@article{SchmidtSirventWollner2017, author = {Schmidt, Martin and Sirvent, Mathias and Wollner, Winnifried}, title = {A Decomposition Method for MINLPs with Lipschitz Continuous Nonlinearities}, series = {Mathematical Programming}, journal = {Mathematical Programming}, number = {178(1)}, pages = {449 -- 483}, year = {2017}, abstract = {Many mixed-integer optimization problems are constrained by nonlinear functions that do not possess desirable analytical properties like convexity or factorability or cannot even be evaluated exactly. This is, e.g., the case for problems constrained by differential equations or for models that rely on black-box simulation runs. For these problem classes, we present, analyze, and test algorithms that solve mixed-integer problems with only Lipschitz continuous nonlinearities. Our theoretical results depend on the assumptions made on the (in)exactness of function evaluations and on the knowledge of Lipschitz constants. If Lipschitz constants are known, we prove finite termination at approximate globally optimal points both for the case of exact and inexact function evaluations. If only approximate Lipschitz constants are known, we prove finite termination and derive additional conditions under which infeasibility can be detected. A computational study for gas transport problems and an academic case study show the applicability of our algorithms to real-world problems and how different assumptions on the constraint functions up- or downgrade the practical performance of the methods.}, language = {en} } @article{SchmidtAssmannBurlacuetal.2017, author = {Schmidt, Martin and Aßmann, Denis and Burlacu, Robert and Humpola, Jesco and Joormann, Imke and Kanelakis, Nikolaos and Koch, Thorsten and Oucherif, Djamal and Pfetsch, Marc E. and Schewe, Lars and Schwarz, Robert and Sirvent, Mathias}, title = {GasLib - A Library of Gas Network Instances}, series = {Data}, volume = {4}, journal = {Data}, number = {2}, doi = {10.3390/data2040040}, pages = {18}, year = {2017}, 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{LeugeringMartinSchmidtetal.2017, author = {Leugering, G{\"u}nter and Martin, Alexander and Schmidt, Martin and Sirvent, Mathias}, title = {Nonoverlapping Domain Decomposition for Optimal Control Problems governed by Semilinear Models for Gas Flow in Networks}, volume = {46}, number = {3}, publisher = {Control and Cybernetics}, pages = {191 -- 225}, year = {2017}, abstract = {We consider optimal control problems for gas flow in pipeline networks. The equations of motion are taken to be represented by a first-order system of hyperbolic semilinear equations derived from the fully nonlinear isothermal Euler gas equations. We formulate an optimal control problem on a network and introduce a tailored time discretization thereof. In order to further reduce the complexity, we consider an instantaneous control strategy. The main part of the paper is concerned with a nonoverlapping domain decomposition of the optimal control problem on the graph into local problems on smaller sub-graphs - ultimately on single edges. We prove convergence of the domain decomposition method on networks and study the wellposedness of the corresponding time-discrete optimal control problems. The point of the paper is that we establish virtual control problems on the decomposed subgraphs such that the corresponding optimality systems are in fact equal to the systems obtained via the domain decomposition of the entire optimality system.}, language = {en} } @article{GugatLeugeringMartinetal.2018, author = {Gugat, Martin and Leugering, G{\"u}nter and Martin, Alexander and Schmidt, Martin and Sirvent, Mathias and Wintergerst, David}, title = {Towards Simulation Based Mixed-Integer Optimization with Differential Equations}, series = {Networks}, journal = {Networks}, doi = {10.1002/net.21812}, pages = {24}, year = {2018}, abstract = {We propose a decomposition based method for solving mixed-integer nonlinear optimization problems with "black-box" nonlinearities, where the latter, e.g., may arise due to differential equations or expensive simulation runs. The method alternatingly solves a mixed-integer linear master problem and a separation problem for iteratively refining the mixed-integer linear relaxation of the nonlinear equalities. The latter yield nonconvex feasible sets for the optimization model but we have to restrict ourselves to convex and monotone constraint functions. Under these assumptions, we prove that our algorithm finitely terminates with an approximate feasible global optimal solution of the mixed integer nonlinear problem. Additionally, we show the applicability of our approach for three applications from optimal control with integer variables, from the field of pressurized flows in pipes with elastic walls, and from steady-state gas transport. For the latter we also present promising numerical results of our method applied to real-world instances that particularly show the effectiveness of our method for problems defined on networks.}, language = {en} } @article{SirventKanelakisGeissleretal.2017, author = {Sirvent, Mathias and Kanelakis, Nikolaos and Geißler, Bj{\"o}rn and Biskas, Pandelis}, title = {A Linearized Model for the Optimization of the Coupled Electricity and Natural Gas System}, series = {Journal of Modern Power Systems and Clean Energy}, volume = {5}, journal = {Journal of Modern Power Systems and Clean Energy}, number = {3}, doi = {10.1007/s40565-017-0275-2}, pages = {364 -- 374}, year = {2017}, abstract = {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.}, language = {en} } @article{GugatLeugeringMartinetal.2017, author = {Gugat, Martin and Leugering, G{\"u}nter and Martin, Alexander and Schmidt, Martin and Sirvent, Mathias and Wintergerst, David}, title = {MIP-Based Instantaneous Control of Mixed-Integer PDE-Constrained Gas Transport Problems}, series = {Computational Optimization and Applications}, volume = {70}, journal = {Computational Optimization and Applications}, number = {1}, doi = {10.1007/s10589-017-9970-1}, pages = {267 -- 294}, year = {2017}, abstract = {We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be well-posed and affine-linear solutions of these systems are derived analytically. As a consequence, finite-dimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using general-purpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network.}, language = {en} } @article{BurlacuEggerGrossetal.2019, author = {Burlacu, Robert and Egger, Herbert and Groß, Martin and Martin, Alexander and Pfetsch, Marc and Schewe, Lars and Sirvent, Mathias and Skutella, Martin}, title = {Maximizing the storage capacity of gas networks: a global MINLP approach}, volume = {20}, edition = {Optimization and Engineering}, doi = {10.1007/s11081-018-9414-5}, pages = {543 -- 573}, year = {2019}, abstract = {In this paper, we study the transient optimization of gas networks, focusing in particular on maximizing the storage capacity of the network. We include nonlinear gas physics and active elements such as valves and compressors, which due to their switching lead to discrete decisions. The former is described by a model derived from the Euler equations that is given by a coupled system of nonlinear parabolic partial differential equations (PDEs). We tackle the resulting mathematical optimization problem by a first-discretize-then-optimize approach. To this end, we introduce a new discretization of the underlying system of parabolic PDEs and prove well-posedness for the resulting nonlinear discretized system. Endowed with this discretization, we model the problem of maximizing the storage capacity as a non-convex mixed-integer nonlinear problem (MINLP). For the numerical solution of the MINLP, we algorithmically extend a well-known relaxation approach that has already been used very successfully in the field of stationary gas network optimization. This method allows us to solve the problem to global optimality by iteratively solving a series of mixed-integer problems (MIPs). Finally, we present two case studies that illustrate the applicability of our approach.}, language = {en} } @article{SchmidtSirventWollner2018, author = {Schmidt, Martin and Sirvent, Mathias and Wollner, Winnifried}, title = {The Cost of Not Knowing Enough: Mixed-Integer Optimization with Implicit Lipschitz Nonlinearities}, series = {Optimization Letters}, journal = {Optimization Letters}, pages = {16}, year = {2018}, abstract = {It is folklore knowledge that nonconvex mixed-integer nonlinear optimization problems can be notoriously hard to solve in practice. In this paper we go one step further and drop analytical properties that are usually taken for granted in mixed-integer nonlinear optimization. First, we only assume Lipschitz continuity of the nonlinear functions and additionally consider multivariate implicit constraint functions that cannot be solved for any parameter analytically. For this class of mixed-integer problems we propose a novel algorithm based on an approximation of the feasible set in the domain of the nonlinear function---in contrast to an approximation of the graph of the function considered in prior work. This method is shown to compute approximate global optimal solutions in finite time and we also provide a worst-case iteration bound. In some first numerical experiments we show that the ``cost of not knowing enough'' is rather high by comparing our approach with the open-source global solver SCIP. This reveals that a lot of work is still to be done for this highly challenging class of problems and we thus finally propose some possible directions of future research.}, language = {en} }