Universidade Federal do Cearà / A teoria de controle robusto evoluiu consideravelmente ao longo das Ãltimas dÃcadas, apresentando soluÃÃes para vÃrios tipos de problemas de anÃlise, desempenho e sÃntese de sistemas lineares incertos. As desigualdades matriciais lineares (LMIs) e suas tÃcnicas surgiram como poderosas ferramentas em diversas Ãreas de engenharia de controle para projetos estruturais. Uma propriedade importante das LMIs reside no fato de que o seu conjunto soluÃÃo à convexo. Esta propriedade à fundamental para que se possam formular problemas em controle robusto como sendo problemas de otimizaÃÃo convexa que minimizam uma funÃÃo objetivo. Diante destas afirmaÃÃes o presente trabalho utiliza um sistema de nÃvel de lÃquido com dois tanques interligados como planta onde a mesma foi modelada, e, em seguida, foi desenvolvido um controlador para garantir a sua estabilidade quadrÃtica, quando submetido a perturbaÃÃes externas incertas definidas em um politopo. Utilizou-se o regulador linear quadrÃtico com aÃÃo integral (LQI) como controlador, porÃm, o conceito Ãtimo do LQR nÃo leva em consideraÃÃo as incertezas paramÃtricas existentes nas plantas de projeto, com isso, foi apresentado um mÃtodo de resoluÃÃo do LQR utilizando otimizaÃÃo convexa. O LQR otimizado via LMIs permite a adiÃÃo de incertezas para a obtenÃÃo do ganho de realimentaÃÃo de estado. Os resultados obtidos comprovaram que a estratÃgia de controle LQI via resoluÃÃo LMI à eficaz como controle robusto, pois à capaz de incluir caracterÃsticas referentes à imprecisÃo do processo, alÃm disso, o controle LQI garante a otimalidade do controle. / The robust control theory has evolved considerably over the past decades, providing solutions for various problems of analysis, synthesis and performance of uncertain linear systems. The linear matrix inequalities (LMI) and its techniques have emerged as powerful tools in various areas of control engineering for structural projects. An important property of LMIs is the fact that its solution set is convex. This property is crucial in order to be able to make robust control problems as convex optimization problems that minimize an objective function. Given these statements the present work uses a liquid level system with two tanks connected to the plant where it was modeled, and then a controller is designed to ensure quadratic stability when subjected to external disturbances defined in an uncertain polytope. We used the linear quadratic regulator with integral action (LQI) as a controller, however, the concept of optimal LQR does not take into account the parametric uncertainties in the existing plant design, with it, was presented a method of solving the LQR using convex optimization. LQR optimized via LMI allows the addition of uncertainty to obtain the state feedback gain. The results obtained proved that the strategy of LQI control via LMI resolution is effective as robust control, because it can include features related to the imprecision of the process, moreover, the LQI control ensures the optimality of control.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2015-03-26T12:20:46Z No. of bitstreams: 2 Dissertação - Vando Antônio Adona - 2014.pdf: 1128475 bytes, checksum: a2d00afcaef383726904cf6e6fd3527d (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-03-27T10:48:07Z (GMT) No. of bitstreams: 2 Dissertação - Vando Antônio Adona - 2014.pdf: 1128475 bytes, checksum: a2d00afcaef383726904cf6e6fd3527d (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-03-27T10:48:07Z (GMT). No. of bitstreams: 2 Dissertação - Vando Antônio Adona - 2014.pdf: 1128475 bytes, checksum: a2d00afcaef383726904cf6e6fd3527d (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2014-12-18 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / We consider an optimization problem for which the objective function is the sum of convex functions, not necessarily differentiable. We study a subgradient method that executes the iterations incrementally selecting each component function sequentially and processing the subgradient iteration individually. We analyze different alternatives for choosing the step length, highlighting the convergence properties for each case. We also analyze the incremental model in other methods, considering proximal iteration and combinations of subgradient and proximal iterations. This incremental approach has been very successful when the number of component functions is large. / Consideramos um problema de otimização cuja função objetivo consiste na soma de funções convexas, não necessariamente diferenciáveis. Estudamos um método subgradiente que executa a iteração de forma incremental, selecionando cada função componente de maneira sequencial e processando a iteração subgradiente individualmente. Analisamos diferentes alternativas para a escolha do comprimento de passo, destacando as propriedades de convergência para cada caso. Abordamos também o modelo incremental em outros métodos, considerando iteração proximal e combinações de iterações subgradiente e proximal. Esta abordagem incremental tem sido muito bem sucedida quando o número de funções componentes é grande.

Traditionally, linear programming (LP) has been used to construct convex relaxations in the context of branch and bound for determining global optimal solutions to nonconvex optimization problems. As second-order cone programming (SOCP) and semidefinite programming (SDP) become better understood by optimization researchers, they become alternative choices for obtaining convex relaxations and producing bounds on the optimal values. In this thesis, we study the use of these convex optimization tools in constructing strong relaxations for several nonconvex problems, including 0-1 integer programming, nonconvex box-constrained quadratic programming (BoxQP), and general quadratic programming (QP). We first study a SOCP relaxation for 0-1 integer programs and a sequential relaxation technique based on this SOCP relaxation. We present desirable properties of this SOCP relaxation, for example, this relaxation cuts off all fractional extreme points of the regular LP relaxation. We further prove that the sequential relaxation technique generates the convex hull of 0-1 solutions asymptotically. We next explore nonconvex quadratic programming. We propose a SDP relaxation for BoxQP based on relaxing the first- and second-order KKT conditions, where the difficulty and contribution lie in relaxing the second-order KKT condition. We show that, although the relaxation we obtain this way is equivalent to an existing SDP relaxation at the root node, it is significantly stronger on the children nodes in a branch-and-bound setting. New advance in optimization theory allows one to express QP as optimizing a linear function over the convex cone of completely positive matrices subject to linear constraints, referred to as completely positive programming (CPP). CPP naturally admits strong semidefinite relaxations. We incorporate the first-order KKT conditions of QP into the constraints of QP, and then pose it in the form of CPP to obtain a strong relaxation. We employ the resulting SDP relaxation inside a finite branch-and-bound algorithm to solve the QP. Comparison of our algorithm with commercial global solvers shows potential as well as room for improvement. The remainder is devoted to new techniques for solving a class of large-scale linear programming problems. First order methods, although not as fast as second-order methods, are extremely memory efficient. We develop a first-order method based on Nesterov's smoothing technique and demonstrate the effectiveness of our method on two machine learning problems.

Cette thèse porte sur l’étude des méthodes inertielles pour résoudre les problèmes de minimisation convexe structurés. Depuis les premiers travaux de Polyak et Nesterov, ces méthodes sont devenues très populaires, grâce à leurs effets d’accélération. Dans ce travail, on étudie une famille d’algorithmes de gradient proximal inertiel de type Nesterov avec un choix spécifique de suites de sur-relaxation. Les différentes propriétés de convergence de cette famille d’algorithmes sont présentées d’une manière unifiée, en fonction du paramètre de sur-relaxation. En outre, on étudie ces propriétés, dans le cas des fonctions lisses vérifiant des hypothèses géométriques supplémentaires, comme la condition de croissance (ou condition de Łojasiewicz). On montre qu’en combinant cette condition de croissance avec une condition de planéité (flatness) sur la géométrie de la fonction minimisante, on obtient de nouveaux taux de convergence. La stratégie adoptée ici, utilise des analogies du continu vers le discret, en passant des systèmes dynamiques continus en temps à des schémas discrets. En particulier, la famille d’algorithmes inertiels qui nous intéresse, peut être identifiée comme un schéma aux différences finies d’une équation/inclusion différentielle. Cette approche donne les grandes lignes d’une façon de transposer les différents résultats et leurs démonstrations du continu au discret. Cela ouvre la voie à de nouveaux schémas inertiels possibles, issus du même système dynamique. / This Thesis focuses on the study of inertial methods for solving composite convex minimization problems. Since the early works of Polyak and Nesterov, inertial methods become very popular, thanks to their acceleration effects. Here, we study a family of Nesterov-type inertial proximalgradient algorithms with a particular over-relaxation sequence. We give a unified presentation about the different convergence properties of this family of algorithms, depending on the over-relaxation parameter. In addition we addressing this issue, in the case of a smooth function with additional geometrical structure, such as the growth (or Łojasiewicz) condition. We show that by combining growth condition and a flatness-type condition on the geometry of the minimizing function, we are able to obtain some new convergence rates. Our analysis follows a continuous-to-discrete trail, passing from continuous-on time-dynamical systems to discrete schemes. In particular the family of inertial algorithms that interest us, can be identified as a finite difference scheme of a differential equation/inclusion. This approach provides a useful guideline, which permits to transpose the different results and their proofs from the continuous system to the discrete one. This opens the way for new possible inertial schemes, derived by the same dynamical system.

Cognitive Radio technologies have been touted to be instrumental in solving resource-allocation problems in resource-constrained radio environments. The adaptive computational intelligence of these radios facilitates the dynamic allocation of network resources--particularly, the spectrum, a scarce physical asset. In addition to consumer-driven innovation that is governing the wireless communication ecosystem, its associated infrastructure is being increasingly viewed by governments around the world as critical national security interests--the US Military instituted the DARPA Spectrum Collaboration Challenge which requires competitors to design intelligent radios that leverage optimization, A.I., and game-theoretic strategies in order to efficiently access the RF spectrum in an environment wherein every other competitor is vying for the same limited resources. In this work, we detail the design of our radio, i.e., the design choices made in each layer of the network protocol stack, strategies rigorously derived from convex optimization, the collaboration API, and heuristics tailor-made to tackle the unique scenarios emulated in this DARPA Grand Challenge. We present performance evaluations of key components of our radio in a variety of military and disaster-relief deployment scenarios that mimic similar real-world situations. Furthermore, specifically focusing on channel access in the MAC, we formulate the spectrum sensing and access problem as a POMDP; derive an optimal policy using approximate value iteration methods; prove that our strategy outperforms the state-of-the-art, and facilitates means to control the trade-off between secondary network throughput and incumbent interference; and evaluate this policy on an ad-hoc distributed wireless platform constituting ESP32 radios, in order to study its implementation feasibility.

Tato diplomová práce se zabývá tématikou konvexní optimalizace a to konkrétně modifikacemi algoritmu ADMM, společně s problematikou proximálních operátorů. Jedna z verzí ADMM je pak implementována v programovacím jazyce Julia s důrazem na obecnost a efektivnost této implementace, a dále aplikována na rozsáhlou úlohu z oblasti odpadového hospodářství.

La modélisation de la langue naturelle est l¿un des défis fondamentaux de l¿intelligence artificielle et de la conception de systèmes interactifs, avec applications dans les systèmes de dialogue, la génération de texte et la traduction automatique. Nous proposons un modèle log-linéaire discriminatif donnant la distribution des mots qui suivent un contexte donné. En raison de la parcimonie des données, nous proposons un terme de pénalité qui code correctement la structure de l¿espace fonctionnel pour éviter le sur-apprentissage et d¿améliorer la généralisation, tout en capturant de manière appropriée les dépendances à long terme. Le résultat est un modèle efficace qui capte suffisamment les dépendances longues sans occasionner une forte augmentation des ressources en espace ou en temps. Dans un modèle log-linéaire, les phases d¿apprentissage et de tests deviennent de plus en plus chères avec un nombre croissant de classes. Le nombre de classes dans un modèle de langue est la taille du vocabulaire, qui est généralement très importante. Une astuce courante consiste à appliquer le modèle en deux étapes: la première étape identifie le cluster le plus probable et la seconde prend le mot le plus probable du cluster choisi. Cette idée peut être généralisée à une hiérarchie de plus grande profondeur avec plusieurs niveaux de regroupement. Cependant, la performance du système de classification hiérarchique qui en résulte dépend du domaine d¿application et de la construction d¿une bonne hiérarchie. Nous étudions différentes stratégies pour construire la hiérarchie des catégories de leurs observations. / Modeling natural language is among fundamental challenges of artificial intelligence and the design of interactive machines, with applications spanning across various domains, such as dialogue systems, text generation and machine translation. We propose a discriminatively trained log-linear model to learn the distribution of words following a given context. Due to data sparsity, it is necessary to appropriately regularize the model using a penalty term. We design a penalty term that properly encodes the structure of the feature space to avoid overfitting and improve generalization while appropriately capturing long range dependencies. Some nice properties of specific structured penalties can be used to reduce the number of parameters required to encode the model. The outcome is an efficient model that suitably captures long dependencies in language without a significant increase in time or space requirements. In a log-linear model, both training and testing become increasingly expensive with growing number of classes. The number of classes in a language model is the size of the vocabulary which is typically very large. A common trick is to cluster classes and apply the model in two-steps; the first step picks the most probable cluster and the second picks the most probable word from the chosen cluster. This idea can be generalized to a hierarchy of larger depth with multiple levels of clustering. However, the performance of the resulting hierarchical classifier depends on the suitability of the clustering to the problem. We study different strategies to build the hierarchy of categories from their observations.

No description available.

Optimization problems are ubiquitous in all quantitative scientific disciplines, from computer science and engineering to operations research and economics. Developing algorithms for solving various optimization problems has been the focus of mathematical research for years. In the last decade, optimization research has become even more popular due to its applications in the rapidly developing field of machine learning. In this thesis, we discuss a few fundamental and well-studied optimization problem classes: decentralized distributed optimization (Chapters 2 to 4), distributed optimization under similarity (Chapter 5), affinely constrained optimization (Chapter 6), minimax optimization (Chapter 7), and high-order optimization (Chapter 8). For each problem class, we develop the first provably optimal algorithm: the complexity of such an algorithm cannot be improved for the problem class given. The proposed algorithms show state-of-the-art performance in practical applications, which makes them highly attractive for potential generalizations and extensions in the future.

This research addresses the concept of radio resource allocation for cellular communications systems operating in congested and contested environments with an emphasis on end-to-end quality of service (QoS). The radio resource allocation is cast under a proportional fairness formulation which translates to a convex optimization problem. Moreover, the resource allocation scheme considers subscription-based and traffic differentiation in order to meet the QoS requirements of the applications running on the user equipment in the system. The devised resource allocation scheme is realized through a centralized and a distributed architecture and solution algorithms for the aforementioned architectures is derived and implemented in the mobile devices and the base stations. The sensitivity of the resource allocation scheme to the temporal dynamics of the quantity of the users in the system is investigated. Furthermore, the sensitivity of the resource allocation scheme to the temporal dynamics in the application usage percentages is accounted for. In addition, a transmission overhead of the centralized and distributed architectures for the resource allocation schemes is performed. Furthermore, the resource allocation scheme is modified to account for a possible additive bandwidth done through spectrum sharing in congested and contested environments, in particular spectrally coexistent radar systems. The radar-spectrum additive portion is devised in a way to ensure fairness of the allocation, high bandwidth utilization, and interference avoidance. In order to justify the aforesaid modification, the interference from radar systems into the Long Term Evolution (LTE) as the predominant 4G technology is studies to confirm the possibility of the spectrum sharing. The preceding interference analysis contains a detailed simulation of radar systems, propagation path loss models, and a third generation partnership project compliant LTE system. The propagation models are Free Space Path Loss (FSPL) and Irregular Terrain Model (ITM). The LTE systems under consideration are macro cell, outdoor small cells, and indoor small cells. Furthermore, the resource allocation under channel consideration is formalized such that the resources are allocated under a congested environment and based on the quality of channel the users have in the network as well as the quality of service requirements of the applications running on the mobile devices. / Ph. D.