The aim of thiswork is to locate a semi-obnoxious facility, i.e. tominimize the distances to a given set of customers in order to save transportation costs on the one hand and to avoid undesirable interactions with other facilities within the region by maximizing the distances to the corresponding facilities on the other hand. Hence, the goal is to satisfy economic and environmental issues simultaneously. Due to the contradicting character of these goals, we obtain a non-convex objective function. We assume that distances can be measured by rectilinear distances and exploit the structure of this norm to obtain a very efficient dual pair of algorithms.

Decentralized multi-agent optimization has emerged as a powerful paradigm that finds broad applications in engineering design including federated machine learning and control of networked systems. In these setups, a group of agents are connected via a network with general topology. Under the communication constraint, they aim to solving a global optimization problem that is characterized collectively by their individual interests. Of particular importance are the computation and communication efficiency of decentralized optimization algorithms. Due to the heterogeneity of local objective functions, fostering cooperation across the agents over a possibly time-varying network is challenging yet necessary to achieve fast convergence to the global optimum. Furthermore, real-world communication networks are subject to congestion and bandwidth limit. To relieve the difficulty, it is highly desirable to design communication-efficient algorithms that proactively reduce the utilization of network resources. This dissertation tackles four concrete settings in decentralized optimization, and develops four provably efficient algorithms for solving them, respectively. Chapter 1 presents an overview of decentralized optimization, where some preliminaries, problem settings, and the state-of-the-art algorithms are introduced. Chapter 2 introduces the notation and reviews some key concepts that are useful throughout this dissertation. In Chapter 3, we investigate the non-smooth cost-coupled decentralized optimization and a special instance, that is, the dual form of constraint-coupled decentralized optimization. We develop a decentralized subgradient method with double averaging that guarantees the last iterate convergence, which is crucial to solving decentralized dual Lagrangian problems with convergence rate guarantee. Chapter 4 studies the composite cost-coupled decentralized optimization in stochastic networks, for which existing algorithms do not guarantee linear convergence. We propose a new decentralized dual averaging (DDA) algorithm to solve this problem. Under a rather mild condition on stochastic networks, we show that the proposed DDA attains an $\mathcal{O}(1/t)$ rate of convergence in the general case and a global linear rate of convergence if each local objective function is strongly convex. Chapter 5 tackles the smooth cost-coupled decentralized constrained optimization problem. We leverage the extrapolation technique and the average consensus protocol to develop an accelerated DDA algorithm. The rate of convergence is proved to be $\mathcal{O}\left( \frac{1}{t^2}+ \frac{1}{t(1-\beta)^2} \right)$, where $\beta$ denotes the second largest singular value of the mixing matrix. To proactively reduce the utilization of network resources, a communication-efficient decentralized primal-dual algorithm is developed based on the event-triggered broadcasting strategy in Chapter 6. In this algorithm, each agent locally determines whether to generate network transmissions by comparing a pre-defined threshold with the deviation between the iterates at present and lastly broadcast. Provided that the threshold sequence is summable over time, we prove an $\mathcal{O}(1/t)$ rate of convergence for convex composite objectives. For strongly convex and smooth problems, linear convergence is guaranteed if the threshold sequence is diminishing geometrically. Finally, Chapter 7 provides some concluding remarks and research directions for future study. / Graduate

Die vorliegende Arbeit befasst sich mit Schnittebenenverfahren, einer Gruppe von iterativen Algorithmen zur Minimierung einer (möglicherweise nicht glatten) konvexen Funktion über einer kompakten konvexen Menge. Wir betrachten zwei prominente Beispiele, nämlich die Ellipsoidmethode und die Methode der Vaidya, und zeigen, dass ihre Konvergenzrate auch bei Verwendung eines ungenauen Orakels erhalten bleibt. Darüber hinaus zeigen wir, dass es möglich ist, diese Methoden im Rahmen der stochastischen Optimierung effizient zu nutzen. Eine andere Richtung, in der Schnittebenenverfahren nützlich sein können, sind duale Probleme. In der Regel können die Zielfunktion und ihre Ableitungen bei solchen Problemen nur näherungsweise berechnet werden. Daher ist die Unempfindlichkeit der Methoden gegenüber Fehlern in den Subgradienten von großem Nutzen. Als Anwendungsbeispiel schlagen wir eine linear konvergierende duale Methode für einen Markow-Entscheidungsprozess mit Nebenbedienungen vor, die auf der Methode der Vaidya basiert. Wir demonstrieren die Leistungsfähigkeit der vorgeschlagenen Methode in einem einfachen RL Problem. Die Arbeit untersucht auch das Konzept der Genauigkeitszertifikate für konvexe Minimierungsprobleme. Zertifikate ermöglichen die Online-Überprüfung der Genauigkeit von Näherungslösungen. In dieser Arbeit verallgemeinern wir den Begriff der Genauigkeitszertifikate für die Situation eines ungenauen Orakels erster Ordnung. Darüber hinaus schlagen wir einen expliziten Weg zur Konstruktion von Genauigkeitszertifikaten für eine große Klasse von Schnittebenenverfahren vor. Als Nebenprodukt zeigen wir, dass die betrachteten Methoden effizient mit einem verrauschten Orakel verwendet werden können, obwohl sie ursprünglich für ein exaktes Orakel entwickelt wurden. Schließlich untersuchen wir die vorgeschlagenen Zertifikate in numerischen Experimenten und zeigen, dass sie eine enge obere Schranke für das objektive Residuum liefern. / The present thesis studies cutting plane methods, which are a group of iterative algorithms for minimizing a (possibly nonsmooth) convex function over a compact convex set. We consider two prominent examples, namely, the ellipsoid method and Vaidya's method, and show that their convergence rate is preserved even when an inexact oracle is used. Furthermore, we demonstrate that it is possible to use these methods in the context of stochastic optimization efficiently. Another direction where cutting plane methods can be useful is Lagrange dual problems. Commonly, the objective and its derivatives can only be computed approximately in such problems. Thus, the methods' insensitivity to error in subgradients comes in handy. As an application example, we propose a linearly converging dual method for a constrained Markov decision process (CMDP) based on Vaidya's algorithm. We demonstrate the performance of the proposed method in a simple RL environment. The work also investigates the concept of accuracy certificates for convex minimization problems. Certificates allow for online verification of the accuracy of approximate solutions. In this thesis, we generalize the notion of accuracy certificates for the setting of an inexact first-order oracle. Furthermore, we propose an explicit way to construct accuracy certificates for a large class of cutting plane methods. As a by-product, we show that the considered methods can be efficiently used with a noisy oracle even though they were originally designed to be equipped with an exact oracle. Finally, we illustrate the work of the proposed certificates in numerical experiments highlighting that they provide a tight upper bound on the objective residual.