Counting Integer flows in Networks

This paper discusses new analytic algorithms and software for the enumeration of all integer flows inside a network. Concrete applications abound in graph theory \cite{Jaeger}, representation theory \cite{kirillov}, and statistics \cite{persi}. Our methods clearly surpass traditional exhaustive enumeration and other algorithms and can even yield formulas when the input data contains some parameters. These methods are based on the study of rational functions with poles on arrangements of hyperplanes

Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometry

We consider a series of optimal control problems with 2-dimensional control lying in an arbitrary convex compact set $\Omega$. The considered problems are well studied for the case when $\Omega$ is a unit disc, but barely studied for arbitrary $\Omega$. We derive extremals to these problems in general case by using machinery of convex trigonometry, which allows us to do this identically and independently on the shape of $\Omega$. The paper describes geodesics in (i) the Finsler problem on the Lobachevsky hyperbolic plane; (ii) left-invariant sub-Finsler problems on all unimodular 3D Lie groups (SU(2), SL(2), SE(2), SH(2)); (iii) the problem of rolling ball on a plane with distance function given by $\Omega$; (iv) a series of "yacht problems" generalizing Euler's elastic problem, Markov-Dubins problem, Reeds-Shepp problem and a new sub-Riemannian problem on SE(2); and (v) the plane dynamic motion problem.Comment: 50 pages, 56 figure

Über Struktur- und Sensitivitätsaussagen in Ganzzahligen Programmen und deren Anwendung in Kombinatorischer Optimierung

In this thesis we investigate properties of integer linear programs (ILPs) and their algorithmic use. Our main focus are ILP-formulations that come from concrete algorthmic problems like the bin packing problem or the scheduling problem on identical machines. Especially for this kind of ILPs we study structural properties as well as properties for their sensitivity. As a result, we are able to answer open algorithmical questions in the area of approximation and parameterized complexity. In the context of sensitivity we analyze how much an ILP solution has to be adjusted when the parameters of the ILP change. There is a classical results by Cook et al. which gave bounds for that question when optimal solutions are considered. However, in this thesis we investigate the sensitivity of ILPs when approximate solutions are allowed, i.e. solutions that differ by a factor of at most (1+ \epsilon) from the optimum value. We could apply the obtained results to the online bin packing problem, when an approximation guarantee with ratio $1+ \epsilon$ has to be fulfilled and repacking of already assigned items (limited by the so called migration factor) is allowed. In the context of structural results, we prove the existence (assuming the ILP is feasible) of solutions of a certain class of ILPs with a certain simplified structure. Specifically, in this thesis, we prove structure properties for ILPs that arise from formulations of bin packing or scheduling problems and natural generalization of those formulations. Based on the those structure properties, we develop an efficient approximation scheme for the scheduling problem on identical machines with a running time of 2^{\tilde{O}(1/\epsilon)} + poly}(n) and furthermore, we develop a structure theorem, which is applied to the bin packing problem when the number of different item sizes d is bounded.In dieser Dissertation werden Eigenschaften von ganzzahligen linearen Programmen (engl. integer linear programs, kurz: ILPs) untersucht. Von Interesse sind dabei hauptsächlich ILP-Formulierungen, welche sich aus dem Kontext von algorithmischen Problemstellungen ergeben, wie beispielsweise dem Bin Packing-Problem und dem Scheduling-Problem auf identischen Maschinen. Insbesondere für diese ILPs zeigen wir Strukturaussagen, sowie Aussagen über die Sensitivität und können so offene algorithmische Fragestellungen im Bereich von Approximation und parametrisierter Komplexität lösen. Im Kontext von Sensitivitätsaussagen wird untersucht, inwiefern Lösung des ILPs angepasst werden können, wenn sich die Parameter des ILPs leicht ändern. Ein klassisches Resultat von Cook u.a. gibt dabei für optimale Lösungen des ILPs Abschätzungen an. In dieser Arbeit betrachten wir Abschätzungen für die Senstivität wenn approximative Lösungen erlaubt sind, d.h. Lösungen deren Zielfunktionswert höchstens um einen Faktor 1+ \epsilon über dem optimalen Zielfunktionswert liegt. Diese Ergebnisse konnten wir auf das Online-Bin Packing-Problem anwenden, wenn eine approximative Lösung mit Güte 1+ \epsilon erreicht werden soll und in beschränktem Maße Items umgepackt werden dürfen. Im Kontext von Strukturaussagen wird in dieser Dissertation die Existenz von ILP-Lösungen bewiesen, welche eine bestimmte vereinfachte Struktur aufweisen. Insbesondere, konnten wir Strukturaussagen für ILPs entwickeln, welche sich aus Formulierungen des Bin Packing-Problems ergeben bzw. natürliche Verallgemeinerungen dieser Formulierung. Dadurch ist es uns zum einen gelungen ein effizientes Approximationsschemata für das Scheduling-Problem auf identischen Maschinen mit einer Laufzeit von 2^{\tilde{O}(1/\epsilon)} + poly(n) zu entwicklen und außerdem konnten wir eine Strukturaussage entwickeln, welche unter anderem Anwendung im Bin Packung-Problem fand, wenn die Anzahl der unterschiedlichen Itemgrößen d beschränkt ist

Mathematical control theory and Finance

Control theory provides a large set of theoretical and computational tools with applications in a wide range of fields, running from ”pure” branches of mathematics, like geometry, to more applied areas where the objective is to find solutions to ”real life” problems, as is the case in robotics, control of industrial processes or finance. The ”high tech” character of modern business has increased the need for advanced methods. These rely heavily on mathematical techniques and seem indispensable for competitiveness of modern enterprises. It became essential for the financial analyst to possess a high level of mathematical skills. Conversely, the complex challenges posed by the problems and models relevant to finance have, for a long time, been an important source of new research topics for mathematicians. The use of techniques from stochastic optimal control constitutes a well established and important branch of mathematical finance. Up to now, other branches of control theory have found comparatively less application in financial problems. To some extent, deterministic and stochastic control theories developed as different branches of mathematics. However, there are many points of contact between them and in recent years the exchange of ideas between these fields has intensified. Some concepts from stochastic calculus (e.g., rough paths) have drawn the attention of the deterministic control theory community. Also, some ideas and tools usual in deterministic control (e.g., geometric, algebraic or functional-analytic methods) can be successfully applied to stochastic control. We strongly believe in the possibility of a fruitful collaboration between specialists of deterministic and stochastic control theory and specialists in finance, both from academic and business backgrounds. It is this kind of collaboration that the organizers of the Workshop on Mathematical Control Theory and Finance wished to foster. This volume collects a set of original papers based on plenary lectures and selected contributed talks presented at the Workshop. They cover a wide range of current research topics on the mathematics of control systems and applications to finance. They should appeal to all those who are interested in research at the junction of these three important fields as well as those who seek special topics within this scope.info:eu-repo/semantics/publishedVersio

Controllability and Stabilization of Kolmogorov Forward Equations for Robotic Swarms

abstract: Numerous works have addressed the control of multi-robot systems for coverage, mapping, navigation, and task allocation problems. In addition to classical microscopic approaches to multi-robot problems, which model the actions and decisions of individual robots, lately, there has been a focus on macroscopic or Eulerian approaches. In these approaches, the population of robots is represented as a continuum that evolves according to a mean-field model, which is directly designed such that the corresponding robot control policies produce target collective behaviours. This dissertation presents a control-theoretic analysis of three types of mean-field models proposed in the literature for modelling and control of large-scale multi-agent systems, including robotic swarms. These mean-field models are Kolmogorov forward equations of stochastic processes, and their analysis is motivated by the fact that as the number of agents tends to infinity, the empirical measure associated with the agents converges to the solution of these models. Hence, the problem of transporting a swarm of agents from one distribution to another can be posed as a control problem for the forward equation of the process that determines the time evolution of the swarm density. First, this thesis considers the case in which the agents' states evolve on a finite state space according to a continuous-time Markov chain (CTMC), and the forward equation is an ordinary differential equation (ODE). Defining the agents' task transition rates as the control parameters, the finite-time controllability, asymptotic controllability, and stabilization of the forward equation are investigated. Second, the controllability and stabilization problem for systems of advection-diffusion-reaction partial differential equations (PDEs) is studied in the case where the control parameters include the agents' velocity as well as transition rates. Third, this thesis considers a controllability and optimal control problem for the forward equation in the more general case where the agent dynamics are given by a nonlinear discrete-time control system. Beyond these theoretical results, this thesis also considers numerical optimal transport for control-affine systems. It is shown that finite-volume approximations of the associated PDEs lead to well-posed transport problems on graphs as long as the control system is controllable everywhere.Dissertation/ThesisDoctoral Dissertation Mechanical Engineering 201