Bibliothek

Beschreibung: This thesis introduces the novel paradigm of constraint integer programming (CIP), which integrates constraint programming (CP) and mixed integer programming (MIP) modeling and solving techniques. It is supplemented by the software SCIP, which is a solver and framework for constraint integer programming that also features SAT solving techniques. SCIP is freely available in source code for academic and non-commercial purposes. Our constraint integer programming approach is a generalization of MIP that allows for the inclusion of arbitrary constraints, as long as they turn into linear constraints on the continuous variables after all integer variables have been fixed. The constraints, may they be linear or more complex, are treated by any combination of CP and MIP techniques: the propagation of the domains by constraint specific algorithms, the generation of a linear relaxation and its solving by LP methods, and the strengthening of the LP by cutting plane separation. The current version of SCIP comes with all of the necessary components to solve mixed integer programs. In the thesis, we cover most of these ingredients and present extensive computational results to compare different variants for the individual building blocks of a MIP solver. We focus on the algorithms and their impact on the overall performance of the solver. In addition to mixed integer programming, the thesis deals with chip design verification, which is an important topic of electronic design automation. Chip manufacturers have to make sure that the logic design of a circuit conforms to the specification of the chip. Otherwise, the chip would show an erroneous behavior that may cause failures in the device where it is employed. An important subproblem of chip design verification is the property checking problem, which is to verify whether a circuit satisfies a specified property. We show how this problem can be modeled as constraint integer program and provide a number of problem-specific algorithms that exploit the structure of the individual constraints and the circuit as a whole. Another set of extensive computational benchmarks compares our CIP approach to the current state-of-the-art SAT methodology and documents the success of our method.

Beschreibung: Diese Arbeit stellt einen integrierten Ansatz aus Constraint Programming (CP) und Gemischt-Ganzzahliger Programmierung (Mixed Integer Programming, MIP) vor, den wir Constraint Integer Programming (CIP) nennen. Sowohl Modellierungs- als auch Lösungstechniken beider Felder fließen in den neuen integrierten Ansatz ein, um die unterschiedlichen Stärken der beiden Gebiete zu kombinieren. Als weiteren Beitrag stellen wir der wissenschaftlichen Gemeinschaft die Software SCIP zur Verfügung, die ein Framework für Constraint Integer Programming darstellt und zusätzlich Techniken des SAT-Lösens beinhaltet. SCIP ist im Source Code für akademische und nicht-kommerzielle Zwecke frei erhältlich. Unser Ansatz des Constraint Integer Programming ist eine Verallgemeinerung von MIP, die zusätzlich die Verwendung beliebiger Constraints erlaubt, solange sich diese durch lineare Bedingungen ausdrücken lassen falls alle ganzzahligen Variablen auf feste Werte eingestellt sind. Die Constraints werden von einer beliebigen Kombination aus CP- und MIP-Techniken behandelt. Dies beinhaltet insbesondere die Domain Propagation, die Relaxierung der Constraints durch lineare Ungleichungen, sowie die Verstärkung der Relaxierung durch dynamisch generierte Schnittebenen. Die derzeitige Version von SCIP enthält alle Komponenten, die für das effiziente Lösen von Gemischt-Ganzzahligen Programmen benötigt werden. Die vorliegende Arbeit liefert eine ausführliche Beschreibung dieser Komponenten und bewertet verschiedene Varianten in Hinblick auf ihren Einfluß auf das Gesamt-Lösungsverhalten anhand von aufwendigen praktischen Experimenten. Dabei wird besonders auf die algorithmischen Aspekte eingegangen. Der zweite Hauptteil der Arbeit befasst sich mit der Chip-Design-Verifikation, die ein wichtiges Thema innerhalb des Fachgebiets der Electronic Design Automation darstellt. Chip-Hersteller müssen sicherstellen, dass der logische Entwurf einer Schaltung der gegebenen Spezifikation entspricht. Andernfalls würde der Chip fehlerhaftes Verhalten aufweisen, dass zu Fehlfunktionen innerhalb des Gerätes führen kann, in dem der Chip verwendet wird. Ein wichtiges Teilproblem in diesem Feld ist das Eigenschafts-Verifikations-Problem, bei dem geprüft wird, ob der gegebene Schaltkreisentwurf eine gewünschte Eigenschaft aufweist. Wir zeigen, wie dieses Problem als Constraint Integer Program modelliert werden kann und geben eine Reihe von problemspezifischen Algorithmen an, die die Struktur der einzelnen Constraints und der Gesamtschaltung ausnutzen. Testrechnungen auf Industrie-Beispielen vergleichen unseren Ansatz mit den bisher verwendeten SAT-Techniken und belegen den Erfolg unserer Methode.

Beschreibung: In this paper, we investigate the interconversion processes of the major flame retardant -- 1,2,5,6,9,10-hexabromocyclododecane (HBCD) -- by the means of statistical thermodynamics based on classical force-fields. Three ideas will be presented. First, the application of classical hybrid Monte-Carlo simulations for quantum mechanical processes will be justified. Second, the problem of insufficient convergence properties of hybrid Monte-Carlo methods for the generation of low temperature canonical ensembles will be solved by an interpolation approach. Furthermore, it will be shown how free energy differences can be used for a rate matrix computation. The results of our numerical simulations will be compared to experimental results.

Beschreibung: In this paper we give an overview of the heuristics which are integrated into the open source branch-cut-and-price-framework SCIP. We briefly describe the fundamental ideas of different categories of heuristics and present some computational results which demonstrate the impact of heuristics on the overall solving process of SCIP.

Beschreibung: Fast nonlinear programming methods following the all-at-once approach usually employ Newton's method for solving linearized Karush-Kuhn-Tucker (KKT) systems. In nonconvex problems, the Newton direction is only guaranteed to be a descent direction if the Hessian of the Lagrange function is positive definite on the nullspace of the active constraints, otherwise some modifications to Newton's method are necessary. This condition can be verified using the signs of the KKT's eigenvalues (inertia), which are usually available from direct solvers for the arising linear saddle point problems. Iterative solvers are mandatory for very large-scale problems, but in general do not provide the inertia. Here we present a preconditioner based on a multilevel incomplete $LBL^T$ factorization, from which an approximation of the inertia can be obtained. The suitability of the heuristics for application in optimization methods is verified on an interior point method applied to the CUTE and COPS test problems, on large-scale 3D PDE-constrained optimal control problems, as well as 3D PDE-constrained optimization in biomedical cancer hyperthermia treatment planning. The efficiency of the preconditioner is demonstrated on convex and nonconvex problems with $150^3$ state variables and $150^2$ control variables, both subject to bound constraints.

Beschreibung: This article describes the main concepts and techniques that have been developed during the last year at ZIB to solve dimensioning and routing optimization problems for IP networks. We discuss the problem of deciding if a given path set corresponds to an unsplittable shortest path routing, the fundamental properties of such path sets, and the computational complexity of some basic network planning problems for this routing type. Then we describe an integer-linear programming approach to solve such problems in practice. This approach has been used successfully in the planning of the German national education and research network for several years.

Beschreibung: In this review article we discuss different techniques to solve numerically the time-dependent Schrödinger equation on unbounded domains. We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works. We conclude with several numerical examples from different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case.

Beschreibung: We discuss first order optimality conditions for state constrained optimal control problems. Our concern is the treatment of problems, where the solution of the state equation is not known to be continuous, as in the case of boundary control in three space dimensions or optimal control with parabolic partial differential equations. We show existence of measure valued Lagrangian multipliers, which have just enough additional regularity to be applicable to all possibly discontinuous solutions of the state equation.

Beschreibung: We address the property checking problem for SoC design verification at the register transfer level (RTL) by integrating techniques from integer programming, constraint programming, and SAT solving. Specialized domain propagation and preprocessing algorithms for individual RTL operations extend a general constraint integer programming framework. Conflict clauses are learned by analyzing infeasible LPs and deductions, and by employing reverse propagation. Experimental results show that our approach outperforms SAT techniques for proving the validity of properties on circuits containing arithmetics.

Beschreibung: An adjustment scheme for the relaxation parameter of interior point approaches to the numerical solution of pointwise state constrained elliptic optimal control problems is introduced. The method is based on error estimates of an associated finite element discretization of the relaxed problems and optimally selects the relaxation parameter in dependence on the mesh size of discretization. The finite element analysis for the relaxed problems is carried out and a numerical example is presented which confirms our analytical findings.

Beschreibung: In this thesis, we study multicommodity routing problems in networks, in which commodities have to be routed from source to destination nodes. Such problems model for instance the traffic flows in street networks, data flows in the Internet, or production flows in factories. In most of these applications, the quality of a flow depends on load dependent cost functions on the edges of the given network. The total cost of a flow is usually defined as the sum of the arc cost of the network. An optimal flow minimizes this cost. A main focus of this thesis is to investigate online multicommodity routing problems in networks, in which commodities have to be routed sequentially. Arcs are equipped with load dependent price functions defining routing costs, which have to be minimized. We discuss a greedy online algorithm that routes (fractionally) each commodity by minimizing a convex cost function that depends on the previously routed flow. We present a competitive analysis of this algorithm and prove upper bounds of (d+1)^(d+1) for polynomial price functions with nonnegative coefficients and maximum degree d. For networks with two nodes and parallel arcs, we show that this algorithm returns an optimal solution. Without restrictions on the price functions and network, no algorithm is competitive. We also investigate a variant in which the demands have to be routed unsplittably. In this case, it is NP-hard to compute the offline optimum. Furthermore, we study selfish routing problems (network games). In a network game, players route demand in a network with minimum cost. In this setting, we study the quality of Nash equilibria compared to the the system optimum (price of anarchy) in network games with nonatomic and atomic players and spittable flow. As a main result, we prove upper bounds on the price of anarchy for polynomial latency functions with nonnegative coefficients and maximum degree d, which improve upon the previous best ones.

Beschreibung: Diese Arbeit befasst sich mit Mehrgüterflussproblemen, in denen Güter mit einer bestimmten Rate durch ein gegebenes Netzwerk geleitet werden müssen. Mithilfe von Mehrgüterflussproblemen können zum Beispiel Verkehrsflüsse in Strassenverkehrsnetzen oder im Internet modelliert werden. In diesen Anwendungen wird die Effizienz von Routenzuweisungen für Güter durch lastabhängige Kostenfunktionen auf den Kanten eines gegebenen Netzwerks definiert. Die Gesamtkosten eines Mehrgüterflüsses sind durch die Summe der Kosten auf den Kanten definiert. Ein optimaler Mehrgüterfluss minimiert diese Gesamtkosten. Ein wesentlicher Bestandteil dieser Arbeit ist die Untersuchung sogenannter Online Algorithmen, die Routen für bekannte Güternachfragen berechnen, ohne vollständiges Wissen über zukünftige Güternachfragen zu haben. Es konnte ein Online Algorithmus gefunden werden, dessen Gesamtkosten für polynomielle Kostenfunktionen mit endlichem Grad nicht beliebig von denen einer optimalen Lösung abweichen. Für die Berechung einer optimalen Lösung müssen alle Güternachfragen a priori vorliegen. Dieses Gütekriterium gilt unabhängig von der gewählten Netzwerktopologie oder der Eingabesequenz von Gütern. Desweiteren befasst sich diese Arbeit mit der Effizienz egoistischer Routenwahl einzelner Nutzer verglichen zu einer optimalen Routenwahl. Egoistisches Verhalten von Nutzern kann mithilfe von nichtkooperativer Spieltheorie untersucht werden. Nutzer werden als strategisch agierende Spieler betrachtet, die ihren Profit maximieren. Als Standardwerkzeug zur Analyse solcher Spiele hat sich das Konzept des Nash Gleichgewichts bewährt. Das Nash Gleichweicht beschreibt eine stabile Strategieverteilung der Spieler, in der kein Spieler einen höheren Profit erzielen kann, wenn er einseitig seine Strategie ändert. Als Hauptergebnis dieser Arbeit konnte für polynomielle Kostenfunktionen mit endlichem Grad gezeigt werden, dass die Gesamtkosten eines Nash Gleichgewichts für sogennante atomare Spieler, die einen diskreten Anteil der gesamten Güternachfrage kontrollieren, nicht beliebig von den Gesamtkosten einer optimalen Lösung abweichen. In this thesis, we study multicommodity routing problems in networks, in which commodities have to be routed from source to destination nodes. Such problems model for instance the traffic flows in street networks, data flows in the Internet, or production flows in factories. In most of these applications, the quality of a flow depends on load dependent cost functions on the edges of the given network. The total cost of a flow is usually defined as the sum of the arc cost of the network. An optimal flow minimizes this cost. A main focus of this thesis is to investigate online multicommodity routing problems in networks, in which commodities have to be routed sequentially. Arcs are equipped with load dependent price functions defining routing costs, which have to be minimized. We discuss a greedy online algorithm that routes (fractionally) each commodity by minimizing a convex cost function that depends on the previously routed flow. We present a competitive analysis of this algorithm and prove upper bounds of (d+1)^(d+1) for polynomial price functions with nonnegative coefficients and maximum degree d. For networks with two nodes and parallel arcs, we show that this algorithm returns an optimal solution. Without restrictions on the price functions and network, no algorithm is competitive. We also investigate a variant in which the demands have to be routed unsplittably. In this case, it is NP-hard to compute the offline optimum. Furthermore, we study selfish routing problems (network games). In a network game, players route demand in a network with minimum cost. In this setting, we study the quality of Nash equilibria compared to the the system optimum (price of anarchy) in network games with nonatomic and atomic players and spittable flow. As a main result, we prove upper bounds on the price of anarchy for polynomial latency functions with nonnegative coefficients and maximum degree d, which improve upon the previous best ones.

Beschreibung: The timetable is the essence of the service offered by any provider of public transport'' (Jonathan Tyler, CASPT 2006). Indeed, the timetable has a major impact on both operating costs and on passenger comfort. Most European agglomerations and railways use periodic timetables in which operation repeats in regular intervals. In contrast, many North and South American municipalities use trip timetables in which the vehicle trips are scheduled individually subject to frequency constraints. We compare these two strategies with respect to vehicle operation costs. It turns out that for short time horizons, periodic timetabling can be suboptimal; for sufficiently long time horizons, however, periodic timetabling can always be done in an optimal way'.

Beschreibung: This thesis describes the algorithm IS-OPT that integrates scheduling of vehicles and duties in public bus transit. IS-OPT is the first algorithm which solves integrated vehicle and duty scheduling problems arising in medium sized carriers such that its solutions can be used in daily operations without further adaptions. This thesis is structured as follows: The first chapter highlights mathematical models of the planning process of public transit companies and examines their potential for integrating them with other planning steps. It also introduces descriptions of the vehicle and the duty scheduling problem. Chapter 2 motivates why it can be useful to integrate vehicle and duty scheduling, explains approaches of the literature, and gives an outline of our algorithm IS-OPT. The following chapters go into the details of the most important techniques and methods of IS-OPT: In Chapter 3 we describe how we use Lagrangean relaxation in a column generation framework. Next, in Chapter 4, we describe a variant of the proximal bundle method (PBM) that is used to approximate linear programs occurring in the solution process. We introduce here a new variant of the PBM which is able to utilize inexact function evaluation and the use of epsilon-subgradients. We also show the convergence of this method under certain assumptions. Chapter 5 treats the generation of duties for the duty scheduling problem. This problem is modeled as a resourceconstraint- shortest-path-problem with non-linear side constraints and nearly linear objective function. It is solved in a two-stage approach. At first we calculate lower bounds on the reduced costs of duties using certain nodes by a new inexact label-setting algorithm. Then we use these bounds to speed up a depth-first-search algorithm that finds feasible duties. In Chapter 6 we present the primal heuristic of IS-OPT that solves the integrated problem to integrality. We introduce a new branch-and-bound based heuristic which we call rapid branching. Rapid branching uses the proximal bundle method to compute lower bounds, it introduces a heuristic node selection scheme, and it utilizes a new branching rule that fixes sets of many variables at once. The common approach to solve the problems occurring in IS-OPT is to trade inexactness of the solutions for speed of the algorithms. This enables, as we show in Chapter 7, to solve large real world integrated problems by IS-OPT. The scheduled produced by IS-OPT save up to 5% of the vehicle and duty cost of existing schedules of regional and urban public transport companies.

Beschreibung: Diese Arbeit beschreibt den Algorithmus IS-OPT, welcher der erste Algorithmus ist, der integrierte Umlauf- und Dienstplanungsprobleme für mittelgroße Verkehrsunternehmen löst und dabei alle betrieblichen Einzelheiten berücksichtigt. Seine Lösungen können daher direkt in den täglichen Betrieb übernommen werden. Im ersten Kapitel werden mathematische Modelle für verschiedenen Probleme aus dem Planungsprozess von Nahverkehrsunternehmen beschrieben. Es werden Ansätze zur Integration der einzelnen Probleme untersucht. In diesem Kapitel werden außerdem das Umlauf- und das Dienstplanungsproblem eingeführt. Kapitel 2 motiviert, warum Integration von Umlauf- und Dienstplanung hilfreich ist oder in welchen Fällen sie sogar unabdingbar ist; es gibt einen Überblick über die vorhanden Literatur zur integrierten Umlauf- und Dienstplanung und umreißt unseren Algorithmus IS-OPT. Die weiteren Kapitel behandeln die in IS-OPT verwendeten Methoden: In Kapitel 3 beschreiben wir, wie Spaltenerzeugung für lineare Programme mit Lagrange-Relaxierung und Subgradienten-Verfahren kombiniert werden kann. In Kapitel 4 wird unsere Variante der proximalen Bündelmethode beschrieben. Diese wird benutzt um näherungsweise primale und duale Lösungen von lineare Programmen zu berechnen. Unsere Variante ist angepasst, um auch mit ungenauer Funktionsauswertung und Epsilon-Subgradienten arbeiten zu können. Wir zeigen die Konvergenz dieser Variante unter bestimmten Annahmen. Kapitel 5 behandelt das Erzeugen von Diensten für das Dienstplanungsproblem. Dieses Problem ist als ein Kürzeste-Wege-Problem mit nichtlinearen Nebenbedingungen und fast linearer Zielfunktion modelliert. Wir lösen es, indem zuerst Schranken für die reduzierten Kosten von Diensten, die bestimmte Knoten benutzen, berechnet werden. Diese Schranken werden benutzt, um in einem Tiefensuchalgorithmus den Suchbaum klein zu halten. Im Kapitel 6 präsentieren wir die neu entwickelte Heuristik "Rapid Branching", die eine ganzzahlige Lösung des integrierten Problems berechnet. Rapid Branching kann als eine spezielle Branch-and-Bound-Heuristik gesehen werden, welche die Bündelmethode benutzt. In den Knoten des Suchbaums können mehrere Variablen auf einmal fixiert werden, die mit Hilfe einer Perturbationsheuristik ausgewählt werden. In Kapitel 7 schließlich zeigen wir, daß wir mit IS-OPT auch große Probleminstanzen aus der Praxis lösen können und dabei bis zu 5% der Fahrzeug- und Dienstkosten sparen können.

Beschreibung: This paper reviews George Dantzig's contribution to integer programming, especially his seminal work with Fulkerson and Johnson on the traveling salesman problem

Beschreibung: Abstract The cost-efficient design of survivable optical telecommunication networks is the topic of this thesis. In cooperation with network operators, we have developed suitable concepts and mathematical optimization methods to solve this comprehensive planning task in practice. Optical technology is more and more employed in modern telecommunication networks. Digital information is thereby transmitted as short light pulses through glass fibers. Moreover, the optical medium allows for simultaneous transmissions on a single fiber by use of different wavelengths. Recent optical switches enable a direct forwarding of optical channels in the network nodes without the previously required signal retransformation to electronics. Their integration creates ongoing optical connections,which are called lightpaths. We study the problem of finding cost-efficient configurations of optical networks which meet specified communication requirements. A configuration comprises the determination of all lightpaths to establish as well as the detailed allocation of all required devices and systems. We use a flexible modeling framework for a realistic representation of the networks and their composition. For different network architectures, we formulate integer linear programs which model the design task in detail. Moreover, network survivability is an important issue due to the immense bandwidths offered by optical technology. Operators therefore request for designs which perpetuate protected connections and guarantee for a defined minimum throughput in case of malfunctions. In order to achieve an effective realization of scalable protection, we present a novel survivability concept tailored to optical networks and integrate several variants into the models. Our solution approach is based on a suitable model decomposition into two subtasks which separates two individually hard subproblems and enables this way to compute cost-efficient designs with approved quality guarantee. The first subtask consists of routing the connections with corresponding dimensioning of capacities and constitutes a common core task in the area of network planning. Sophisticated methods for such problems have already been developed and are deployed by appropriate integration. The second subtask is characteristic for optical networks and seeks for a conflict-free assignment of available wavelengths to the lightpaths using a minimum number of involved wavelength converters. For this coloring-like task, we derive particular models and study methods to estimate the number of unavoidable conversions. As constructive approach, we develop heuristics and an exact branch-and-price algorithm. Finally, we carry out an extensive computational study on realistic data, provided by our industrial partners. As twofold purpose, we demonstrate the potential of our approach for computing good solutions with quality guarantee, and we exemplify its flexibility for application to network design and analysis.

Beschreibung: Parabolic reaction-diffusion systems may develop sharp moving reaction fronts which pose a challenge even for adaptive finite element methods. We propose a method to transform the equation into an equivalent form that usually exhibits solutions which are easier to discretize, giving higher accuracy for a given number of degrees of freedom. The transformation is realized as an efficiently computable pointwise nonlinear scaling that is optimized for prototypical planar travelling wave solutions of the underlying reaction-diffusion equation. The gain in either performance or accuracy is demonstrated on different numerical examples.

Beschreibung: We propose and analyse an interior point path-following method in function space for state constrained optimal control. Our emphasis is on proving convergence in function space and on constructing a practical path-following algorithm. In particular, the introduction of a pointwise damping step leads to a very efficient method, as verified by numerical experiments.

Beschreibung: wird nachgereicht

Beschreibung: Multi-scale phenomena are abundant in many application fields. Representing and numerically simulating such processes is a challenging task since quite different scales have to be resolved, which often requires enormous amounts of storage and computational power. An important strategy in this context is adaptivity, i.e. local adjustment of the spatio-temporal resolution to the details to be resolved. A standard representation therefore are hierarchical, locally refined grids. A specific adaptive approach for solving partial differential equations, usually called AMR (Adaptive Mesh Refinement), was introduced in 1984. The basic idea is to combine the simplicity of structured grids and the advantages of local refinement. In this numerical scheme the computations are started on a set of coarse, potentially overlapping structured grids, that cover the computational domain. Local error criteria are applied to detect regions that require higher resolution. These are covered by subgrids with decreasing mesh spacing, which do not replace, but rather overlap the refined regions of the coarser patches. The equations are advanced on the finer subgrids and the refinement procedure recursively continues until all cells fulfill the considered error criteria, giving rise to a hierarchy of nested levels of refinement. In 1989 a variant of this scheme, called Structured Adaptive Mesh Refinement (SAMR), which reduces some of the complexity of the original approach, was proposed. While the separate subgrids in the AMR scheme could be rotated against each other, in SAMR they are aligned with the major axes of the coordinate system, which for example simplifies the computation of fluxes of (conserved) quantities through the cell faces. SAMR has become more and more popular in the last decade, and nowadays it is applied in many domains like hydrodynamics, meteorology and in particular in cosmology and relativistic astrophysics. Due to this growing popularity, an increasing number of scientists is in need of appropriate interactive visualization techniques to interpret and analyze AMR simulation data. Tools for both, 2D analysis to quantitatively convey the information within single slices and 3D representations to apprehend the overall structure are required. In this thesis we develop direct and indirect volume visualization algorithms for scalar fields that are defined on structured Adaptive Mesh Refinement (SAMR) grids. In particular algorithms for planar slicing and the display of height fields, C0-continuous isosurface extraction, software-, and hardware-based direct volume rendering and temporal interpolation for cell-, and vertex-centered data on unrestricted SAMR grids are proposed. Additionally we investigate the applicability of SAMR data structures for accelerated software-, and hardware-based volume rendering of large 3D scalar data.