented into the commercial software CPLEX. Second, we study the optimization of a quadratic objective over the standard simplex or a knapsack constraint. These basic structures appear as part of many complex models. Exploiting connections to the maximum clique problem and RLT, we derive new valid inequalities. Using exact and heuristic separation methods, we demonstrate the impact of the new inequalities on the relaxation and the global optimization of these problems. Third, we strengthen the state-of-the-art relaxation for the pooling problem, a well-known non-convex quadratic problem, which is, for example, relevant in the petrochemical industry. We propose a novel relaxation that captures the essential non-convex structure of the problem but is small enough for an in-depth study. We provide a complete inner description in terms of the extreme points as well as an outer description in terms of inequalities defining its convex hull (which is not a polyhedron). We show that the resulting valid convex inequalities significantly strengthen the standard relaxation of the pooling problem. The second part of this thesis focuses on a common challenge in real world applications, namely, the uncertainty entailed in the input data. We study the extension of a gas transport network, e.g., from our project partner Open Grid Europe GmbH. For a single scenario this maps to a challenging non-convex MINLP. As the future transport patterns are highly uncertain, we propose a robust model to best prepare the network operator for an array of scenarios. We develop a custom decomposition approach that makes use of the hierarchical structure of network extensions and the loose coupling between the scenarios. The algorithm used the single-scenario problem as black-box subproblem allowing the generalization of our approach to problems with the same structure. The scenario-expanded version of this problem is out of reach for today's general-purpose MINLP solvers. Yet our approach provides primal and dual bounds for instances with up to 256 scenarios and solves many of them to optimality. Extensive computational studies show the impact of our work. N2 - Der bemerkenswerte Erfolg der angewandten mathematischen Optimierung in den letzten Dekaden ist mehr auf Einsichten in mathematische Strukturen zurückzuführen, als auf eine Steigerung der Rechenleistung. In diesem Sinne adressieren wir Anwendungen, in denen Nichtkonvexität und Unsicherheit in den Daten die Hauptschwierigkeiten darstellen. Der erste Teil dieser Arbeit beschäftigt sich mit nichtkonvexen quadratischen Optimierungsproblemen. Relaxierungen sind integraler Bestandteil von \BranchAndBound{}-Lösungsmethoden für diese Problemkategorie. Wir leisten folgende Beiträge: Erstens beschreiben wir eine neue Art fehlende Linearisierungsvariablen, in der so genannten Reformulation-Linearization-Technique (RLT), zu behandeln. Diese wird inzwischen in der kommerziellen Software CPLEX verwendet. Zweitens beschäftigen wir uns mit der Optimierung einer quadratischen Zielfunktion über die Standardsimplex oder einen so genannten Knapsack-Constraint. Solche grundlegenden Strukturen sind Teil vieler komplexer Modelle. Wir benutzen bekannte Verbindungen zum maximalen Cliquenproblem sowie zu RLT, um neue gültige Ungleichungen herzuleiten, die die Relaxierung verstärken. Drittens beschäftigen wir uns mit dem Pooling Problem, das z.B. in der Erdölindustrie relevant ist. Wie leiten eine neue Relaxierung her, die die wesentliche nicht-konvexe Struktur des Problems erfasst, aber klein genug für eine grundlegende Untersuchung ist. Wir geben eine innere Beschreibung in Form der Extrempunkte, sowie eine äußere Beschreibung in Form von Ungleichungen, die die konvexe Hülle (welche im Allgemeinen kein Polyeder ist) beschreiben, an. Wir zeigen, dass neuen die Ungleichungen die Relaxierung des Pooling Problems erheblich verstärken. Der zweite Teil der Arbeit befasst sich mit einer weiteren Herausforderung in realen Anwendungen, nämlich Unsicherheit in den Eingabedaten. Konkret untersuchen wir die Optimierung des Ausbaus eines Gastransportnetzes, wie z.B. von unserem Projektpartner Open Grid Europe GmbH. Dieses Problem ist bereits bei gegebenen Eingabedaten ein schweres nicht-konvexes gemischt-ganzzahliges Optimierungsproblem. Da zukünftige Nutzungsmuster des Netzes mit großer Unsicherheit behaftet sind, beschreiben wir ein robustes Modell, um den Netzbetreiber gegen verschiedene Szenarien abzusichern. Wir entwickeln einen speziellen Dekompositionsalgorithmus unter Berücksichtigung der hierarchischen Struktur der Ausbauten und der schwachen Kopplung zwischen den Szenarien. Unser Ansatz liefert primale und duale Schranken für Instanzen mit bis zu 256 Szenarien und löst viele beweisbar optimal. Umfangreiche Rechnungen bestätigen die Effizient der vorgestellten Methoden. KW - Nonconvexity KW - Uncertainty KW - Quadratic Programming KW - Relaxations KW - Cutting Planes KW - Standard Quadratic Programming KW - Pooling Problem KW - Gas Network Planning KW - Robust Optimization KW - Decomposition KW - Computations Y1 - 2017 UR - http://dx.doi.org/10.14279/depositonce-6015 ER - TY - JOUR A1 - Schmidt, Martin A1 - Assmann, Denis A1 - Burlacu, Robert A1 - Humpola, Jesco A1 - Joormann, Imke A1 - Kanelakis, Nikolaos A1 - Koch, Thorsten A1 - Oucherif, Djamal A1 - Pfetsch, Marc A1 - Schewe, Lars A1 - Schwarz, Robert A1 - Sirvent, Matthias T1 - GasLib – A Library of Gas Network Instances JF - Data Y1 - 2017 U6 - https://doi.org/10.3390/data2040040 VL - 2 IS - 4 ER - TY - JOUR A1 - Humpola, Jesco A1 - Lehmann, Thomas A1 - Fügenschuh, Armin T1 - A primal heuristic for optimizing the topology of gas networks based on dual information JF - EURO Journal on Computational Optimization N2 - We present a novel heuristic 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 of a nonlinear relaxation. Based on the information from the KKT point we alter some of the binary variables in a locally promising way exploiting 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. Y1 - 2015 U6 - https://doi.org/10.1007/s13675-014-0029-0 VL - 3 IS - 1 SP - 53 EP - 78 ER - TY - GEN A1 - Hiller, Benjamin A1 - Walther, Tom T1 - Modelling compressor stations in gas networks N2 - Gas networks are an important application area for optimization. When considering long-range transmission, compressor stations play a crucial role in these applications. The purpose of this report is to collect and systematize the models used for compressor stations in the literature. The emphasis is on recent work on simple yet accurate polyhedral models that may replace more simplified traditional models without increasing model complexity. The report also describes an extension of the compressor station data available in GasLib (http://gaslib.zib.de/) with the parameters of these models. T3 - ZIB-Report - 17-67 KW - MINLP model KW - compressor station KW - gas compressor modeling KW - nonlinear model KW - polyhedral model Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-67443 SN - 1438-0064 ER -