TY - INPR A1 - Burlacu, Robert A1 - Geißler, Björn A1 - Schewe, Lars T1 - Solving Mixed-Integer Nonlinear Programs using Adaptively Refined Mixed-Integer Linear Programs N2 - We propose a method for solving mixed-integer nonlinear programs (MINLPs) to global optimality by discretization of occuring nonlinearities. The main idea is based on using piecewise linear functions to construct mixed-integer linear program (MIP) relaxations of the underlying MINLP. In order to find a global optimum of the given MINLP we develope an iterative algorithm which solves MIP relaxations that are adaptively refined. We are able to give convergence results for a wide range of MINLPs requiring only continuous nonlinearities with bounded domains and an oracle computing maxima of the nonlinearities on their domain. Moreover, the practicalness of our approach is shown numerically by an application from the field of gas network optimization. KW - Mixed-Integer Nonlinear Programming KW - Piecewise Linearization Y1 - 2017 ER - TY - JOUR A1 - Geißler, Björn A1 - Morsi, Antonio A1 - Schewe, Lars A1 - Schmidt, Martin T1 - Solving Highly Detailed Gas Transport MINLPs: Block Separability and Penalty Alternating Direction Methods JF - INFORMS Journal on Computing N2 - Detailed modeling of gas transport problems leads to nonlinear and nonconvex mixed-integer optimization or feasibility models (MINLPs) because both the incorporation of discrete controls of the network as well as accurate physical and technical modeling is required in order to achieve practical solutions. Hence, ignoring certain parts of the physics model is not valid for practice. In the present contribution we extend an approach based on linear relaxations of the underlying nonlinearities by tailored model reformulation techniques yielding block-separable MINLPs. This combination of techniques allows us to apply a penalty alternating direction method and thus to solve highly detailed MINLPs for large-scale real-world instances. The practical strength of the proposed method is demonstrated by a computational study in which we apply the method to instances from steady-state gas transport including both pooling effects with respect to the mixing of gases of different composition and a highly detailed compressor station model. Y1 - 2016 IS - 30(2) SP - 309 EP - 323 ER - TY - JOUR A1 - Geißler, Björn A1 - Morsi, Antonio A1 - Schewe, Lars A1 - Schmidt, Martin T1 - Penalty Alternating Direction Methods for Mixed-Integer Optimization: A New View on Feasibility Pumps JF - SIAM Journal on Optimization N2 - Feasibility pumps are highly effective primal heuristics for mixed-integer linear and nonlinear optimization. However, despite their success in practice there are only few works considering their theoretical properties. We show that feasibility pumps can be seen as alternating direction methods applied to special reformulations of the original problem, inheriting the convergence theory of these methods. Moreover, we propose a novel penalty framework that encompasses this alternating direction method, which allows us to refrain from random perturbations that are applied in standard versions of feasibility pumps in case of failure. We present a convergence theory for the new penalty based alternating direction method and compare the new variant of the feasibility pump with existing versions in an extensive numerical study for mixed-integer linear and nonlinear problems. Y1 - 2017 U6 - https://doi.org/10.1137/16M1069687 VL - 27 IS - 3 SP - 1611 EP - 1636 ER - TY - JOUR A1 - Sirvent, Mathias A1 - Kanelakis, Nikolaos A1 - Geißler, Björn A1 - Biskas, Pandelis T1 - A Linearized Model for the Optimization of the Coupled Electricity and Natural Gas System JF - Journal of Modern Power Systems and Clean Energy N2 - In the following paper a combined optimization of a coupled electricity and gas system is presented. For the electricity network a unit commitment problem with optimization of energy and reserves under a power pool, considering all system operational and unit technical constraints is solved. The gas network subproblem is a medium-scale mixed-integer nonconvex and nonlinear programming problem. The coupling constraints between the two networks are nonlinear as well. The resulting mixed-integer nonlinear program is linearized with the extended incremental method and an outer approximation technique. The resulting model is evaluated using the Greek power and gas system comprising fourteen gas-fired units under four different approximation accuracy levels. The results indicate the efficiency of the proposed MIP model and the interplay between computational requirements and accuracy. KW - Electricity System KW - Natural Gas System KW - Mixed-Integer (Non)Linear Programming KW - Extended Incremental Method KW - Outer Approximation Y1 - 2017 U6 - https://doi.org/10.1007/s40565-017-0275-2 VL - 5 IS - 3 SP - 364 EP - 374 ER - TY - JOUR A1 - Geißler, Björn A1 - Morsi, Antonio A1 - Schewe, Lars A1 - Schmidt, Martin T1 - Solving Power-Constrained Gas Transportation Problems using an MIP-based Alternating Direction Method JF - Computers & Chemical Engineering N2 - We present a solution algorithm for problems from steady-state gas transport optimization. Due to nonlinear and nonconvex physics and engineering models as well as discrete controllability of active network devices, these problems lead to difficult nonconvex mixed-integer nonlinear optimization models. The proposed method is based on mixed-integer linear techniques using piecewise linear relaxations of the nonlinearities and a tailored alternating direction method. Most other publications in the field of gas transport optimization only consider pressure and flow as main physical quantities. In this work, we additionally incorporate heat power supplies and demands as well as a mixing model for different gas qualities. We demonstrate the capabilities of our method on Germany's largest transport networks and hereby present numerical results on the largest instances that were ever reported in the literature for this problem class. Y1 - 2016 U6 - https://doi.org/10.1016/j.compchemeng.2015.07.005 VL - 82 IS - 2 SP - 303 EP - 317 ER - TY - INPR A1 - Schmidt, Martin A1 - Hiller, Benjamin A1 - Koch, Thorsten A1 - Pfetsch, Marc A1 - Geißler, Björn A1 - Henrion, René A1 - Joormann, Imke A1 - Martin, Alexander A1 - Morsi, Antonio A1 - Römisch, Werner A1 - Schewe, Lars A1 - Schultz, Rüdiger A1 - Steinbach, Marc C. T1 - Capacity Evaluation for Large-Scale Gas Networks N2 - Natural gas is important for the energy turnaround in many countries like in Germany, where it serves as a "bridging energy" towards a fossil-free energy supply in the future. About 20% of the total German energy demand is provided by natural gas, which is transported through a complex pipeline network with a total length of about 30000 km and the efficient use of the given transport infrastructure for natural gas is of political, economic, and societal importance. As a consequence of the liberalization of the European gas market in the last decades, gas trading and transport have been decoupled. This has led to new challenges for gas transport companies, and mathematical optimization is perfectly suited for tackling many of these challenges. However, the underlying mathematical problems are by far too hard to be solved by today's general-purpose software so that novel mathematical theory and algorithms are needed. The industrial research project "ForNe: Research Cooperation Network Optimization" has been initiated and funded by Open Grid Europe in 2009 and brought together experts in mathematical optimization from seven German universities and research institutes, which cover almost the entire range of mathematical optimization: integer and nonlinear optimization as well as optimization under uncertainty. The mathematical research results have been put together in a software package that has been delivered to Open Grid Europe at the end of the project. Moreover, the research is still continuing - e.g., in the Collaborative Research Center/Transregio 154 "Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks" funded by the German Research Foundation. Y1 - 2019 ER -