L'objectif de cette thèse consiste à présenter un nouvel algorithme pour la programmation non linéaire en nombres entiers, inspirée par la méthode Multiplicative Weights Update et qui compte sur une nouvelle classe de reformulations, appelées les reformulations ponctuelles.La programmation non linéaire en nombres entiers est un sujet très difficile et fascinant dans le domaine de l'optimisation mathématique à la fois d'un point de vue théorique et computationnel. Il est possible de formuler de nombreux problèmes dans ce schéma général et, habituellement, ils posent de réels défis en termes d'efficacité et de précision de la solution obtenue quant aux procédures de résolution.La thèse est divisée en trois parties principales : une introduction composée par le Chapitre 1, une définition théorique du nouvel algorithme dans le Chapitre 2 et l'application de cette nouvelle méthodologie à deux problèmes concrets d'optimisation, tels que la sélection optimale du portefeuille avec le critère moyenne-variance dans le Chapitre 3 et le problème du sac à dos non linéaire dans le Chapitre 4. Conclusions et questions ouvertes sont présentées dans le Chapitre 5. / This thesis presents a new algorithm for Mixed Integer NonLinear Programming, inspired by the Multiplicative Weights Update framework and relying on a new class of reformulations, called the pointwise reformulations.Mixed Integer NonLinear Programming is a hard and fascinating topic in Mathematical Optimization both from a theoretical and a computational viewpoint. Many real-word problems can be cast this general scheme and, usually, are quite challenging in terms of efficiency and solution accuracy with respect to the solving procedures.The thesis is divided in three main parts: a foreword consisting in Chapter 1, a theoretical foundation of the new algorithm in Chapter 2, and the application of this new methodology to two real-world optimization problems, namely the Mean-Variance Portfolio Selection in Chapter 3, and the Multiple NonLinear Separable Knapsack Problem in Chapter 4. Conclusions and open questions are drawn in Chapter 5.

[pt] Otimização Robusta é uma das formas mais comuns de considerar in- certeza nos parâmetros de um problema de otimização. A forma tradicional de achar soluções robustas consiste em resolver a contraparte robusta de um problema, o que em muitos casos, na prática, pode ter um custo computacional proibitivo. Neste trabalho, estudamos métodos iterativos para resolver problemas de Otimização Convexa Robusta de forma aproximada, que não exigem a formulação da contraparte robusta. Utilizamos conceitos de Online Learning para propor um novo algoritmo que utiliza agregação de restrições, demonstrando garantias teóricas de convergência. Desenvolvemos ainda uma modificação deste algoritmo que, apesar de não possuir tais garantias, obtém melhor performance prática. Por fim, implementamos outros métodos iterativos conhecidos da literatura de Otimização Robusta e fazemos uma análise computacional de seus desempenhos. / [en] Robust Optimization is a common paradigm to consider uncertainty in the parameters of an optimization problem. The traditional way to find robust solutions requires solving the robust counterpart of an optimiza- tion problem, which, in practice, can often be prohibitively costly. In this work, we study iterative methods to approximately solve Robust Convex Optimization problems, which do not require solving the robust counter- part. We use concepts from the Online Learning framework to propose a new algorithm that performs constraint aggregation, and we demonstrate theoretical convergence guarantees. We then develop a modification of this algorithm that, although without such guarantees, obtains better practical performance. Finally, we implement other classical iterative methods from the Robust Optimization literature and present a computational study of their performances.

Em diversas análises estatísticas, nos deparamos com dados multinomiais, dos quais precisamos analisar o comportamento ao longo do tempo e sua relação com fatores determinantes. Os métodos clássicos para modelos de regressão multinomiais consistem em utilizar a estrutura de modelos lineares generalizados para desenvolver tais modelos McCullagh & Nelder (1989). No entanto, este enfoque apresenta algumas desvantagens como não admiter a incidência de zeros em nenhuma categoria, a hipótese da proporcionalidade da razão de chances e o fato de não serem modelos adequados para análise de dados censurados. Com o objetivo de analisar dados multinomiais com essas características propomos um modelo que é uma extensão do modelo de intensidade multiplicativo desenvolvido por Aalen (1978) e apresentado em Fleming & Harrington (2005), para variáveis aleatórias multinomiais. Com isso, ao invés de modelarmos as probabilidades associadas às categorias, como nos métodos clássicos, modelamos a função intensidade associada à variável aleatória multinomial. Através do critério martingale, estimamos os parâmetros do modelo ajustado e propomos testes de hipóteses para estes parâmetros para uma e duas populações. O teste para comparação de duas populações é baseado na estatística de logrank / In several applications, we want to analyze the behavior of multinomial datas over the time and its relationship with important factors. The classic methods commonly used for multinomial regression models are based in the generalized linear model framework. However, this models presents some disadvantages such that: it does not admit the incidence of zeros in any category, the assumption of proportionality of odds ratio and the fact that they are not appropriate models to analyze censored data. For multinomial data analyses with this characteristics, we propose a model that it is an extension of the multiplicative intensity model developed by Aalen to random multinomial variables. Therefore, instead of modeling the categorical probabilities, as in the classics methods, we modeled the intensity fuction associated with the multinomial variable. Using the martingale criterion, we estimate the models parameters and propose hypothesis testing for these parameters for one and two populations. The test for comparing two populations is based in the logrank statistics

Em diversas análises estatísticas, nos deparamos com dados multinomiais, dos quais precisamos analisar o comportamento ao longo do tempo e sua relação com fatores determinantes. Os métodos clássicos para modelos de regressão multinomiais consistem em utilizar a estrutura de modelos lineares generalizados para desenvolver tais modelos McCullagh & Nelder (1989). No entanto, este enfoque apresenta algumas desvantagens como não admiter a incidência de zeros em nenhuma categoria, a hipótese da proporcionalidade da razão de chances e o fato de não serem modelos adequados para análise de dados censurados. Com o objetivo de analisar dados multinomiais com essas características propomos um modelo que é uma extensão do modelo de intensidade multiplicativo desenvolvido por Aalen (1978) e apresentado em Fleming & Harrington (2005), para variáveis aleatórias multinomiais. Com isso, ao invés de modelarmos as probabilidades associadas às categorias, como nos métodos clássicos, modelamos a função intensidade associada à variável aleatória multinomial. Através do critério martingale, estimamos os parâmetros do modelo ajustado e propomos testes de hipóteses para estes parâmetros para uma e duas populações. O teste para comparação de duas populações é baseado na estatística de logrank / In several applications, we want to analyze the behavior of multinomial datas over the time and its relationship with important factors. The classic methods commonly used for multinomial regression models are based in the generalized linear model framework. However, this models presents some disadvantages such that: it does not admit the incidence of zeros in any category, the assumption of proportionality of odds ratio and the fact that they are not appropriate models to analyze censored data. For multinomial data analyses with this characteristics, we propose a model that it is an extension of the multiplicative intensity model developed by Aalen to random multinomial variables. Therefore, instead of modeling the categorical probabilities, as in the classics methods, we modeled the intensity fuction associated with the multinomial variable. Using the martingale criterion, we estimate the models parameters and propose hypothesis testing for these parameters for one and two populations. The test for comparing two populations is based in the logrank statistics

Submitted by Filipe dos Santos (fsantos@pucsp.br) on 2017-01-16T13:11:25Z No. of bitstreams: 1 Edvonete Souza de Alencar.pdf: 1644964 bytes, checksum: a34f0a8fd40fe1cdbff542f352d371c7 (MD5) / Made available in DSpace on 2017-01-16T13:11:25Z (GMT). No. of bitstreams: 1 Edvonete Souza de Alencar.pdf: 1644964 bytes, checksum: a34f0a8fd40fe1cdbff542f352d371c7 (MD5) Previous issue date: 2016-11-29 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This research aims to investigate the theoretical frameworks that support Brazilian research published between 1997-2015, regarding the training of teachers in the early elementary years on the Multiplicative Conceptual Field. In order to achieve it, research review procedures called qualitative metasynthesis was used. The search of the investigation was carried out on the website of Capes Bank of Theses using the expressions "Teacher Training" and "Multiplicative Conceptual Field" in addition to the use of a filter for investigations that would discourse on "Continuing Education" and "Early ElementaryYears". A group of research in which we seek the most used theoretical frameworks was selected; Shulman and Vergnaud were found. A new selection was carried out, focusing on the research which had presented both authors as references. With this smaller group of research, a qualitative metasynthesis was held regarding the use of these theoretical references for the investigations. The qualitative metasynthesis was performed presenting categories which were taken from these reports, in which similarities were observed as well as differences and complementarities of the most discussed theoretical frameworks. In general, the use of the qualitative metasynthesis of the studies concerning Shulman through the investigations made it possible to identify that these investigations used the three categories of knowledge as those presented by Shulman (specific, pedagogical and curricular). It is worth mentioning that there were not any differences observed in relation to what is mentioned by the author. However, it was pointed out some aspects which were addressed by some research that complement Shulman studies regarding the specific and pedagogical knowledge used to encourage reflection on teacher training. The qualitative metasynthesis of the use of Vergnaud's theory by research has shown similarities in terms of approaches on the explanation of the Theory of Conceptual Fields. It was observed that most of the investigations focus on explaining what characterizes the Theory of Vergnaud. It was found that three of the selected research deepen their explanations of the Multiplicative Conceptual Field and that such investigations do not show different aspects of what is proposed by Vergnaud. In addition to that, it was stated that one of the research complements Vergnaud studies when explaining relational and numerical calculations in the Multiplicative Conceptual Field. This thesis led us to infer the need to conduct more studies on this topic and to reflect on the use of theoretical frameworks applied by the investigations / Esta pesquisa teve como objetivo investigar os referenciais teóricos que sustentam pesquisas brasileiras, publicadas entre 1997 e 2015, a respeito da formação contínua de professores dos anos iniciais do Ensino Fundamental sobre o Campo Conceitual Multiplicativo. Para a sua realização, utilizamos uma das modalidades de revisão de pesquisas denominada metassíntese qualitativa. A busca das investigações foi realizada no site do Banco de Teses da Capes por meio das expressões “Formação de Professores” e “Campo Conceitual Multiplicativo” e com a aplicação do filtro para investigações que discorressem sobre a “Formação Contínua” e “Anos Iniciais”. Selecionamos assim um grupo de pesquisas no qual buscamos os referenciais teóricos mais utilizados: Shulman e Vergnaud. Fizemos novamente uma seleção, centrando a investigação nas pesquisas que tiveram os estudos desenvolvidos pelos dois autores. Com esse grupo menor de pesquisas, realizamos a metassíntese qualitativa do uso desses referencias teóricos pelas investigações. A metassíntese qualitativa foi realizada apresentando categorias retiradas dos fichamentos dessas pesquisas, nos quais foram observadas semelhanças, diferenças e complementariedades dos referenciais teóricos mais abordados. De modo geral, a metassíntese qualitativa do uso dos estudos de Shulman pelas investigações permitiu identificar que estas utilizaram as três categorias do conhecimento de modo semelhante ao que é apresentado por Shulman (específico, pedagógico e curricular). Cabe ressaltar que não percebemos diferenças em relação ao que é mencionado pelo autor. No entanto, notamos aspectos abordados por algumas pesquisas que complementam os estudos de Shulman quanto ao conhecimento específico e pedagógico utilizado para incentivar a reflexão nas formações de professores. A metassíntese qualitativa do uso da teoria de Vergnaud pelas pesquisas nos mostrou semelhança quanto às abordagens sobre a explicação da Teoria dos Campos Conceituais. Pudemos perceber que a maioria das investigações se preocuparam em explicar o que caracteriza a Teoria de Vergnaud. Verificamos que três das pesquisas selecionadas aprofundam suas explanações sobre o Campo Conceitual Multiplicativo ainda que essas investigações não apresentam aspectos diferentes do que é proposto por Vergnaud. Constatamos também que uma das pesquisas complementa os estudos de Vergnaud ao explicar os cálculos relacional e numérico no Campo Conceitual Multiplicativo. Esta Tese nos levou a inferir a necessidade de se realizar mais estudos sobre essa temática e a refletir sobre o uso dos referenciais teóricos utilizados pelas investigações

Este estudo considera o problema de controle ótimo multi-período de média-variância para sistemas em tempo discreto com saltos markovianos e ruídos multiplicativos. Inicialmente considera-se um critério de desempenho formado por uma combinação linear da variância nal e valor esperado da saída do sistema. É apresentada uma solução analítica na obtenção da estratégia ótima para este problema. Em seguida são considerados os casos onde os critérios de desempenho são minimizar a variância nal sujeito a uma restrição no valor esperado ou maximizar o valor esperado nal sujeito a uma restrição na variância nal da saída do sistema. As estratégias ótimas de controle são obtidas de um conjunto de equações de diferenças acopladas de Riccati. Os resultados obtidos neste estudo generalizam resultados anteriores da literatura para o problema de controle ótimo com saldos markovianos e ruídos multiplicativos, apresentando condições explícitas e sucientes para a otimalidade da estratégia de controle. São apresentados modelos e simulações numéricas em otimização de carteiras de investimento e estratégias de gestão de ALM (asset liabilities management). / This thesis focuses on the stochastic optimal control problem of discrete-time linear systems subject to Markov jumps and multiplicative noise under three kinds of performance criterions related to the nal value of the expectation and variance of the output. In the first problem it is desired to minimize the nal variance of the output subject to a restriction on its nal expectation, in the second one it is desired to maximize the nal expectation of the output subject to a restriction on its nal variance, and in the third one it is considered a performance criterion composed by a linear combination of the nal variance and expectation of the output of the system. The optimal control strategies are obtained from a set of interconnected Riccati dierence equations and explicit sufficient conditions are presented for the existence of an optimal control strategy for these problems, generalizing previous results in the literature. Numerical simulations of investment portfolios and asset liabilities management models for pension funds with regime switching are presented.

On se place dans le cadre de l'estimation non paramétrique dans le modèle de régression. Dans un premier temps, on dispose des observations Y dont la densité $g$ est connue et dépend d'une fonction de régression $f(X)$ inconnue. Dans cette thèse, cette fonction est supposée régulière, i.e. appartenant à une boule de Hölder. Le but est d'estimer la fonction $f$ à un point $y$ (estimation ponctuelle). Pour cela, nous développons un estimateur local de type {\it bayésien}, construit à partir de la densité $g$ des observations. Nous proposons une procédure adaptative s'appuyant sur la méthode de Lepski, qui permet de construire un estimateur adaptatif choisi dans la famille des estimateurs bayésiens locales indexés par la fenêtre. Sous certaines hypothèses suffisantes sur la densité $g$, notre estimateur atteint la vitesse adaptative optimale (en un certain sens). En outre, nous constatons que dans certains modèles, l'estimateur bayésien est plus performant que les estimateurs linéaires. Ensuite, une autre approche est considérée. Nous nous plaçons dans le modèle de régression additive, où la densité du bruit est inconnue, mais supposée symétrique. Dans ce cadre, nous développons un estimateur dit de {\it Huber} reposant sur l'idée de la médiane. Cet estimateur permet d'estimer la fonction de régression, quelque soit la densité du bruit additif (par exemple, densité gaussienne ou densité de Cauchy). Avec la méthode de Lepski, nous sélectionnons un estimateur qui atteint la vitesse adaptative classique des estimateurs linéaires sur les espaces de Hölder.

While there has been research on development of multiplicative reasoning, and how to teach multiplication facts, there is little research on how children consider they learn these. This study explores the children's learning as they consider how they commit their multiplication facts to memory, discover calculation strategies and develop multiplicative thinking. A group of eleven Year 4 children (8 years old) participated in a series of 13 lessons where they became coresearchers in the exploration of their learning. A contextually based thematic approach was provided through 'Crocodilian Studies'. The mixed-method approach to this study included formal assessment, participant observation, individual interviews, the children's written ideas, and individual case studies. The most significant finding of this study was the powerful influence of peer learning. The children enriched and directed each other's learning as they shared ideas and reflected on their own mathematical learning as they observed and critiqued the thinking of peers. As the children were involved in thinking about how they learn they were able to identify gaps and construct their own learning pathways. A significant finding was that children can develop their multiplicative strategies while they commit their multiplication facts to memory, in a relatively short time provided that the learning process facilitates strategy development and understanding. By exposing the children to multiplication facts in sequenced clusters provided them with a manageable number of facts to be learnt at one time. Another finding related to how children develop calculation strategies through lesson activities rather than being explicitly taught them. The children considered practice important for memorisation. Parental support was significant in enriching the children's learning.

In this study, theorem-based, generally applicable stochastic parameterisation schemes are developed and applied to a quasi-geostrophic model of extratropical atmospheric low-frequency variability (LFV). Hasselmann’s method is developed from limiting theorems for slow-fast systems of ordinary differential equations (ODEs) and applied to this high-dimensional model of intermediate complexity comprised of partial differential equations (PDEs) with complicated boundary conditions. Seamless, efficient algorithms for integrating the parameterised models are developed, which require only minimal changes to the full model algorithm. These algorithms may be readily adapted to a range of climate models of greater complexity in parameterising the effects of fast, sub-grid scale processes on the resolved scales. For comparison, the Majda-Timofeyev-Vanden-Eijnden (MTV) parameterisation method is applied to this model. The seamless algorithms are first adapted to probe the multiple regime behaviour that characterises the full model LFV. In contrast to the conclusions of a previous study, it is found that the multiple regime behaviour is not the result of a nonlinear interaction between the leading two planetary-scale modes, but rather is the result of interactions among these two modes and the leading synoptic-scale mode. The low-dimensional Hasselmann stochastic models perform well in simulating the statistics of the planetary-scale modes. In particular, a model with only one resolved (planetary-scale) mode captures the multiple regime behaviour of the full model. Although a fast-evolving synoptic-scale mode is of primary importance to the multiple regime behaviour, deterministic averaged forcing and not multiplicative noise is responsible for the regime behaviour in this model. The MTV models generate non-Gaussian statistics, but generally do not perform as well in capturing the climate statistics.

Este trabalho apresenta investigação acerca das concepções de professores dos Anos Inicias sobre o Campo Multiplicativo à luz da Teoria dos Campos Conceituais de Gérard Vergnaud. O campo multiplicativo foi escolhido como tema de estudo por permear todo o ensino básico, sendo assim de grande importância para a formação matemática escolar dos estudantes. A coleta de dados se dá em um curso de formação continuada, que se desenvolveu ao longo de 5 encontros de 4 horas, com 11 professoras da Escola Estadual de Ensino Médio Célia Flores Lavra Pinto localizada no município de Viamão/RS. Utilizou-se a pesquisa-ação como metodologia da pesquisa. Ao final da pesquisa, concluímos que o grupo de professores pesquisado se mantém restrito ao trabalho voltado para o algoritmo de multiplicação e a memorização da tabuada, quando se trata do Campo Multiplicativo das Operações. O produto desta pesquisa é um conjunto de atividades sobre o tema Campo Multiplicativo das Operações, dentre outros temas relacionados, que poderão auxiliar professores que ensinam matemática interessados em pensar sobre a prática docente e ampliar suas possibilidades de intervenção junto aos estudantes. / This paper presents the development of a training course for teachers of the early years in order to investigate teachers' conceptions about Multiplicative Field Operations to the Theory of Conceptual Fields Gérard Vergnaud. The multiplicative field was chosen as the subject of study not only because it permates the whole basic education, but also because it has a great importance to the training school mathematics students. Data collection occurs in a continuing education course, which was developed in 5 meetings of 4 hours, with 11 teachers from State Preparatory School High School Celia Flores Lavra Pinto localized in Viamão/RS. We used the action research as the research methodology. At the end of the paper, we conclude that the studied group of teachers remains restricted to the work facing the multiplication algorithm and memorizing multiplication tables when it comes to the field Multiplicative Operations . The product of this research is a set of activities on the theme Multiplicative Field Operations. Among other issues, it may help teachers to teach mathematics for those who are interested in thinking about teaching practice and expand their possibilities of intervention with students.