On Limits of Dense Packing of Equal Spheres in a Cube

We examine packing of $n$ congruent spheres in a cube when $n$ is close but less than the number of spheres in a regular cubic close-packed (ccp) arrangement of $\lceil p^{3}/2\rceil$ spheres. For this family of packings, the previous best-known arrangements were usually derived from a ccp by omission of a certain number of spheres without changing the initial structure. In this paper, we show that better arrangements exist for all $n\leq\lceil p^{3}/2\rceil-2$. We introduce an optimization method to reveal improvements of these packings, and present many new improvements for $n\leq4629$

A new verified optimization technique for the “packing circles in a unit square” problems

Abstract. The paper presents a new verified optimization method for the problem of finding the densest packings of non-overlapping equal circles in a square. In order to provide reliable numerical results, the developed algorithm is based on interval analysis. As one of the most efficient parts of the algorithm, an interval-based version of a previous elimination procedure is introduced. This method represents the remaining areas still of interest as polygons fully calculated in a reliable way. Currently the most promising strategy of finding optimal circle packing configurations is to partition the original problem into subproblems. Still as a result of the highly increasing number of subproblems, earlier computer-aided methods were not able to solve problem instances where the number of circles was greater than 27. The present paper provides a carefully developed technique resolving this difficulty by eliminating large groups of subproblems together. As a demonstration of the capabilities of the new algorithm the problems of packing 28, 29, and 30 circles were solved within very tight tolerance values. Our verified procedure decreased the uncertainty in the location of the optimal packings by more than 700 orders of magnitude in all cases

Véhicules connectés contributions à la communication véhicule-réseau mobile et la localisation coopérative

RÉSUMÉ Véhicules connectés, ou « connected vehicles », est un nouveau paradigme des systèmes de transport intelligents (STI) qui vise à améliorer la sécurité et l’efficacité du trafic routier en utilisant les communications sans fil. Les communications des véhicules connectés (ou, V2X) englobent les communications sans fil entre véhicules et infrastructures (V2I), entre véhicule et véhicule (V2V), et entre les véhicules et les dispositifs sans fil (V2D). Considéré comme la norme de facto pour les communications V2X, le DSRC/WAVE est le principal standard de communication sans fil spécifiquement conçu pour les communications véhiculaires. L’efficacité du DSRC/WAVE pour les communications V2V et V2I a été prouvée par de nombreuses études et bancs d'essai dans le monde réel. En ce qui concerne la communication V2V, le passage au stade de déploiement à grande échelle est prévu à l’horizon 2020. En ce qui a trait à la communication V2I, bien que le déploiement d’une infrastructure DSRC (RSU) soit critique pour plusieurs applications STI, il n'y a toujours pas de plan pour son déploiement à grande échelle, essentiellement en raison de la nécessité d'investissements publics considérables. Avec les progrès réalisés au niveau des dernières versions du réseau mobile 4G LTE-A, les réseaux mobiles émergent comme l'une des principales technologies pour les communications V2I. En effet, le réseau mobile LTE-A permet aujourd’hui un plus grand débit (100Mb/s - 1Gb/s) avec relativement un faible délai (10ms) grâce à une évolution au niveau de l’architecture du réseau et l’introduction de nouvelles technologies telles que la densification du réseau à l'aide de petites cellules, relais (fixes et mobiles), la connectivité double (Dual Connectivity, DC), l’agrégation de porteuses (Carrier Aggregation, CA), etc.; des évolutions qui ouvrent la voie vers la 5G à l’horizon 2020 avec la promesse d’un débit encore plus élevé (10Gb/s) et d’un plus faible délai (1ms), ce qui renforce ainsi la tendance pour une future intégration véhicule et réseau mobile. Afin d’atteindre une plus grande efficacité spectrale, les petites cellules sont largement adoptées par les opérateurs de réseaux mobiles, dans les réseaux dits hétérogènes (HetNets), comme une solution clé pour désengorger le trafic au niveau des macrocellules et améliorer la capacité et la couverture du réseau d’accès. Cependant, bien que l’utilisation des petites cellules soit une solution intéressante pour les communications V2I, étant donné leur faible portée, cela provoque des relèves fréquentes qui mènent à une surcharge élevée de signalisation vers le réseau cœur. De plus, étant donné que les petites cellules sont généralement connectées au réseau cœur via une connexion Internet, celle-ci devient le goulot d'étranglement pour la relève et le transfert de données. D’où l’importance de compléter un maximum de relève localement. Cette thèse s'inscrit dans le cadre de l’étude de l’intégration véhicule infrastructure. L'objectif général est de proposer une architecture pour véhicule connecté basée sur la localisation coopérative, la communication V2I et la gestion de relève pour une meilleure intégration VANET – réseau mobile. Cette thèse fait état de trois principales contributions. 1. La première contribution concerne la proposition d’algorithmes de localisation coopérative, basés sur une approche ensembliste, qui permettent d’améliorer la précision de localisation. Le premier algorithme appelé (CLES) est un algorithme générique pour la localisation coopérative basée sur une approche ensembliste. Le deuxième algorithme, appelé (CLEF), est une application de l’algorithme CLES à la localisation par approche des signatures. De plus, nous caractérisons leur précision en évaluant la réduction du diamètre maximal et l’aire du polygone en fonction de différents paramètres tels que le nombre de polygones, la configuration géométrique, la proximité du nœud par rapport à la frontière de son polygone, et l’incertitude sur les mesures de distances. 2. La deuxième contribution porte sur la sélection des passerelles mobiles pour connecter efficacement les véhicules aux petites cellules du réseau mobile. Nous formulons le problème de sélection des passerelles mobiles sous forme d’un problème de programmation linéaire binaire multi-objectif (MO-BIP). Ensuite, nous évaluons l’efficacité de l’algorithme au niveau du temps de calcul pour différents degrés de connectivité et un nombre variable de véhicules. 3. La troisième contribution concerne la gestion de la relève dans les petites cellules du réseau mobile (LTE-A) afin de supporter efficacement les communications des véhicules connectés et réduire la surcharge de signalisation vers le réseau cœur. Pour ce faire, nous proposons un nouveau schéma basé sur le transfert local de trafic en utilisant les liens X2 et les nœuds d’ancrage. Trois procédures sont proposées, à savoir: 1) intra-domaine, 2) inter-domaines, 3) et K-sauts inter-domaines. Ensuite, en utilisant un modèle analytique, nous évaluons l’efficacité du schéma proposé.----------ABSTRACT Connected Vehicles are a new intelligent transportation paradigm that uses wireless communications to improve traffic safety and efficiency. It has received a great deal of attention in recent years, across many communities. While the DSRC is widely recognized as the de facto standard for V2V, other wireless technologies are required for large-scale deployment of V2I communications. Thanks to its high data rates and large scale deployment, the LTE-A enhanced by small cells densification, is positioned as one of the major candidate technologies for V2I communications. However, using LTE-A small cells for V2I communications is challenging due to their small coverage which lead to frequent handoffs and more signaling overhead. Thanks to recent advances in LTE-A Releases-10/11/12, the 4G LTE-Advanced (LTE-A) mobile network appears as one of the major candidate technologies for V2I communications. In fact, the LTE-A promises to deliver reduced connection setup time and lower latency (10ms) and higher data rates (up to 1Gbps) by using new physical layer technologies and new network elements and functions such as, network densification using Small Cells (SCs), Dual Connectivity (DC), Relaying functionality, Carrier Aggregation (CA), Device to Device (D2D) communication, etc.; developments that pave the way to 5G in the Horizon 2020, with the promise of an even higher data rates (more than 10 Gbps) and even much lower latency (1ms), which reinforces the trend for future integration between VANET and mobile networks for V2I communications. Although the macrocell will remain the major Radio Access Network (RAN) element for wide-area coverage and high-mobility users, it is no longer sufficient to meet user's demand in many high-density areas. Indeed, due to the proliferation of mobile devices and applications, mobile data demand continues to grow exponentially. Small cells, which include microcells, picocells, and femtocells, are widely recognized as a key solution for enhancing RAN capacity and coverage. They are increasingly used by mobile operators, in the so-called Heterogeneous Network (HetNet), to offload traffic from their macrocells. A HetNet is typically composed of several layers (macrocells, small cells), and in some cases different access technologies (e.g., LTE-A, UMTS, WiFi). SCs densification involves deploying more small coverage base stations in high demand areas to bring higher spectral efficiency per coverage area. Nevertheless, the SCs deployment faces a number of problems relevant to mobility handling that have to be addressed. More specifically, the use of SCs with limited coverage causes frequent handovers that lead to high signaling overhead toward the core network. In addition, since the small cells are generally connected to the EPC via a network Internet connection, this one becomes the bottleneck for handovers and data forwarding, hence the importance of completing a maximum of handover locally. This thesis therefore aims to propose solutions for VANETs and mobile networks integration. The main contributions of this thesis are summarized as follows: The first contribution concerns the proposed cooperative localization algorithms, based on a set-membership approach which improves the location accuracy. The first algorithm called (CLES) is a generic algorithm for cooperative localization based on a set-membership approach. The second algorithm called (CLEF) is an application of CLES algorithm to fingerprinting localization. In addition, we characterize their accuracy by evaluating the reduction of the maximum diameter and the area of the polygon depending on various parameters such as the number of polygons, the geometric configuration, the nearest node in relation to the boundary of the polygon, and the uncertainty of distance measurements. The second contribution concerns the selection of mobile gateways to effectively connect vehicles to small cells of the mobile network. In fact, while each vehicle may directly uses its LTE-A interface for V2I communications, we argue that by selecting a limited number of GWs, we can effectively reduce the mobility signaling overhead. Hence, we propose a new network-based mobile gateway selection scheme with one-hop clustering to efficiently relay the traffic from neighbouring vehicles toward the serving SC. The selection problem is formulated as a multi-objective binary linear programming problem. Using linear programming solver, we show that, for realistic number of vehicles per small cell and GW connectivity degree, the execution time is relatively short. As a third contribution of this thesis, we focus on challenges relevant to mobility for VANETs using LTE-A network. Specifically, a novel architecture that integrates VANET and 4G LTE-A Heterogeneous Network for enhanced mobility in LTE-A small cells is introduced. First, we propose a new network-based mobile gateway selection scheme with one-hop clustering to efficiently relay traffic from neighbouring vehicles toward the serving SC. The problem is formulated as a multi-objective binary programming problem. Then, for seamless mobility of connected vehicles, we propose a local k-hops anchor-based mobility scheme with three procedures, namely intra-domain, k-hops inter-domain and inter-domain procedures. Numerical results show the effectiveness of the proposed mobility schemes for reducing the generated signaling load towards the core network

Rigorous techniques for continuous constraint satisfaction problems

Diese Arbeit beschäftigt sich mit rigorosen Techniken für das Lösen kontinuierlicher Zulässigkeitsprobleme. Das heißt, wir suchen nach einem oder allen Punkte, genannt zulässige Punkte, die eine Familie von Gleichungen und/oder Ungleichungen erfüllen, die wir im Weiteren Nebenbedingungen nennen werden. Zahlreiche Anwendungen führen auf kontinuierliche Zulässigkeitsprobleme. Neue und bereits existierende moderne Methoden werden präsentiert und integriert in GloptLab, eine neue, leicht bedienbare Test- und Entwicklungsplattform zum Lösen quadratischer Zulässigkeitsprobleme. Der Lösungsalgorithmus beruht auf dem Grundprinzip von Branch-and-Prune und auf Filterung. Filterungsmethoden dienen zur Verkleinerung/Reduktion einer Box, definiert als das kartesische Produkt der Intervalle, die die Schranken an die Variablen festlegen. Um den Verlust zulässiger Punkte zu vermeiden, werden alle Fehlerabschätzungen rigoros mittels Intervallarithmetik und gerichteter Rundung durchgeführt. Das stellt sicher, dass alle Rechnungen auch in Gleitkommaarithmetik gültig sind. In der Doktorarbeit werden die folgenden Themen diskutiert: der mathematische Hintergrund, Algorithmen und Tests für Constraint-Propagation, strikt konvexe Einschließungen, lineare Relaxationen, das Berechnen, korrekte Benutzen und Verifizieren approximativ zulässiger Punkte, optimale Skalierung und diverse Hilfsmethoden. Insbesondere: - Constraint-Propagation basiert auf einer Folge von Schritten, die jeweils eine einzelne Nebenbedingung verwenden. Traditionelle Techniken werden durch eine spezielle quadratische Methode erweitert, die neue Verfahren für die Eliminierung bilinearer Einträge und für das Berechnen optimaler Einschließungen für separable quadratische Ausdrücke verwendet. - Eine quadratische Ungleichungsnebenbedingung, die eine positiv definite Hesse-Matrix besitzt, definiert ein Ellipsoid. Eine spezielle rundungsfehlerkontrollierte Version der Cholesky-Zerlegung wird verwendet, um die strikt konvexe quadratische Nebenbedingungen in Norm-Ungleichungen zu transformieren. Für diese ist es dann einfach, die Intervall-Hülle analytisch zu bestimmen. - Diverse Methoden für die Erzeugung linearer Relaxationen werden diskutiert, kombiniert und erweitert. Teilweise verbesserte, existierende und neue Verfahren für das rigorose Einschließen der Lösungsmenge linearer Systeme werden präsentiert. - Eine Vielzahl von Beispielen demonstrieren, dass die präsentierten Verfahren einander ergänzen. Außerdem zeigen sie, wie man Lösungsstrategien entwickelt, die Zulässigkeitsprobleme global und effizient lösen.This thesis contributes rigorous techniques for solving continuous constraint satisfaction problems, i.e., finding one or all points (called feasible points) satisfying a given family of equations and/or inequalities (called constraints). Many real word problems are continuous constraint satisfaction problems. New and old state of the art methods are presented, integrated in GloptLab, a new easy-to-use testing and development platform for solving quadratic constraint satisfaction problems. The basic solution principle is branch and prune and filtering. Filtering techniques tighten a box -- the Cartesian product of intervals defined by the bounds on the variables. In order to avoid a loss of feasible points, rigorous error estimation using interval arithmetic and directed rounding are used, to take care that all calculations are valid even though the calculations are performed in floating-point arithmetic only. Discussed are the mathematical background, algorithms and tests of constraint propagation, strictly convex enclosures, linear relaxations, finding, using and verifying approximately feasible points, optimal scaling and other auxiliary techniques. In particular: - Constraint propagation is based on a sequence of steps, each using a single constraint only. Traditional techniques are extended by special quadratic constraint propagation methods using new techniques for eliminating bilinear entries and finding optimal enclosures for separable quadratic expressions. - A quadratic inequality constraint with a positive definite Hessian defines an ellipsoid. A rounding error controlled version of the Cholesky factorization is used to transform a strictly convex quadratic constraint into a norm inequality, for which the interval hull is easy to compute analytically. - Different methods for computing linear relaxations are discussed, combined and extended. Partially improved and existing methods, as well as new approaches for rigorously enclosing the solution set of linear systems of inequalities are presented. - Numerous examples show that the above methods complement each other and how to create solution strategies that solve constraint satisfaction problems globally and efficiently

Some applications of continuous variable neighbourhood search metaheuristic (mathematical modelling)

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.In the real world, many problems are continuous in nature. In some cases, finding the global solutions for these problems is di±cult. The reason is that the problem's objective function is non convex, nor concave and even not differentiable. Tackling these problems is often computationally too expensive. Although the development in computer technologies are increasing the speed of computations, this often is not adequate, particularly if the size of the problem's instance are large. Applying exact methods on some problems may necessitate their linearisation. Several new ideas using heuristic approaches have been considered particularly since they tackle the problems within reasonable computational time and give an approximate solution. In this thesis, the variable neighbourhood search (VNS) metaheuristic (the framework for building heuristic) has been considered. Two variants of variable neighbourhood search metaheuristic have been developed, continuous variable neighbourhood search and reformulation descent variable neighbourhood search. The GLOB-VNS software (Drazic et al., 2006) hybridises the Microsoft Visual Studio C++ solver with variable neighbourhood search metaheuristics. It has been used as a starting point for this research and then adapted and modified for problems studied in this thesis. In fact, two problems have been considered, censored quantile regression and the circle packing problem. The results of this approach for censored quantile regression outperforms other methods described in the literature, and the near-optimal solutions are obtained in short running computational time. In addition, the reformulation descent variable neighbourhood search variant in solving circle packing problems is developed and the computational results are provided

Some applications of continuous variable neighbourhood search metaheuristic (mathematical modelling)

In the real world, many problems are continuous in nature. In some cases, finding the global solutions for these problems is di±cult. The reason is that the problem's objective function is non convex, nor concave and even not differentiable. Tackling these problems is often computationally too expensive. Although the development in computer technologies are increasing the speed of computations, this often is not adequate, particularly if the size of the problem's instance are large. Applying exact methods on some problems may necessitate their linearisation. Several new ideas using heuristic approaches have been considered particularly since they tackle the problems within reasonable computational time and give an approximate solution. In this thesis, the variable neighbourhood search (VNS) metaheuristic (the framework for building heuristic) has been considered. Two variants of variable neighbourhood search metaheuristic have been developed, continuous variable neighbourhood search and reformulation descent variable neighbourhood search. The GLOB-VNS software (Drazic et al., 2006) hybridises the Microsoft Visual Studio C++ solver with variable neighbourhood search metaheuristics. It has been used as a starting point for this research and then adapted and modified for problems studied in this thesis. In fact, two problems have been considered, censored quantile regression and the circle packing problem. The results of this approach for censored quantile regression outperforms other methods described in the literature, and the near-optimal solutions are obtained in short running computational time. In addition, the reformulation descent variable neighbourhood search variant in solving circle packing problems is developed and the computational results are provided.EThOS - Electronic Theses Online ServiceGBUnited Kingdo