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Solving Mixed-Integer Linear and Nonlinear Network Optimization Problems by Local Reformulations and Relaxations

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Solving Mixed-Integer Linear and Nonlinear Network Optimization Problems by Local Reformulations and Relaxations Solving Mixed-Integer Linear and Nonlinear Network Optimization Problems by Local Reformulations and Relaxations Lösungsmethoden für gemischt-ganzzahlige lineare und nichtlineare Netzwerkoptimierungsprobleme basierend auf lokalen Reformulierungen und Relaxierungen Der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg zur Erlangung des Doktorgrades Dr. rer. nat. vorgelegt von Maximilian Merkert aus Kaiserslautern Als Dissertation genehmigt von der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg Tag der mündlichen Prüfung: 02.11.2017 Vorsitzender des Promotionsorgans: Prof. Dr. Georg Kreimer Gutachterin: Prof. Dr. Frauke Liers Gutachter: Prof. Dr. Rüdiger Schultz Gutachter: Prof. Dr. Christoph Helmberg Acknowledgements First of all, I would like to thank my supervisors Frauke Liers and Alexander Martin. They enabled me to work in a fruitful environment on a number of interesting projects with various academic and industrial partners. I also wish to express my deepest gratitude for their guidance not only on mathematical issues but also on more general topics regarding life in academia. I am also grateful to Johannes Jahn, Rüdiger Schultz and Christoph Helmberg for agreeing to be involved in the examination process. During my time in Erlangen, I have experienced that mathematical research is team work to a great extend. Therefore, I wish to thank my coauthors Andreas Bärmann, Thorsten Gellermann, Frauke Liers, Alexander Martin, Nick Mertens, Dennis Micha- els, Oskar Schneider, Christoph Thurner, Robert Weismantel and Dieter Weninger for the pleasant and productive collaborations. Moreover, many thanks to all my collea- gues at FAU Erlangen-Nürnberg and from the Collaborative Research Center TRR 154 for many valuable discussions as well as the supportive atmosphere and many social events. It was a pleasure to work with you! Furthermore, I want to thank Nina Gunkelmann, Andreas Bärmann, Lena Hupp, Nick Mertens, Dennis Michaels and Dieter Weninger for proof-reading parts of this thesis and helpful remarks. I gratefully acknowledge the computing resources provided by the group of Mi- chael Jünger and the technical support by Thomas Lange in Cologne as well as by Denis Aßmann and Thorsten Gellermann in Erlangen. Further thanks go to Christina Weber, Beate Kirchner and Gabriele Bittner for administrative aid. Last but not least, my special thanks go to my parents for all their support ever since I can remember, and to my partner for her great continual believe in me. 3 Abstract Since the beginnings of network optimization, the number of use cases has grown enormously and can be expected to further expand in an increasingly interconnected world. The wide range of modern applications include optimization tasks on energy networks, telecommunication networks and in public transport, just to name a few. Although many traditional network optimization problems are NP-hard in their ba- sic version, applications pose additional challenges due to more complicated—often nonlinear—dependencies or the sheer size of the network. In this thesis, we develop methods that help to cope with those challenges. A com- mon strategy will be to improve mathematical programming formulations locally by modeling substructures in an integrated way. The resulting reformulations and relax- ations will allow for global methods that either solve the problem to exact optimality or up to a predefined precision. For large-scale network expansion problems, a solution method is proposed that is based on iterative aggregation. Starting with an initial aggregation, we solve a se- quence of network design problems over increasingly fine-grained representations of the original network. This is done until the whole network is represented sufficiently well in the sense that an optimal solution to the aggregated problem can easily be ex- tended to an optimal solution of the original problem. Global optimality is guaranteed by a subproblem that computationally is less expensive and either proves optimality or gives an indication of where to refine the representation. In this algorithmic scheme, locally relaxing the problem allows us to focus on the critical part of the network. In many optimization problems on transportation networks—especially those ari- sing from energy applications—the main challenge is connected to the problem’s non- linear features, arising, for example, from laws of physics. Gas networks represent a typical example for such a nonlinear network flow setting that we will repeatedly re- fer to throughout this work. A common and established solution approach consists of constructing a piecewise linear approximation or relaxation. We study how to streng- then the resulting mixed-integer programming formulation for specific substructures in the network. We find effective cutting planes and derive a complete description for induced paths of arbitrary length—using graph-theoretic arguments related to perfect graphs. A generalization of key properties of this special case leads to an abstract defini- tion in terms of clique problems on a specific type of graph. This abstract setting also comprises a basic version of the project scheduling problem and still allows us to give totally unimodular reformulations that are of linear size. Moreover, questions regar- ding recognizability of this structure will be discussed. We also discuss the concept of simultaneous convexification that can be seen as a continuous counterpart to our approach for piecewise linearized problems. The re- sulting reformulations can improve relaxations employed by general-purpose MINLP solvers, which usually rely on convexifying nonlinear functions separately. Computational results demonstrate the practical impact of the methods developed in this thesis, in many cases using real-world data sets. 5 Zusammenfassung Seit den Anfängen der Netzwerkoptimierung ist die Zahl der Anwendungsfälle im- mens gewachsen und angesichts einer zunehmend vernetzten Welt ist ein weiterer Anstieg zu erwarten. Die Spannbreite moderner Anwendungen umfasst Optimie- rungsprobleme auf Energienetzen, Telekommunikationsnetzen und Verkehrsnetzen, um nur einige zu nennen. Auch wenn viele traditionelle Netzwerkoptimierungspro- bleme bereits in ihrer Grundversion NP-schwer sind, stellen Anwendungen weitere Anforderungen aufgrund komplexerer - oftmals nichtlinearer - Abhängigkeiten oder der schieren Größe der zugrunde liegenden Netzwerke. In dieser Arbeit werden Methoden entwickelt, um mit diesen Herausforderungen umzugehen. Die wesentliche Strategie wird darin bestehen, mathematische Problem- formulierungen lokal zu verstärken, indem ausgewählte Substrukturen als Ganzes er- fasst und modelliert werden. Die resultierenden Reformulierungen und Relaxierun- gen unterstützen globale Methoden, die entweder exakte oder bis auf eine vordefi- nierte Genauigkeit optimale Lösungen finden. Für große Netzausbauprobleme wird eine Lösungsmethodik basierend auf iterati- ver Aggregation entworfen. Beginnend mit einer Startaggregation lösen wir eine Folge zunehmend detaillierter Vergröberungen des ursprünglichen Netzwerks, bis die Dar- stellung hinreichend genau ist, sodass seine Optimallösung leicht auf das ursprüngli- che Problem übertragen werden kann. Exaktheit wird dabei durch ein Subproblem si- chergestellt, das entweder Optimalität bestätigt oder Ansatzpunkte zur Verfeinerung der Darstellung liefert. In diesem Schema erlaubt somit eine lokale Relaxierung die Fokussierung auf kritische Teile des Netzwerks. In vielen Optimierungsproblemen auf Transportnetzen, insbesondere für Energie- träger, besteht die wesentliche Herausforderung in den auftretenden Nichtlinearitä- ten, die beispielsweise physikalischen Gesetzen geschuldet sind. Gasnetwerke sind hierfür ein typisches Beispiel, auf das wir uns mehrfach in dieser Arbeit beziehen werden. Ein etablierter Lösungsansatz besteht in der Konstruktion stückweise line- arer Approximationen oder Relaxierungen. Es wird untersucht, wie die entstehende Formulierung für bestimmte Substrukturen verstärkt werden kann. Dabei finden wir effektive Schnittebenen und leiten mittels graphentheoretischer Argumente eine voll- ständige Beschreibung für induzierte Pfade her. Eine Abstraktion wesentlicher Eigenschaften dieses Spezialfalls führt auf Cliquen- probleme auf bestimmten Graphen. Dieser abstrakte Rahmen umfasst auch eine Ba- sisversion des Projektplanungsproblems und erlaubt es weiterhin, eine total unimo- dulare Reformulierung von linearer Größe nachzuweisen. Weiterhin werden Fragen zur Erkennbarkeit dieser Struktur behandelt. Außerdem wird das Konzept der simultanen Konvexifizierung diskutiert, das als kontinuierliches Gegenstück zu unserem Ansatz für stückweise linearisierte Probleme angesehen werden kann. Die entstehenden Reformulierungen verstärken Relaxierun- gen, auf die allgemeine MINLP-Löser typischerweise angewiesen sind. Rechenergebnisse unter Einbeziehung realer Datensätze zeigen den praktischen Einfluss der in dieser Arbeit entwickelten Methoden. 7 Contents 1 Introduction 17 2 Preliminaries 21 2.1 A Selection of Problems on Transportation Networks . 21 2.1.1 The Basic Linear Network Flow problem—Notations . 22 2.1.2 Analyzing Infeasibility of the Linear Network Flow Problem . 25 2.1.3 The Network Design Problem . 28 2.1.4 Further Extensions of Linear Network Design Problems . 29 2.1.5 Gas Network Optimization . 32 2.2 Modeling Piecewise Linear Functions . 37 2.2.1 The Multiple Choice Method . 38 2.2.2 The Convex Combination Method
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