In wireless networks, communication links may be subject to random fatal impacts: for example, sensor networks under sudden power losses or cognitive radio networks with unpredictable primary user spectrum occupancy. Under such circumstances, it is critical to quantify how fast and reliably the information can be collected over attacked links. For a single point-to-point channel subject to a random attack, named as a dying channel, we model it as a block-fading (BF) channel with a finite and random channel length. First, we study the outage probability when the coding length K is fixed and uniform power allocation is assumed. Furthermore, we discuss the optimization over K and the power allocation vector PK to minimize the outage probability. In addition, we extend the single point to-point dying channel case to the parallel multi-channel case where each sub-channel is a dying channel, and investigate the corresponding asymptotic behavior of the overall outage probability with two different attack models: the independent-attack case and the m-dependent-attack case. It can be shown that the overall outage probability diminishes to zero for both cases as the number of sub-channels increases if the rate per unit cost is less than a certain threshold. The outage exponents are also studied to reveal how fast the outage probability improves over the number of sub-channels. Besides the information-theoretical results, we also study a practical coding scheme for the dying binary erasure channel (DBEC), which is a binary erasure channel (BEC) subject to a random fatal failure. We consider the rateless codes and optimize the degree distribution to maximize the average recovery probability. In particular, we first study the upper bound of the average recovery probability, based on which we define the objective function as the gap between the upper bound and the average recovery probability achieved by a particular degree distribution. We then seek the optimal degree distribution by minimizing the objective function. A simple and heuristic approach is also proposed to provide a suboptimal but good degree distribution.

Numerous control and decision problems in networked systems can be posed as optimization problems. Examples include the framework of network utility maximization for resource allocation in communication networks, multi-agent coordination in robotics, and collaborative estimation in wireless sensor networks (WSNs). In contrast to classical distributed optimization, which focuses on improving computational efficiency and scalability, these new applications require simple mechanisms that can operate under limited communication. In this thesis, we develop several novel mechanisms for distributed optimization under communication constraints, and apply these to several challenging engineering problems. In particular, we devise three tailored optimization algorithms relying only on nearest neighbor, also known as peer-to-peer, communication. Two of the algorithms are designed to minimize a non-smooth convex additive objective function, in which each term corresponds to a node in a network. The first method is an extension of the randomized incremental subgradient method where the update order is given by a random walk on the underlying communication graph, resulting in a randomized peer-to-peer algorithm with guaranteed convergence properties. The second method combines local subgradient iterations with consensus steps to average local update directions. The resulting optimization method can be executed in a peer-to-peer fashion and analyzed using epsilon-subgradient methods. The third algorithm is a center-free algorithm, which solves a non-smooth resource allocation problem with a separable additive convex objective function subject to a constant sum constraint. Then we consider cross-layer optimization of communication networks, and demonstrate how optimization techniques allow us to engineer protocols that mimic the operation of distributed optimization algorithms to obtain an optimal resource allocation. We describe a novel use of decomposition methods for cross-layer optimization, and present a flowchart that can be used to categorize and visualize a large part of the current literature on this topic. In addition, we devise protocols that optimize the resource allocation in frequency-division multiple access (FDMA) networks and spatial reuse time-division multiple access (TDMA) networks, respectively. Next we investigate some variants of the consensus problem for multi-robot coordination, for which it is usually standard to assume that agents should meet at the barycenter of the initial states. We propose a negotiation strategy to find an optimal meeting point in the sense that the agents' trajectories to the meeting point minimize a quadratic cost criterion. Furthermore, we also demonstrate how an augmented state vector can be used to boost the convergence rate of the standard linear distributed averaging iterations, and we present necessary and sufficient convergence conditions for a general version of these iterations. Finally, we devise a generic optimization software component for WSNs. To this end, we implement some of the most promising optimization algorithms developed by ourselves and others in our WSN testbed, and present experimental results, which show that the proposed algorithms work surprisingly well. / QC 20100813

Many problems in fluid mechanics and material sciences deal with liquid-vapour flows. In these flows, the ideal gas assumption is not accurate and the van der Waals equation of state is usually used. This equation of state is non-convex and causes the solution domain to have two hyperbolic regions separated by an elliptic region. Therefore, the governing equations of these flows have a mixed elliptic-hyperbolic nature. Numerical oscillations usually appear with standard finite-difference space discretization schemes, and they persist when the order of accuracy of the semi-discrete scheme is increased. In this study, we propose to use a Chebyshev pseudospectral method for solving the governing equations. A comparison of the results of this method with very high-order (up to tenth-order accurate) finite difference schemes is presented, which shows that the proposed method leads to a lower level of numerical oscillations than other high-order finite difference schemes, and also does not exhibit fast-traveling packages of short waves which are usually observed in high-order finite difference methods. The proposed method can thus successfully capture various complex regimes of waves and phase transitions in both elliptic and hyperbolic regimes

The concept of cooperation in communications has drawn a lot of research attention in recent years due to its potential to improve the efficiency of wireless networks. This new form of communications allows some users to act as relays and assist the transmission of other users' information signals. The aim of this thesis is to apply optimization techniques in the design of multi-relay wireless networks employing cooperative communications. In general, the thesis is organized into two parts: ``Distributed space-time coding' (DSTC) and ``Distributed beamforming', which cover two main approaches in cooperative communications over multi-relay networks. <br><br> In Part I of the thesis, various aspects of distributed implementation of space-time coding in a wireless relay network are treated. First, the thesis proposes a new fully-diverse distributed code which allows noncoherent reception at the destination. Second, the problem of coordinating the power allocation (PA) between source and relays to achieve the optimal performance of DSTC is studied and a novel PA scheme is developed. It is shown that the proposed PA scheme can obtain the maximum diversity order of DSTC and significantly outperform other suboptimal PA schemes. Third, the thesis presents the optimal PA scheme to minimize the mean-square error (MSE) in channel estimation during training phase of DSTC. The effect of imperfect channel estimation to the performance of DSTC is also thoroughly studied. <br><br> In Part II of the thesis, optimal distributed beamforming designs are developed for a wireless multiuser multi-relay network. Two design criteria for the optimal distributed beamforming at the relays are considered: (i) minimizing the total relay power subject to a guaranteed Quality of Service (QoS) measured in terms of signal-to-noise-ratio (SNR) at the destinations, and (ii) jointly maximizing the SNR margin at the destinations subject to power constraints at the relays. Based on convex optimization techniques, it is shown that these problems can be formulated and solved via second-order conic programming (SOCP). In addition, this part also proposes simple and fast iterative algorithms to directly solve these optimization problems.

In this thesis, various numerical methods are developed to solve nonsmooth and in particular, nonconvex optimization problems. More speci

Many problems in fluid mechanics and material sciences deal with liquid-vapour flows. In these flows, the ideal gas assumption is not accurate and the van der Waals equation of state is usually used. This equation of state is non-convex and causes the solution domain to have two hyperbolic regions separated by an elliptic region. Therefore, the governing equations of these flows have a mixed elliptic-hyperbolic nature. Numerical oscillations usually appear with standard finite-difference space discretization schemes, and they persist when the order of accuracy of the semi-discrete scheme is increased. In this study, we propose to use a Chebyshev pseudospectral method for solving the governing equations. A comparison of the results of this method with very high-order (up to tenth-order accurate) finite difference schemes is presented, which shows that the proposed method leads to a lower level of numerical oscillations than other high-order finite difference schemes, and also does not exhibit fast-traveling packages of short waves which are usually observed in high-order finite difference methods. The proposed method can thus successfully capture various complex regimes of waves and phase transitions in both elliptic and hyperbolic regimes

One of the most essential objectives in WSNs is to determine the spatial coordinates of a source or a sensor node having information. In this study, the problem of range measurement-based localization of a signal source or a sensor is revisited. The main challenge of the problem results from the non-convexity associated with range measurements calculated using the distances from the set of nodes with known positions to a xed sen- sor node. Such measurements corresponding to certain distances are non-convex in two and three dimensions. Attempts recently proposed in the literature to eliminate the non- convexity approach the problem as a non-convex geometric minimization problem, using techniques to handle the non-convexity. This study proposes a new fuzzy range-free sensor localization method. The method suggests using some notions of Euclidean geometry to convert the problem into a convex geometric problem. The convex equivalent problem is built using convex fuzzy sets, thus avoiding multiple stable local minima issues, then a gradient based localization algorithm is chosen to solve the problem. Next, the proposed algorithm is simulated considering various scenarios, including the number of available source nodes, fuzzi cation level, and area coverage. The results are compared with an algorithm having similar fuzzy logic settings. Also, the behaviour of both algorithms with noisy measurements are discussed. Finally, future extensions of the algorithm are suggested, along with some guidelines.

En esta memoria se incluyen diversas alternativas a una teoría de krein-milman no arquimediana las cuales vienen sugeridas bien por intentos anteriores de otros autores bien por conseguir una teoría unificada en los casos arquimediano o no o bien por lograr una teoría independiente del cuerpo valuado y que en condiciones de comparación dan lugar a resultados muy similares / This monography provides several alternatives to a non-Archimedan Krein-Milman Theory. They are suggested by some previous attempts to this subject carried out by other authors, as well as by the aim of getting an unified theory that works in the Archimede and in the non-Archimedean cases, in the sense that in the Archimedean context, this theory coincides with the well-known one existing in the classical literature

Tradicionalment, les tècniques d'accés múltiple en sistemes de comunicacions multi-usuari han estat desenvolupades o bé orientades a la connexió o bé orientades al tràfic. En el primer cas, l'objectiu és establir tants canals ortogonals com sigui possible per tal d'assignar-los als usuaris. Aquesta idea va motivar el disseny de les estratègies més conegudes, com són FDMA, TDMA i CDMA. Per altra banda, però, els mètodes d'accés aleatori que s'iniciaren amb el conegut ALOHA pretenen compartir estadísticament un mateix medi de comunicació aprofitant la necessitat de transmetre la informació a ràfegues que s'origina en les xarxes de dades. Així, molts dels actuals sistemes es poden encabir dins d'aquest esquema si a més a més, tenim en compte combinacions d'aquestes. No obstant, sistemes moderns com el DVB-RCS en l'entorn de comunicacions digitals per satèl·lit o el WiMAX en l'accés terrestre de banda ampla implementen mecanismes de petició i assignació de recursos, els quals requereixen una gestió dinàmica d'aquests en el sistema (és el que s'anomena distribució dinàmica de l'amplada de banda en un sentit ampli).L'anterior concepte inclou múltiples variables, configuracions i protocols tant de capa física com de capa d'enllaç. En aquesta tesi s'exploren en primer lloc les bases matemàtiques que permeten coordinar les diferents capes de la divisió OSI dels sistemes i els distints nodes dins la xarxa. Ens referim a les tècniques de descomposició focalitzades en problemes d'optimització convexa, els quals han aportat, durant els últims anys, solucions elegants a molts problemes dins dels camps del processament del senyal i les comunicacions. Revisarem els esquemes coneguts i proposarem una nova metodologia. Acte seguit, es comparen les diferents possibilitats de descomposició, cadascuna de les quals implica diferents maneres d'establir la senyalització. A la pràctica, són aquestes diverses opcions de descomposició les que infereixen les diferents interaccions entre capes o els protocols de control entre elements de la xarxa. Els resultats en quant a nombre d'iteracions requerides per a convergir a la solució òptima són favorables al nou mètode proposat, la qual cosa obra noves línies d'investigació.Finalment, es contribueix també amb dos exemples d'aplicació, en DVB-RCS i en WiMAX. Formulem el problema de gestió de recursos resultant de l'accés múltiple dissenyat per cadascun dels sistemes com un problema de maximització d'utilitat de xarxa (conegut com a NUM en la bibliografia) i el solucionarem aplicant les tècniques anteriors. L'objectiu serà garantir l'equitativitat entre els usuaris i preservar, al mateix temps, la seva qualitat de servei. Per aconseguir-ho, cal seleccionar funcions d'utilitat adequades que permetin balancejar l'assignació de recursos cap als serveis més prioritaris. Mostrarem que en els escenaris considerats, l'ús del mètode proposat comporta guanys significatius ja que requereix menys iteracions en el procés (i per tant, menys senyalització) o bé menys temps de càlcul en un enfoc centralitzat (que es tradueix en la possibilitat d'incloure més usuaris). També es mostren els avantatges de considerar interaccions entre capes, ja que es poden ajustar els paràmetres de capa física per tal d'afavorir els tràfics més prioritaris o bé extreure els requeriments de servei de valors típicament disponibles en capes superiors.En general, la implementació eficient de tècniques de gestió dinàmica de recursos treballant en l'accés múltiple dels sistemes pot aportar guanys significatius però implica establir una bona coordinació entre capes i elements de xarxa. L'eina matemàtica que ho possibilita són les tècniques de descomposició. Cada nou escenari i sistema introdueix un nou repte d'optimització i la capacitat que tinguem de coordinar totes les variables del sistema cap al punt òptim en determinarà el rendiment global. / Traditionally, multiple access schemes in multi-user communications systems have been designed either connection-oriented or traffic-oriented. In the first ones, the goal was to provide as many orthogonal channels as possible, each one serving a different connection. That is the motivation of the so-called FDMA, TDMA and CDMA solutions. On the other hand, random access techniques, which started with the so-called ALOHA protocol, aim to statistically multiplex a shared communication medium by means of exploiting the random and bursty nature of transmission needs in data networks. Most of the multiple access solutions can be interpreted according to that classification or as a combination of those approaches. Notwithstanding, modern systems, such as the digital satellite communications standard DVB-RCS or the broadband wireless access WiMAX, have implemented a multiple access technique where users request for transmission opportunities and receive grants from the network, therefore requiring dynamic bandwidth allocation techniques. The concept of dynamic bandwidth allocation is wide and involves a number of physical and link layer variables, configurations and protocols. In this Ph.D. dissertation we first explore the mathematical foundation that is required to coordinate the distinct layers of the OSI protocol stack and the distinct nodes within the network. We talk about decomposition techniques focused on the resolution of convex programs, which have elegantly solved many problems in the signal processing and communications fields during the last years. Known schemes are reviewed and a novel decomposition methodology is proposed. Thereafter, we compare the four resulting strategies, each one having its own particular signalling needs, which results in distinct cross-layer interactions or signalling protocols at implementation level. The results in terms of iterations required to converge are favourable to the proposed method, thus opening a new line of research.Finally, we contribute with two practical application examples in the DVB-RCS and WiMAX systems. First, we formulate the dynamic bandwidth allocation problem that is derived from the multiple access schemes of both systems. Thereafter, the resulting Network Utility Maximization (NUM) based problem is solved by means of the previous decomposition mechanisms. The goal is to guarantee fairness among the users at the same time that Quality of Service (QoS) is preserved. In order to achieve that, we choose adequate utility functions that allow to balance the allocation towards the most priority traffic flows under a common fairness framework. We show that in the scenarios considered, the novel proposed coupled-decomposition method reports significant gains since it reduces significantly the iterations required (less iterations implies less signalling) or it reduces the time needed to obtain the optimal allocation when it is centrally computed (more users can be managed). We further show the advantages of cross-layer interactions with the physical and upper layers, which allow to benefit from more favourable adjustments of the transmission parameters and to consider the QoS requirements at upper layers. In general, an efficient implementation of dynamic bandwidth allocation techniques in Demand Assignment Multiple Access (DAMA) schemes may report significant performance gains but it requires proper coordination among system layers and network nodes, which is attained thanks to decomposition techniques. Each new scenario and system adds another optimization challenge and, as far as we are able to coordinate all the variables in the system towards that optimal point, the highest will be the revenue.

It is well known that many of the optimization problems which arise in applications are <I>hard</I>, which usually means that they are NP-hard. Hence much research has been devoted to finding <I>good</I> relaxations for these hard problems. Usually a <I>good</I> relaxation is one which can be solved (either exactly or within a prescribed numerical tolerance) in polynomial-time. Nesterov and Nemirovskii showed that by this criterion, many convex optimization problems are good relaxations. This thesis presents new convex relaxations for two such hard problems, namely the Maximum-Cut (Max-Cut) problem and the VLSI (Very Large Scale Integration of electronic circuits) layout problem. We derive and study the properties of two new strengthened semidefinite programming relaxations for the Max-Cut problem. Our theoretical results hold for every instance of Max-Cut; in particular, we make no assumptions about the edge weights. The first relaxation provides a strengthening of the well-known Goemans-Williamson relaxation, and the second relaxation is a further tightening of the first. We prove that the tighter relaxation automatically enforces the well-known triangle inequalities, and in fact is stronger than the simple addition of these inequalities to the Goemans-Williamson relaxation. We further prove that the tighter relaxation fully characterizes some low dimensional faces of the cut polytope via the rank of its feasible matrices. We also address some practical issues arising in the solution of these relaxations and present numerical results showing the remarkably good bounds computed by the tighter relaxation. For the VLSI layout problem, we derive a new relaxation by extending the <I>target distance</I> concept recently introduced by Etawil and Vannelli. The resulting AR (Attractor-Repeller) model improves on the NLT (Non-linear optimization Layout Technique) method of van Camp et al. by efficiently finding a good initial point for the NLT solver. Because the AR model is not a convex optimization problem, we isolate the source of the non-convexity and thereby derive a convex version of the model. This derivation leads to the definition of certain generalized target distances with an interesting practical interpretation. Finally, we present numerical results that demonstrate the potential of the AR model.