Consider a lossy compression system with l distributed encoders and a centralized decoder. Each encoder compresses its observed source and forwards the compressed data to the decoder for joint reconstruction of the target signals under the mean squared error distortion constraint. It is assumed that the observed sources can be expressed as the sum of the target signals and the corruptive noises, which are generated independently from two (possibly di erent) symmetric multivariate Gaussian distributions. Depending on the parameters of such Gaussian distributions, the rate-distortion limit of this lossy compression system is characterized either completely or for a subset of distortions (including, but not necessarily limited to, those su fficiently close to the minimum distortion achievable when the observed sources are directly available at the decoder). The results are further extended to the robust distributed compression setting, where the outputs of a subset of encoders may also be used to produce a non-trivial reconstruction of the corresponding target signals. In particular, we obtain in the high-resolution regime a precise characterization of the minimum achievable reconstruction distortion based on the outputs of k + 1 or more encoders when every k out of all l encoders are operated collectively in the same mode that is greedy in the sense of minimizing the distortion incurred by the reconstruction of the corresponding k target signals with respect to the average rate of these k encoders. / Thesis / Master of Applied Science (MASc)

This work is concerned with the construction of preconditioners for indefinite linear systems. The systems under investigation arise in the numerical solution of quadratic programming problems, for example in the form of Karush--Kuhn--Tucker (KKT) optimality conditions or in interior--point methods. Therefore, the system matrix is referred to as a KKT matrix. It is not the purpose of this thesis to investigate systems arising from general quadratic programming problems, but to study systems arising in linear quadratic control problems governed by partial differential equations. The KKT matrix is symmetric, nonsingular, and indefinite. For the solution of the linear systems generalizations of the conjugate gradient method, MINRES and SYMMLQ, are used. The performance of these iterative solution methods depends on the eigenvalue distribution of the matrix and of the cost of the multiplication of the system matrix with a vector. To increase the performance of these methods, one tries to transform the system to favorably change its eigenvalue distribution. This is called preconditioning and the nonsingular transformation matrices are called preconditioners. Since the overall performance of the iterative methods also depends on the cost of matrix--vector multiplications, the preconditioner has to be constructed so that it can be applied efficiently. The preconditioners designed in this thesis are positive definite and they maintain the symmetry of the system. For the construction of the preconditioners we strongly exploit the structure of the underlying system. The preconditioners are composed of preconditioners for the submatrices in the KKT system. Therefore, known efficient preconditioners can be readily adapted to this context. The derivation of the preconditioners is motivated by the properties of the KKT matrices arising in optimal control problems. An analysis of the preconditioners is given and various cases which are important for interior point methods are treated separately. The preconditioners are tested on a typical problem, a Neumann boundary control for an elliptic equation. In many important situations the preconditioners substantially reduce the number of iterations needed by the solvers. In some cases, it can even be shown that the number of iterations for the preconditioned system is independent of the refinement of the discretization of the partial differential equation. / Master of Science

Neste trabalho é apresentada uma abordagem do Método de Newton associado à função penalidade quadrática e ao método dos conjuntos ativos na solução do problema de Fluxo de Potência Ótimo (FPO). A formulação geral do problema de FPO é apresentada, assim como a técnica utilizada na resolução do sistema de equações. A fatoração da matriz Lagrangeana é feita por elementos ao invés das estruturas em blocos. A característica de esparsidade da matriz Lagrangeana é levada em consideração. Resultados dos testes realizados em 4 sistemas (3, 14, 30 e 118 barras) são apresentados. / This work presents an approach on Newton\'s Method associated with the quadratic penalty function and the active set methods in the solution of Optimal Power Flow Problem (OPF). The general formulation of the OPF problem is presented, as will as the technique used in the equation systems resolution. The Lagrangean matrix factorization is carried out by elements instead of structures in blocks. The characteristic of sparsity of the Lagrangean matrix is taken in to account. Numerical results of tests realized in systems of 3, 14, 30 and 118 buses are presented to show the efficiency of the method.

De nos jours, les grandes masses de données se retrouvent dans de nombreux domaines relatifs aux applications multimédia, sociologiques, biomédicales, radio astronomiques, etc. On parle alors du phénomène ‘Big Data’ qui nécessite le développement d’outils appropriés pour la manipulation et l’analyse appropriée de telles masses de données. Ce travail de thèse est dédié au développement de méthodes efficaces pour la décomposition rapide et adaptative de tenseurs ou matrices de grandes tailles et ce pour l’analyse de données multidimensionnelles. Nous proposons en premier une méthode d’estimation de sous espaces qui s’appuie sur la technique dite ‘divide and conquer’ permettant une estimation distribuée ou parallèle des sous-espaces désirés. Après avoir démontré l’efficacité numérique de cette solution, nous introduisons différentes variantes de celle-ci pour la poursuite adaptative ou bloc des sous espaces principaux ou mineurs ainsi que des vecteurs propres de la matrice de covariance des données. Une application à la suppression d’interférences radiofréquences en radioastronomie a été traitée. La seconde partie du travail a été consacrée aux décompositions rapides de type PARAFAC ou Tucker de tenseurs multidimensionnels. Nous commençons par généraliser l’approche ‘divide and conquer’ précédente au contexte tensoriel et ce en vue de la décomposition PARAFAC parallélisable des tenseurs. Ensuite nous adaptons une technique d’optimisation de type ‘all-at-once’ pour la décomposition robuste (à la méconnaissance des ordres) de tenseurs parcimonieux et non négatifs. Finalement, nous considérons le cas de flux de données continu et proposons deux algorithmes adaptatifs pour la décomposition rapide (à complexité linéaire) de tenseurs en dimension 3. Malgré leurs faibles complexités, ces algorithmes ont des performances similaires (voire parfois supérieures) à celles des méthodes existantes de la littérature. Au final, ce travail aboutit à un ensemble d’outils algorithmiques et algébriques efficaces pour la manipulation et l’analyse de données multidimensionnelles de grandes tailles. / Large volumes of data are being generated at any given time, especially from transactional databases, multimedia content, social media, and applications of sensor networks. When the size of datasets is beyond the ability of typical database software tools to capture, store, manage, and analyze, we face the phenomenon of big data for which new and smarter data analytic tools are required. Big data provides opportunities for new form of data analytics, resulting in substantial productivity. In this thesis, we will explore fast matrix and tensor decompositions as computational tools to process and analyze multidimensional massive-data. We first aim to study fast subspace estimation, a specific technique used in matrix decomposition. Traditional subspace estimation yields high performance but suffers from processing large-scale data. We thus propose distributed/parallel subspace estimation following a divide-and-conquer approach in both batch and adaptive settings. Based on this technique, we further consider its important variants such as principal component analysis, minor and principal subspace tracking and principal eigenvector tracking. We demonstrate the potential of our proposed algorithms by solving the challenging radio frequency interference (RFI) mitigation problem in radio astronomy. In the second part, we concentrate on fast tensor decomposition, a natural extension of the matrix one. We generalize the results for the matrix case to make PARAFAC tensor decomposition parallelizable in batch setting. Then we adapt all-at-once optimization approach to consider sparse non-negative PARAFAC and Tucker decomposition with unknown tensor rank. Finally, we propose two PARAFAC decomposition algorithms for a classof third-order tensors that have one dimension growing linearly with time. The proposed algorithms have linear complexity, good convergence rate and good estimation accuracy. The results in a standard setting show that the performance of our proposed algorithms is comparable or even superior to the state-of-the-art algorithms. We also introduce an adaptive nonnegative PARAFAC problem and refine the solution of adaptive PARAFAC to tackle it. The main contributions of this thesis, as new tools to allow fast handling large-scale multidimensional data, thus bring a step forward real-time applications.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work some classical methods for constrained nonlinear optimization are studied. The mathematical formulations for the optimization problem with equality and inequality constrained, convergence properties and algorithms are presented. Furthermore, optimality conditions of rst order (Karush-Kuhn-Tucker conditions) and of second order. These conditions are essential for the demonstration of many results. Among the methods studied, some techniques transform the original problem into an unconstrained problem (Penalty Methods, Augmented Lagrange Multipliers Method). In others methods, the original problem is modeled as one or as a sequence of quadratic subproblems subject to linear constraints (Quadratic Programming Method, Sequential Quadratic Programming Method). In order to illustrate and compare the performance of the methods studied, two nonlinear optimization problems are considered: a bi-dimensional problem and a problem of mass minimization of a coil spring. The obtained results are analyzed and confronted with each other. / Neste trabalho são estudados alguns métodos clássicos de otimização restrita não linear. São abordadas a formulação matemática para o problema de otimização com restrições de igualdade e desigualdade, propriedades de convergência e algoritmos. Além disso, são relatadas as condições de otimalidade de primeira ordem (condições de Karush-Kuhn-Tucker) e de segunda ordem. Estas condições são essenciais para a demonstração de muitos resultados. Dentre os métodos estudados, algumas técnicas transformam o problema original em um problema irrestrito (Métodos de Penalidade, Método dos Multiplicadores de Lagrange Aumentado). Em outros métodos, o problema original é modelado como um ou uma seqüência de subproblemas quadráticos sujeito _a restrições lineares (Método de Programação Quadrática, Método de Programação Quadrática Seqüencial). A fim de ilustrar e comparar o desempenho dos métodos estudados são considerados dois problemas de otimização não linear: um problema bidimensional e o problema de minimização da massa de uma mola helicoidal. Os resultados obtidos são examinados e confrontados entre si. / Mestre em Matemática

Neste trabalho é apresentada uma abordagem do Método de Newton associado à função penalidade quadrática e ao método dos conjuntos ativos na solução do problema de Fluxo de Potência Ótimo (FPO). A formulação geral do problema de FPO é apresentada, assim como a técnica utilizada na resolução do sistema de equações. A fatoração da matriz Lagrangeana é feita por elementos ao invés das estruturas em blocos. A característica de esparsidade da matriz Lagrangeana é levada em consideração. Resultados dos testes realizados em 4 sistemas (3, 14, 30 e 118 barras) são apresentados. / This work presents an approach on Newton\'s Method associated with the quadratic penalty function and the active set methods in the solution of Optimal Power Flow Problem (OPF). The general formulation of the OPF problem is presented, as will as the technique used in the equation systems resolution. The Lagrangean matrix factorization is carried out by elements instead of structures in blocks. The characteristic of sparsity of the Lagrangean matrix is taken in to account. Numerical results of tests realized in systems of 3, 14, 30 and 118 buses are presented to show the efficiency of the method.

The thesis Racism and New Dimensions of Projecting the Multicultural Experience in Contemporary British Drama analyzes multiculturalism in contemporary Britain and questions its discursive boundaries through the works of some black and Asian contemporary playwrights such as Roy Williams, debbie tucker green and Tanika Gupta. The works of these playwrights articulate a set of experiences that reflects an image of the contemporary issues of bigotry and violence in Britain. Williams, Gupta and green present new approaches on the multicultural Britain concerning the issues of racism, discrimination and knife crime, shedding light on the cruelly racist world from the 'white and black' perspectives. Rethinking the questions of identity, Britishness, social agency and national affiliation from new proportions. The second chapter Roy Williams's Sing Yer Heart Out for the Lads (2002), Sucker Punch (2010) and The No Boys Cricket Club (1996). Williams stages sport in all its complexity as a rich ground for contemplating the issues of racism, belonging, nationalism and identity. He portrays an image of the conflict among the ethnic communities in a multicultural space, highlighting that conflict in its larger context. The third chapter discusses Tanika Gupta's White Boy (2008) and Sugar Mummies (2006). Both of...

Orientador: Valeriano Antunes de Oliveira / Resumo: Neste trabalho, primeiramente, serão apresentados problemas de otimização nos quais a função objetivo é de múltiplas variáveis e de valor intervalar e as restrições de desigualdade são dadas por funcionais clássicos, isto é, de valor real. Serão dadas as condições de otimalidade usando a E−diferenciabilidade e, depois, a gH−diferenciabilidade total das funções com valor intervalar de várias variáveis. As condições necessárias de otimalidade usando a gH−diferenciabilidade total são do tipo KKT e as suficientes são do tipo de convexidade generalizada. Em seguida, serão estabelecidos problemas de controle ótimo nos quais a funçãao objetivo também é com valor intervalar de múltiplas variáveis e as restrições estão na forma de desigualdades e igualdades clássicas. Serão fornecidas as condições de otimalidade usando o conceito de Lipschitz para funções intervalares de várias variáveis e, logo, a gH−diferenciabilidade total das funções com valor intervalar de várias variáveis. As condições necessárias de otimalidade, usando a gH−diferenciabilidade total, estão na forma do célebre Princípio do Máximo de Pontryagin, mas desta vez na versão intervalar. / Abstract: In this work, firstly, it will be presented optimization problems in which the objective function is interval−valued of multiple variables and the inequality constraints are given by classical functionals, that is, real−valued ones. It will be given the optimality conditions using the E−differentiability and then the total gH−differentiability of interval−valued functions of several variables. The necessary optimality conditions using the total gH−differentiability are of KKT−type and the sufficient ones are of generalized convexity type. Next, it will be established optimal control problems in which the objective function is also interval−valued of multiple variables and the constraints are in the form of classical inequalities and equalities. It will be furnished the optimality conditions using the Lipschitz concept for interval−valued functions of several variables and then the total gH−differentiability of interval−valued functions of several variables. The necessary optimality conditions using the total gH−differentiability is in the form of the celebrated local Pontryagin Maximum Principle, but this time in the intervalar version. / Doutor

This thesis focuses on the development of guidance algorithms in order to avoid a prescribed obstacle primarily using the Collision Cone Method (CCM). The Collision Cone Method is a geometric approach to obstacle avoidance, which forms an avoidance zone around the obstacles for the vehicle to pass the obstacle around this zone. The method is reactive as it helps to avoid the pop-up obstacles as well as the known obstacles and local as it passes the obstacles and continue to the prescribed trajectory. The algorithm is first developed for a 2D (planar) avoidance in 3D environment and then extended for 3D scenarios. The algorithm is formed for the optimized CCM as well. The avoidance zone radius and velocity are optimized using constraint optimization, Lagrange multipliers with Karush-Kuhn-Tucker conditions and direct experimentation.

Sattelpunkte und Optimalitätsbedingungen bei restringierten Optimierungsproblemen Ausarbeitung im Rahmen des Seminars "Optimierung", WS 2008/2009 Die Dualitätstheorie für restringierte Optimierungsaufgaben findet in der Spieltheorie und in der Ökonomik eine interessante Anwendung. Mit Hilfe von Sattelpunkteigenschaften werden diverse Interpretationsmöglichkeiten der Lagrange-Dualität vorgestellt. Anschließend gilt das Augenmerk den Optimalitätsbedingungen solcher Probleme. Grundlage für die Ausarbeitung ist das Buch "Convex Optimization" von Stephen Boyd und Lieven Vandenberghe.