Esta tese é um estudo acerca da análise de convergência de vários métodos numéricos de primeira e de segunda ordem para resolver problemas de programação matemática e as condições de otimalidade associadas. Nossas principais ferramentas são as condições sequenciais de otimalidade. As condições sequenciais de otimalidade oferecem um quadro teórico para a análise de convergência para várias famílias de métodos de primeira ordem sob condições de qualificações fracas. Nesta tese, apresentamos, para cada condição sequencial de otimalidade, a condição de qualificação mínima associada e mostramos as relações com outras condições de qualificação conhecidas. Este fato tem implicações práticas, uma vez que enfraquece as hipóteses requeridas para a convergência de vários métodos numéricos cujos critérios de paradas estão associados às condições sequenciais de otimalidade. Ainda mais, esse tipo de resultado não pode ser melhorado usando outras condições de qualificações. Nós estendemos a noção de condições sequenciais de otimalidade de primeira ordem, para incorporar informações de segunda ordem. Apresentamos, segundo nosso conhecimento, a primeira condição sequencial de otimalidade de segunda ordem, adequada para a análise de convergência de vários métodos numéricos com convergência a pontos estacionários de segunda ordem, como por exemplo métodos baseados no Lagrangeano aumentado, regiões de confiança e SQP regularizado. Associada com a nova condição sequencial de segunda ordem, temos uma nova condição de qualificação, mais fraca que as outras condições de qualificações utilizadas para a análise de convergência para métodos numéricos de segunda ordem. Nós situamos essa nova condição de qualificação com respeito a outras condições de qualificação usadas em análise de convergência. Finalmente apresentamos outra razão pela qual a condição fraca necessária de segunda ordem é a condição de segunda ordem adequada quando lidarmos com a convergência de algoritmos práticos / This thesis deals with the convergence analysis for several rst-and-second-order numerical methods used to solve mathematical programming problems. Our main tools are the sequential optimality conditions. First-order sequential optimality conditions oer a framework to the study of the convergence analysis of several families of rst-order methods, under weak constraint qualications. In this thesis, we will introduce, for each sequential optimality condition the minimal constraint qualications associated with it and we will show their relationships with other constraint qualications. This fact has a practical aspect, since, we improve the convergence analysis of practical methods with stopping criteria associated with sequential optimality conditions. This results can not be improved by using another weak constraint qualications. We will extend the notion of rst-order sequential optimality conditions to incorporate secondorder information. We will introduce, to the best of our knowledge, the rst second-order sequential optimality condition, suitable to the study of the convergence analysis of several second-order methods including methods based on the augmented lagrangian, trust-region and regularized SQP. Associated with the second-order sequential optimality condition, we have a new constraint qualication weaker than all constraint qualications used for the convergence analysis of second-order methods. We show the relationships of this new constraint qualications with other constraint qualications used for algorithmic purposes. We will also present a new reason why the weak secondorder necessary condition is the natural second-order condition when we are dealing with practical numerical methods

Esta tese é um estudo acerca da análise de convergência de vários métodos numéricos de primeira e de segunda ordem para resolver problemas de programação matemática e as condições de otimalidade associadas. Nossas principais ferramentas são as condições sequenciais de otimalidade. As condições sequenciais de otimalidade oferecem um quadro teórico para a análise de convergência para várias famílias de métodos de primeira ordem sob condições de qualificações fracas. Nesta tese, apresentamos, para cada condição sequencial de otimalidade, a condição de qualificação mínima associada e mostramos as relações com outras condições de qualificação conhecidas. Este fato tem implicações práticas, uma vez que enfraquece as hipóteses requeridas para a convergência de vários métodos numéricos cujos critérios de paradas estão associados às condições sequenciais de otimalidade. Ainda mais, esse tipo de resultado não pode ser melhorado usando outras condições de qualificações. Nós estendemos a noção de condições sequenciais de otimalidade de primeira ordem, para incorporar informações de segunda ordem. Apresentamos, segundo nosso conhecimento, a primeira condição sequencial de otimalidade de segunda ordem, adequada para a análise de convergência de vários métodos numéricos com convergência a pontos estacionários de segunda ordem, como por exemplo métodos baseados no Lagrangeano aumentado, regiões de confiança e SQP regularizado. Associada com a nova condição sequencial de segunda ordem, temos uma nova condição de qualificação, mais fraca que as outras condições de qualificações utilizadas para a análise de convergência para métodos numéricos de segunda ordem. Nós situamos essa nova condição de qualificação com respeito a outras condições de qualificação usadas em análise de convergência. Finalmente apresentamos outra razão pela qual a condição fraca necessária de segunda ordem é a condição de segunda ordem adequada quando lidarmos com a convergência de algoritmos práticos / This thesis deals with the convergence analysis for several rst-and-second-order numerical methods used to solve mathematical programming problems. Our main tools are the sequential optimality conditions. First-order sequential optimality conditions oer a framework to the study of the convergence analysis of several families of rst-order methods, under weak constraint qualications. In this thesis, we will introduce, for each sequential optimality condition the minimal constraint qualications associated with it and we will show their relationships with other constraint qualications. This fact has a practical aspect, since, we improve the convergence analysis of practical methods with stopping criteria associated with sequential optimality conditions. This results can not be improved by using another weak constraint qualications. We will extend the notion of rst-order sequential optimality conditions to incorporate secondorder information. We will introduce, to the best of our knowledge, the rst second-order sequential optimality condition, suitable to the study of the convergence analysis of several second-order methods including methods based on the augmented lagrangian, trust-region and regularized SQP. Associated with the second-order sequential optimality condition, we have a new constraint qualication weaker than all constraint qualications used for the convergence analysis of second-order methods. We show the relationships of this new constraint qualications with other constraint qualications used for algorithmic purposes. We will also present a new reason why the weak secondorder necessary condition is the natural second-order condition when we are dealing with practical numerical methods

Distribution system plays a significant role in the overall electrical power system due to its impact on electricity costs, reliability as well as security of supplied energy. Optimal development planning of modern distribution system is mainly required to satisfy continuous change in customer demands and generations in a cost-effective manner, utilizing the available smart solutions. All these aspects need to be addressed in modern distribution planning methodology that can be applied today in real-life. Review has shown that there are no distributions planning models that adequately model security of supply of radially operated networks. Moreover, the optimal development planning models still do not consider multiple operating regimes, which has become a necessity due to connection of low carbon technologies. Numerous techniques published on this subject tend to ignore the regulations and planning standards that must be complied during system development, resulting in methodology that is not in-tuned with business practices. Furthermore, a comprehensive model that integrates all major components of today’s real-life distribution planning is still lacking, even though many of them have been addressed individually. In this thesis, integrated planning methodology for development of distribution system is proposed, incorporating utility practices in the UK. The overall methodology built on two independent stages, investment stage and operation stage. The operation stage is further cast into two sub-stages, quality of supply planning and minimization of operation costs planning. The overall planning methodology incorporates the novel probabilistic decision tree concept for distribution system planning to consider probable network uncertainties. The first model which is the investment stage determines the new construction and reinforcement of circuits and switchgear, along with circuit decommissioning. Multiple operating regimes due to fluctuation in generation and load profiles are considered, in addition to explicit modelling of N-1 security constraint according to P2/6 planning standards. The quality of supply planning determines the allocation of switchgear and its automation to maximise the reliability benefits from the regulatory incentive regime. Finally, the operation model determines the optimal network configuration that minimises the total operation costs of distribution system. The final outputs are list of cables and switchgear for construction, reinforcement, and decommission, benefits harvested due to quality of supply investments on switchgear, optimal network running arrangement, etc. These studies have proven to be important in formulating effective strategies for development of distribution system, in compliance to the planning standards and resulted in higher network operation capabilities.

O presente trabalho possui como metas o estudo, simulação, identificação de parâmetros e comparação estatística de modelos compartimentais utilizados em tomografia por emissão de pósitrons (PET). Para tanto, propõe-se utilizar a metodologia de equações de sensibilidade e o método de Levenberg-Marquardt para a tarefa de estimação de parâmetros característicos das equações diferenciais descritoras dos referidos sistemas. Para comparação entre modelos, foi empregado o critério de informação de Akaike. São consideradas três estruturas compartimentais compostas, respectivamente, por dois compartimentos e duas constantes características, três compartimentos e quatro constantes características e quatro compartimentos e seis constantes características. Os dados considerados neste texto foram sintetizados preocupando-se em reunir as principais características de um exame de tomografia real, tais como tipo e nível de ruído e morfologia de função de excitação do sistema. Para tanto, foram utilizados exames de pacientes do setor de Medicina Nuclear do Instituto do Coração da Faculdade de Medicina da Universidade de São Paulo. Aplicando-se a metodologia proposta em três níveis de ruído (baixo, médio e alto), obteve-se concordância do melhor modelo em graus forte e considerável (com índices de Kappa iguais a 0.95, 0.93 e 0.63, respectivamente). Observou-se que, com elevado nível de ruído e modelos mais complexos (quatro compartimentos), a classificação se deteriora devido ao pequeno número de dados para a decisão. Foram desenvolvidos programas e uma interface gráfica que podem ser utilizadas na investigação, elaboração, simulação e identificação de parâmetros de modelos compartimentais para apoio e análise de diagnósticos clínicos e práticas científicas. / This work has as goals the study, simulation, parameter identification and statistical comparison of compartmental models used in positron emission tomography (PET). We propose to use the methodology of sensitivity equations and the method of Levenberg-Marquardt for the task of estimating the characteristic parameters of the differential equations describing such systems. For model comparison, Akaikes information criterion is applied. We have considered three compartmental structures represented, respectively, by two compartments and two characteristic constants, three compartments and four characteristic constants and four compartments and six characteristics constants. The data considered in this work were synthesized taking into account key features of a real tomography exam, such as type and level of noise and morphology of the input function of the system. To this end, we used tests of patients in the sector of Nuclear Medicine of the Heart Institute of the Faculty of Medicine, University of São Paulo. Applying the proposed methodology with three noise levels (low, medium and high), we obtained agreement of the best model with strong and considerable degrees (with Kappa indexes equal to 0.95, 0.93 and 0.63, respectively). It was observed that, with high noise level and more complex models (four compartments), the classification is deteriorated due to lack of data for the decision. Programs have been developed and a graphical interface that can be used in research, development, simulation and parameter identification of compartmental models, supporting analysis of clinical diagnostics and scientific practices.

Neste estudo, consideramos um problema transiente de fluxo unidimensional vertical de água em meio poroso insaturado, modelado pela equação Richards não-linear. As reações constitutivas de Van Genuchten são empregadas para representar a capacidade hidráulica e a condutividade. A fórmula da solução é otimizada e avaliada usando a equação governante em um critério de autoconsciente. Os resultados são apresentados para alguns tipos de solo e seus parâmetros relacionados, que são mencionados em literatura. / In this study, we consider a transiente vertical one-dimensional flow problem of water in unsaturated porus media, modelled by the non-linear Richards equation. Constitutive relations of Van Genutchten are employed to represent the hydraulic capacity and conductivity. The solution formula is optimized and evaluated using to governing equation for a self-consistency criterion. The results are presented for some oil types and its related soil parameters, that are reported in the literature.

This thesis concerns the development of novel feasible direction type algorithms for constrained nonlinear optimization. The new algorithms are based upon enhancements of the search direction determination and the line search steps. The Frank-Wolfe method is popular for solving certain structured linearly constrained nonlinear problems, although its rate of convergence is often poor. We develop improved Frank--Wolfe type algorithms based on conjugate directions. In the conjugate direction Frank-Wolfe method a line search is performed along a direction which is conjugate to the previous one with respect to the Hessian matrix of the objective. A further refinement of this method is derived by applying conjugation with respect to the last two directions, instead of only the last one. The new methods are applied to the single-class user traffic equilibrium problem, the multi-class user traffic equilibrium problem under social marginal cost pricing, and the stochastic transportation problem. In a limited set of computational tests the algorithms turn out to be quite efficient. Additionally, a feasible direction method with multi-dimensional search for the stochastic transportation problem is developed. We also derive a novel sequential linear programming algorithm for general constrained nonlinear optimization problems, with the intention of being able to attack problems with large numbers of variables and constraints. The algorithm is based on inner approximations of both the primal and the dual spaces, which yields a method combining column and constraint generation in the primal space. / The articles are note published due to copyright rextrictions.

Recent developments show that derivative free methods are highly demanded by researches for solving optimization problems in various practical contexts. Although well-known optimization methods that employ derivative information can be very effcient, a derivative free method will be more effcient in cases where the objective function is nondifferentiable, the derivative information is not available or is not reliable. Derivative Free Optimization (DFO) is developed for solving small dimensional problems (less than 100 variables) in which the computation of an objective function is relatively expensive and the derivatives of the objective function are not available. Problems of this nature more and more arise in modern physical, chemical and econometric measurements and in engineering applications, where computer simulation is employed for the evaluation of the objective functions. In this thesis, we give an example of the implementation of DFO in an approach for optimizing stirrer configurations, including a parametrized grid generator, a flow solver, and DFO. A derivative free method, i.e., DFO is preferred because the gradient of the objective function with respect to the stirrer&rsquo / s design variables is not directly available. This nonlinear objective function is obtained from the flow field by the flow solver. We present and interpret numerical results of this implementation. Moreover, a contribution is given to a survey and a distinction of DFO research directions, to an analysis and discussion of these. We also state a derivative free algorithm used within a clustering algorithm in combination with non-smooth optimization techniques to reveal the effectiveness of derivative free methods in computations. This algorithm is applied on some data sets from various sources of public life and medicine. We compare various methods, their practical backgrounds, and conclude with a summary and outlook. This work may serve as a preparation of possible future research.

Le problème de la gestion éco-responsable du trafic urbain est adressé. Ce type de gestion du trafic vise à réduire les arrêts des véhicules, les accélérations, la consommation énergétique, ainsi que la congestion. L'éco-management du trafic dans les réseaux urbains peut être catégorisé dans deux classes principales : contrôle du véhicule et contrôle de l'infrastructure. Les deux domaines de contrôle peuvent présenter caractéristiques soit isolées soit coordonnées, en dépendant du type d'information utilisée dans l'optimisation.La gestion du trafic côté véhicule influe sur chaque véhicule en fonction de ses propres caractéristiques et position. Le contrôle isolé du véhicule vise principalement à optimiser la transmission et/ou le profil de conduite des véhicules, en utilisant éventuellement des informations sur les caractéristiques de la route, mais sans communiquer avec les autres agents du réseau. Le contrôle coordonné du véhicule, d'autre part, fait usage de la communication entre les véhicules et avec l'infrastructure pour obtenir des bénéfices plus importants en termes de consommation d'énergie et de fluidité de la circulation.En revanche, la gestion du côté infrastructure influe sur les feux et les panneaux de signalisation, afin d'améliorer les performances de l'ensemble du trafic. Le contrôle isolé de l'infrastructure régule essentiellement les feux de signalisation pour une seule intersection, ou bien les limites de vitesse dans un seul tronçon de route, sans prendre en compte les interactions avec les jonctions et/ou les sections voisines. Le contrôle coordonné de l'infrastructure surmonte cette limitation en utilisant des informations sur les conditions de circulation dans d'autres sections de la route, afin de réduire la congestion.Les contributions de ce travail peuvent être résumées comme suit.Tout d'abord, une solution pour le contrôle coordonné du véhicule a été proposée, dans laquelle la communication avec l'infrastructure est exploitée pour réduire la consommation d'énergie. En particulier, les plans des feux de signalisation sont supposés être communiqués au véhicule et connus, et une vitesse optimale est suggérée au véhicule afin de traverser une séquence de carrefours à feux sans s'arrêter, tout en suivant une trajectoire d'énergie minimale. La stratégie proposée, appliquée indépendamment à chaque véhicule, a été testée dans un simulateur de trafic microscopique afin d'évaluer l'impact sur les performances du trafic. L'analyse a montré que la consommation d'énergie et le nombre d'arrêts peuvent être considérablement réduits sans affecter le temps de parcours.Ensuite, une solution pour le contrôle isolé de l'infrastructure a été proposée. Un modèle macroscopique du trafic urbain a été introduit, et les limites de vitesse variables ont été utilisées pour améliorer les performances de la circulation. L'optimisation vise à trouver un compromis entre la réduction de consommation énergétique et le temps de parcours moyen des véhicules dans le tronçon de route considéré. Des expériences ont démontré qu'il existe une limite de vitesse optimale qui améliore les performances du trafic, et qui réduit la longueur de la file d'attente au feu de signalisation.Enfin, une solution pour le contrôle coordonné de l'infrastructure a été proposée. La synchronisation des feux de signalisation sur les grands axes de circulation a été prouvée efficace pour réduire le temps de parcours. Notre analyse a démontré qu'un problème d'optimisation peut être formalisé pour prendre en compte également les aspects énergétiques. Des expériences approfondies dans un simulateur de trafic microscopique ont montré qu'il existe une corrélation entre la progression du trafic et ses performances. La stratégie de contrôle proposée a montré qu'une réduction significative de la consommation d'énergie peut être atteinte, en éliminant presque complètement les arrêts et le temps d'arrêt, sans affecter le temps de parcours. / The problem of energy-aware traffic management in urban environment is addressed. Such traffic management aims at reducing vehicle stops, accelerations, energy consumption, and ultimately congestion. The eco-management in urban traffic networks may be divided in two broad categories: vehicle-side control and infrastructure-side control. Both control domains can feature isolated or coordinated characteristics, depending on the type of information used in the optimization.The vehicle-side traffic management influences each single vehicle according to its own characteristics and position. Isolated vehicle control aims primarily at optimizing the powertrain and/or the driving profile of the vehicles, possibly using information about the road characteristics, but without communicating with the other agents of the traffic network. Coordinated vehicle control makes use of communication among vehicles and with the infrastructure in order to achieve larger benefits in terms of energy consumption and traffic fluidity.The infrastructure-side management, on the other hand, influences traffic lights and road side panels in order to improve the performance of the traffic as a whole. Isolated infrastructure control regulates essentially the traffic lights at a single signalized intersection, or the speed limits in a single stretch of road, without taking into account the interactions with the neighboring junctions and/or road sections. Coordinated infrastructure control overcomes this limitation by using information about traffic conditions in other road sections to alleviate congestion.The contributions of this work to the energy-aware traffic management may be summarized as follows.Firstly, a solution for the coordinated vehicle control has been proposed, in which communication with the infrastructure is exploited to reduce energy consumption. In particular, the traffic lights timings are assumed to be communicated to the vehicle and known, and the vehicle is suggested an optimal speed to drive through a sequence of signalized intersections without stopping, while following a minimum-energy trajectory. The proposed strategy, independently applied to each vehicle, has been tested in a microscopic traffic simulator in order to assess the impact on the traffic performance. The analysis has demonstrated that the energy consumption and the number of stops can be drastically reduced without affecting the travel time.Then, a solution for the isolated infrastructure control has been proposed. A macroscopic urban traffic model has been introduced, and the variable speed limits have been used as actuation to improve traffic performance. In particular, the analysis has been carried out at saturated traffic conditions, with given and fixed traffic lights scheduling. The optimization aims at reducing the energy consumption in trade-off with the average travel time of the vehicles in the considered road section. Experiments have demonstrated that there exists an optimal speed limit that improves traffic performance and reduces the length of the queue at the traffic light.Lastly, a solution for the coordinated infrastructure control has been proposed. Traffic lights coordination on arterials has been proved to be effective in terms of traffic delay reduction. Our analysis has demonstrated that an optimization problem can be cast to take into account also energetic aspects. Extensive experiments in a microscopic traffic simulator have showed that a correlation exists between traffic progression and traffic performance indexes, such as energy consumption, travel time, idling time, and number of stops. The proposed control strategy has showed that a significant reduction of energy consumption can be achieved, almost completely eliminating number of stops and idling time, without affecting the travel time.

Bundle adjustment is the process of simultaneously optimizing camera poses and 3D structure given image point tracks. In structure-from-motion, it is typically used as the final refinement step due to the nonlinearity of the problem, meaning that it requires sufficiently good initialization. Contrary to this belief, recent literature showed that useful solutions can be obtained even from arbitrary initialization for fixed-rank matrix factorization problems, including bundle adjustment with affine cameras. This property of wide convergence basin of high quality optima is desirable for any nonlinear optimization algorithm since obtaining good initial values can often be non-trivial. The aim of this thesis is to find the key factor behind the success of these recent matrix factorization algorithms and explore the potential applicability of the findings to bundle adjustment, which is closely related to matrix factorization. The thesis begins by unifying a handful of matrix factorization algorithms and comparing similarities and differences between them. The theoretical analysis shows that the set of successful algorithms actually stems from the same root of the optimization method called variable projection (VarPro). The investigation then extends to address why VarPro outperforms the joint optimization technique, which is widely used in computer vision. This algorithmic comparison of these methods yields a larger unification, leading to a conclusion that VarPro benefits from an unequal trust region assumption between two matrix factors. The thesis then explores ways to incorporate VarPro to bundle adjustment problems using projective and perspective cameras. Unfortunately, the added nonlinearity causes a substantial decrease in the convergence basin of VarPro, and therefore a bootstrapping strategy is proposed to bypass this issue. Experimental results show that it is possible to yield feasible metric reconstructions and pose estimations from arbitrary initialization given relatively clean point tracks, taking one step towards initialization-free structure-from-motion.

Neste estudo, consideramos um problema transiente de fluxo unidimensional vertical de água em meio poroso insaturado, modelado pela equação Richards não-linear. As reações constitutivas de Van Genuchten são empregadas para representar a capacidade hidráulica e a condutividade. A fórmula da solução é otimizada e avaliada usando a equação governante em um critério de autoconsciente. Os resultados são apresentados para alguns tipos de solo e seus parâmetros relacionados, que são mencionados em literatura. / In this study, we consider a transiente vertical one-dimensional flow problem of water in unsaturated porus media, modelled by the non-linear Richards equation. Constitutive relations of Van Genutchten are employed to represent the hydraulic capacity and conductivity. The solution formula is optimized and evaluated using to governing equation for a self-consistency criterion. The results are presented for some oil types and its related soil parameters, that are reported in the literature.