TY - GEN A1 - Schweiger, Jonas T1 - Exploiting structure in non-convex quadratic optimization 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 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. T3 - ZIB-Report - 18-35 KW - Nonconvexity KW - Quadratic Programming KW - Relaxations KW - Cutting Planes KW - Standard Quadratic Programming KW - Pooling Problem Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-69476 SN - 1438-0064 ER - TY - GEN A1 - Luedtke, James A1 - D'Ambrosio, Claudia A1 - Linderoth, Jeff A1 - Schweiger, Jonas T1 - Strong Convex Nonlinear Relaxations of the Pooling Problem: Extreme Points N2 - 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. T3 - ZIB-Report - 18-13 KW - Pooling Problem KW - Relaxation KW - Extreme Points Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-67801 SN - 1438-0064 ER - TY - GEN A1 - Luedtke, James A1 - D'Ambrosio, Claudia A1 - Linderoth, Jeff A1 - Schweiger, Jonas T1 - Strong Convex Nonlinear Relaxations of the Pooling Problem N2 - 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. T3 - ZIB-Report - 18-12 KW - Pooling Problem KW - Relaxation KW - Valid Inequalities Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-67824 SN - 1438-0064 ER - TY - GEN A1 - Hiller, Benjamin A1 - Koch, Thorsten A1 - Schewe, Lars A1 - Schwarz, Robert A1 - Schweiger, Jonas T1 - A System to Evaluate Gas Network Capacities: Concepts and Implementation N2 - In 2005 the European Union liberalized the gas market with a disruptive change and decoupled trading of natural gas from its transport. The gas is now transported by independent so-called transmissions system operators or TSOs. The market model established by the European Union views the gas transmission network as a black box, providing shippers (gas traders and consumers) the opportunity to transport gas from any entry to any exit. TSOs are required to offer the maximum possible capacities at each entry and exit such that any resulting gas flow can be realized by the network. The revenue from selling these capacities more than one billion Euro in Germany alone, but overestimating the capacity might compromise the security of supply. Therefore, evaluating the available transport capacities is extremely important to the TSOs. This is a report on a large project in mathematical optimization, set out to develop a new toolset for evaluating gas network capacities. The goals and the challenges as they occurred in the project are described, as well as the developments and design decisions taken to meet the requirements. T3 - ZIB-Report - 18-11 Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-67438 SN - 1438-0064 N1 - An earlier version of this report is available as ZR 17-03 at https://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/6193. ER - TY - GEN A1 - Hiller, Benjamin A1 - Koch, Thorsten A1 - Schewe, Lars A1 - Schwarz, Robert A1 - Schweiger, Jonas T1 - A System to Evaluate Gas Network Capacities: Concepts and Implementation N2 - Since 2005, the gas market in the European Union is liberalized and the trading of natural gas is decoupled from its transport. The transport is done by so-called transmissions system operators or TSOs. The market model established by the European Union views the gas transmission network as a black box, providing shippers (gas traders and consumers) the opportunity to transport gas from any entry to any exit. TSOs are required to offer maximum independent capacities at each entry and exit such that the resulting gas flows can be realized by the network without compromising security of supply. Therefore, evaluating the available transport capacities is extremely important to the TSOs. This paper gives an overview of the toolset for evaluating gas network capacities that has been developed within the ForNe project, a joint research project of seven research partners initiated by Open Grid Europe, Germany's biggest TSO. While most of the relevant mathematics is described in the book "Evaluating Gas Network Capacities", this article sketches the system as a whole, describes some developments that have taken place recently, and gives some details about the current implementation. T3 - ZIB-Report - 17-03 KW - operations research in energy KW - gas network optimization KW - entry-exit model KW - freely allocable capacity KW - large-scale mixed-integer nonlinear programming Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-61931 SN - 1438-0064 N1 - A revised and extended version is available as ZIB-Report 18-11. ER - TY - GEN A1 - Schweiger, Jonas T1 - Gas network extension planning for multiple demand scenarios N2 - 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 one bottleneck scenario, but should increase the flexibility to fulfill different demand scenarios. We formulate a model for the network extension problem for multiple demand scenarios and propose a scenario decomposition. We solve MINLP single-scenario sub-problems and obtain valid bounds even without solving them to optimality. Heuristics prove capable of improving the initial solutions substantially. Results of computational experiments are presented. T3 - ZIB-Report - 14-29 KW - Gas Network Planning KW - Scenario Decomposition Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-51030 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 - Martin, Alexander A1 - Geißler, Björn A1 - Hayn, Christine A1 - Hiller, Benjamin A1 - Humpola, Jesco A1 - Koch, Thorsten A1 - Lehmann, Thomas A1 - Morsi, Antonio A1 - Pfetsch, Marc A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Schultz, Rüdiger A1 - Schwarz, Robert A1 - Schweiger, Jonas A1 - Steinbach, Marc A1 - Willert, Bernhard T1 - Optimierung Technischer Kapazitäten in Gasnetzen N2 - Die mittel- und längerfristige Planung für den Gastransport hat sich durch Änderungen in den regulatorischen Rahmenbedingungen stark verkompliziert. Kernpunkt ist die Trennung von Gashandel und -transport. Dieser Artikel diskutiert die hieraus resultierenden mathematischen Planungsprobleme, welche als Validierung von Nominierungen und Buchungen, Bestimmung der technischen Kapazität und Topologieplanung bezeichnet werden. Diese mathematischen Optimierungsprobleme werden vorgestellt und Lösungsansätze skizziert. T3 - ZIB-Report - 11-56 KW - Gasnetzplanung KW - Technische Kapazitäten KW - Nominierungsvalidierung KW - Buchungsvalidierung KW - Topologieplanung Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-15121 SN - 1438-0064 ER - TY - GEN A1 - Eisenblätter, Andreas A1 - Schweiger, Jonas T1 - Multistage Stochastic Programming in Strategic Telecommunication Network Planning N2 - Mobile communication is nowadays taken for granted. Having started primarily as a service for speech communication, data service and mobile Internet access are now driving the evolution of network infrastructure. Operators are facing the challenge to match the demand by continuously expanding and upgrading the network infrastructure. However, the evolution of the customer's demand is uncertain. We introduce a novel (long-term) network planning approach based on multistage stochastic programming, where demand evolution is considered as a stochastic process and the network is extended as to maximize the expected profit. The approach proves capable of designing large-scale realistic UMTS networks with a time-horizon of several years. Our mathematical optimization model, the solution approach, and computational results are presented in this paper. T3 - ZIB-Report - 11-06 KW - UMTS KW - Network Evolution KW - Multistage KW - Stochastic Programming Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-12232 ER - TY - GEN A1 - Fügenschuh, Armin A1 - Hiller, Benjamin A1 - Humpola, Jesco A1 - Koch, Thorsten A1 - Lehmann, Thomas A1 - Schwarz, Robert A1 - Schweiger, Jonas A1 - Szabó, Jácint T1 - Gas Network Topology Optimization for Upcoming Market Requirements N2 - 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. T3 - ZIB-Report - 11-09 KW - Mathematical Optimization KW - Gas Distribution Networks KW - Topology Planning Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-12348 ER -