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Applications of accuracy certificates for problems with convex structure   This dissertation addresses the efficient generation and potential applications of accuracy certificates in the framework of “black-box-represented” convex optimization problems - convex problems where the objective and the constraints are represented by  “black boxes” which, given on input a value x of the argument, somehow (perhaps in a fashion unknown to the user) provide on output the values and the derivatives of the objective and the constraints at x. The main body of the dissertation can be split into three parts.  In the first part, we provide our background --- state of the art of the theory of accuracy certificates for black-box-represented convex optimization. In the second part, we extend the toolbox of black-box-oriented convex optimization algorithms with accuracy certificates by equipping with these certificates a state-of-the-art algorithm for large-scale nonsmooth black-box-represented problems with convex structure, specifically, the Non-Euclidean Restricted Memory Level (NERML) method. In the third part, we present several novel academic applications of accuracy certificates. The dissertation is organized as follows: In Chapter 1, we motivate our research goals and present a detailed summary of our results. In Chapter 2, we outline the relevant background, specifically, describe four generic black-box-represented generic problems with convex structure (Convex Minimization, Convex-Concave Saddle Point, Convex Nash Equilibrium, and Variational Inequality with Monotone Operator), and outline the existing theory of accuracy certificates for these problems. In Chapter 3, we develop techniques for equipping with on-line accuracy certificates the state-of-the-art NERML algorithm for large-scale nonsmooth problems with convex structure, both in the cases when the domain of the problem is a simple solid and in the case when the domain is given by Separation oracle. In Chapter 4, we develop  several novel academic applications of accuracy certificates, primarily to (a) efficient certifying emptiness of the intersection of finitely many solids given by Separation oracles, and (b) building efficient algorithms for convex minimization over solids given by Linear Optimization oracles (both precise and approximate). In Chapter 5, we apply accuracy certificates to efficient decomposition of “well structured” convex-concave saddle point problems, with applications to computationally attractive decomposition of a large-scale LP program with the constraint matrix which becomes block-diagonal after eliminating a relatively small number of possibly dense columns (corresponding to “linking variables”) and possibly dense rows (corresponding to “linking constraints”).

Online Convex Optimization (OCO) is a field in the intersection of game theory, optimization, and machine learning which has been receiving increasing attention due to its recent applications to a wide range of topics such as complexity theory and graph sparsification. Besides the usually simple description and implementation of OCO algorithms, a lot of this recent success is due to a deepening of our understanding of the OCO setting and their algorithms by using cornerstone ideas from convex analysis and optimization such as the powerful results from convex duality theory. In this text we present a mostly self-contained introduction to the field of online convex optimization. We first describe the online learning and online convex optimization settings, proposing an alternative way to formalize both of them so we can make formal claims in a clear and unambiguous fashion while not cluttering the readers understanding. We then present an overview of the main concepts of convex analysis we use, with a focus on building intuition. With respect to algorithms for OCO, we first present and analyze the Adaptive Follow the Regularized Leader (AdaFTRL) together with an analysis which relies mainly on the duality between strongly convex and strongly smooth functions. We then describe the Adaptive Online Mirror Descent (AdaOMD) and the Adaptive Dual Averaging (AdaDA) algorithms and analyze both by writing them as special cases of the AdaFTRL algorithm. Additionally, we show simple sufficient conditions for Eager and Lazy Online Mirror Descent (the non-adaptive counter-parts of AdaOMD and AdaDA) to be equivalent. We also present the well-known AdaGrad and Online Newton Step algorithms as special cases of the AdaReg algorithm, proposed by Gupta, Koren, and Singer, which is itself a special case of the AdaOMD algorithm. We conclude by taking a bird\'s-eyes view of the connections shown throughout the text, forming a ``genealogy\'\' of OCO algorithms, and discuss some possible path for future research. / Otimização Convexa Online (OCO) é uma área na intersecção de teoria dos jogos, otimização e aprendizado de máquina que tem recebido maior atenção recentemente devido a suas recentes aplicações em uma grande gama de áreas como complexidade computacional e esparsificação de grafos. Além dos algoritmos de OCO usualmente terem descrições diretas e poderem ser implementados de forma relativamente simples, muito do recente sucesso da área foi possível graças a um melhor entendimento do cenário e dos algoritmos de OCO que se deu com uso de conhecidas ideias de análise e otimização convexa como a poderosa teoria de dualidade convexa. Nesse texto nós apresentamos uma introdução (em sua maioria auto-contida) à área de otimização convexa online. Primeiro, descrevemos os cenários de aprendizado online e de otimização convexa online, propondo uma forma alternativa de formalizar ambos os modelos de forma que conseguimos enunciar afirmações claras e formais de forma que não atrapalha o entendimento do leitor. Nós então apresentamos um resumo dos principais conceitos e resultados de análise convexa que usamos no texto com um foco em criar intuição sobre os mesmos. Com relação a algoritmos para OCO, nós começamos apresentando o algoritmo Adaptive Follow the Regularized Leader (AdaFTRL) e analisamos sua eficácia com um resultado sobre a dualidade de funções strongly convex e strongly smooth. Na sequência, descrevemos os algoritmos Adaptive Online Mirror Descent (AdaOMD) e Adaptive Dual Averaging (AdaDA), analisando a eficácia de cada um escrevendo eles como instâncias do algoritmo AdaFTRL. Além disso, nós mostramos condições simples para que as versões Eager e Lazy do Online Mirror Descent (que são as versões não adaptativas do AdaOMD e do AdaDA, respectivamente) sejam equivalentes. Também apresentamos os algoritmos AdaGrad e Online Newton Step, bem conhecidos na literatura sobre OCO, como casos especiais do algoritmo AdaReg, esse último um algoritmo proposto por Gupta, Koren, and Singer, que, por sua vez, é um caso especial do algoritmo AdaOMD. Nós concluímos o texto com uma visão global das conexões entre os algoritmos que mostramos durante o texto, formando uma \"genealogia\" de algoritmos para OCO, além de discutirmos possíveis direções futuras de pesquisa.

The main focus of this thesis is to develop full parametric and semiparametric Bayesian inference for data arising from triangular distributions. A natural consequence of working with such distributions is it allows one to consider regression models where the response variable is now the mode of the data distribution. A new family of nonparametric prior distributions is developed for a certain class of convex densities of particular relevance to mode regressions. Triangular distributions arise in several contexts such as geosciences, econometrics, finance, health care management, sociology, reliability engineering, decision and risk analysis, etc. In many fields, experts, typically, have a reasonable idea about the range and most likely values that define a data distribution. Eliciting these quantities is thus, generally, easier than eliciting moments of other commonly known distributions. Using simulated and actual data, applications of triangular distributions, with and without mode regressions, in some of the aforementioned areas are tackled. / text

Metric topologies can be viewed as one-dimensional measures. This dissertation is a topological study of two-dimensional measures. Attention is focused on locally convex vector topologies on infinite dimensional real spaces. A nabla (referred to in the literature as a 2-norm) is the analogue of a norm which assigns areas to the parallelograms. Nablas are defined for the classical normed spaces and techniques are developed for defining nablas on arbitrary spaces. The work here brings out a strong connection with tensor and wedge products. Aside from the normable theory, it is shown that nabla topologies need not be metrizable or Mackey. A class of concretely given non-Mackey nablas on the ℓp and Lp spaces is introduced and extensively analyzed. Among other results it is found that the topological dual of ℓ₁ with respect to these nabla topologies is C₀, one of the spaces infamous for having no normed predual. Also, a connection is made with the theory of two-norm convergence (not to be confused with 2-norms). In addition to the hard analysis on the classical spaces, a duality framework from which to study the softer aspects is introduced. This theory is developed in analogy with polar duality. The ideas corresponding to barrelledness, quasi-barrelledness, equicontinuity and so on are developed. This dissertation concludes with a discussion of angles in arbitrary normed spaces and a list of open questions.

The focus of the present master thesis is the automation of an existing controllerdesign for a missile using two aerodynamic actuating systems. The motivation isto evaluate more missile concepts in a shorter period of time.The option used is trimming and linearization of a highly nonlinear missile at specic conditions. According to these conditions, either a two-dimensional operatingpoint grid dened by Mach number and height or three-dimensional operatingpoint grid dened by Mach number, height and angle of attack is generated forthe whole operating range of the missile. The controllers are designed at thesepoints using convex optimization. The convex set denes the pole placement areawhich is constrained by linear matrix inequalities according to the dynamic behaviorof the missile at the operating point conditions. These controllers describea validity area where the missile can be stabilized. This area consists all neighboringoperating points and denes therefore the grid density which can dier atspecic regions of the operating range. Controlling the missile to the target makesit necessary to apply gain-scheduling in order to get the manipulated variable byinterpolation of adjacent operating points. During this blending of the controllersa problem called windup can occur when an actuator is saturated. This mightlead to instability in worst case but can be counteracted by a model-recovery antiwindupnetwork which guarantees stability in the presence of saturation. Thisanti-windup design is automated by an ane linear parameter dependency of thegrid parameters and has the same validity area like the controllers.The whole design was successfully developed and tested in MATLAB/Simulink onmissiles using one or two aerodynamic actuating systems. The controllers have agood performance at small and high acceleration steps and the anti-windup keepsthe missile stable even though the actuators are saturated. Stability and robustnessof the controllers and anti-windup networks was veried as well as an airdefense maneuver where the missile starts at the ground and intercepts a targetat high altitude was successfully simulated for dierent grids and missiles.

A thesis submitted to the Faculty of Science, School of Statistics and Actuarial Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, January 25, 2017. / The classical Wishart distribution, was first derived byWishart (1928) as the distribution of the maximum likelihood estimator of the covariance matrix of the multivariate normal distribution. It is a matrix variate generalization of the gamma distribution. In high dimensional settings,Wishart distributions defined within the framework of graphical models are of particular importance. [No abstract provided. Information taken from introduction] / MT2017

Master of Science - Science / Axial line placement is one step in a method known as space syntax which is used in town planning to analyse architectural structures. This is becoming increasingly important in the quickly growing urban world of today. The field of axial line placement is an area of space syntax that has previously been done manually which is becoming increasingly impractical. Research is underway to automate the process and this research forms a large part of the automation. The general problem of axial line placement has been shown to be NP-complete. For this reason, previous research in this field has been focused on finding special cases where this is not the case or finding heuristics that approximate a solution. The majority of the research conducted has been on the simpler case of axial line placement in configurations of orthogonal rectangles and the only work done with convex polygons has been in the restricted case of deformed urban grids. This document presents research that finds two non-trivial special cases of convex polygons that have polynomial solutions and finds the first heuristic for general configurations of convex polygons.