Let fd(n), n > d ≥ 2, be the smallest positive integer such that any set of fd(n) points, in general position in Rd , contains n points in convex position. Let hd(n, k), n > d ≥ 2 and k ≥ 0, denote the smallest number with the property that in any set of hd(n, k) points, in general position in Rd , there are n points in convex position whose convex hull contains at most k other points. Previous result of Valtr states that h4(n, 0) does not exist for all n ≥ 249. We show that h4(n, 0) does not exist for all n ≥ 137. We show that h3(8, k) ≤ f3(8) for all k ≥ 26, h4(10, k) ≤ f4(10) for all k ≥ 147 and h5(12, k) ≤ f5(12) for all k ≥ 999. Next, let fd(k, n) be the smallest number such that in every set of fd(k, n) points, in general position in Rd , there are n points whose convex hull has at least k vertices. We show that, for arbitrary integers n ≥ k ≥ d + 1, d ≥ 2, fd(k, n) ≥ (n − 1) (k − 1)/(cd logd−2 (n − 1)) , where cd > 0 is a constant dependent only on the dimension d. 1

Convex optimization has developed a wide variety of useful tools critical to many applications in machine learning. However, unlike linear and quadratic programming, general convex solvers have not yet reached sufficient maturity to fully decouple the convex programming model from the numerical algorithms required for implementation. Especially as datasets grow in size, there is a significant gap in speed and scalability between general solvers and specialized algorithms. This thesis addresses this gap with a new model for convex programming based on an intermediate representation of convex problems as a sum of functions with efficient proximal operators. This representation serves two purposes: 1) many problems can be expressed in terms of functions with simple proximal operators, and 2) the proximal operator form serves as a general interface to any specialized algorithm that can incorporate additional `2-regularization. On a single CPU core, numerical results demonstrate that the prox-affine form results in significantly faster algorithms than existing general solvers based on conic forms. In addition, splitting problems into separable sums is attractive from the perspective of distributing solver work amongst multiple cores and machines. We apply large-scale convex programming to several problems arising from building the next-generation, information-enabled electrical grid. In these problems (as is common in many domains) large, high-dimensional datasets present opportunities for novel data-driven solutions. We present approaches based on convex models for several problems: probabilistic forecasting of electricity generation and demand, preventing failures in microgrids and source separation for whole-home energy disaggregation.

On se place dans le cadre du traitement des signaux audio multicanaux et multi-sources. À partir du mélange de plusieurs sources sonores enregistrées en milieu réverbérant, on cherche à estimer les réponses acoustiques (ou filtres de mélange) entre les sources et les microphones. Ce problème inverse ne peut être résolu qu'en prenant en compte des hypothèses sur la nature des filtres. Notre approche consiste d'une part à identifier mathématiquement les hypothèses nécessaires sur les filtres pour pouvoir les estimer et d'autre part à construire des fonctions de coût et des algorithmes permettant de les estimer effectivement. Premièrement, nous avons considéré le cas où les signaux sources sont connus. Nous avons développé une méthode d'estimation des filtres basée sur une régularisation convexe prenant en compte à la fois la nature parcimonieuse des filtres et leur enveloppe de forme exponentielle décroissante. Nous avons effectué des enregistrements en environnement réel qui ont confirmé l'efficacité de cet algorithme. Deuxièmement, nous avons considéré le cas où les signaux sources sont inconnus, mais statistiquement indépendants. Les filtres de mélange peuvent alors être estimés à une indétermination de permutation et de gain près à chaque fréquence par des techniques d'analyse en composantes indépendantes. Nous avons apporté une étude exhaustive des garanties théoriques par lesquelles l'indétermination de permutation peut être levée dans le cas où les filtres sont parcimonieux dans le domaine temporel. Troisièmement, nous avons commencé à analyser les hypothèses sous lesquelles notre algorithme d'estimation des filtres pourrait être étendu à l'estimation conjointe des signaux sources et des filtres et montré un premier résultat négatif inattendu : dans le cadre de la déconvolution parcimonieuse aveugle, pour une famille assez large de fonctions de coût régularisées, le minimum global est trivial. Des contraintes supplémentaires sur les signaux sources ou les filtres sont donc nécessaires. / This work is focused on the processing of multichannel and multisource audio signals. From an audio mixture of several audio sources recorded in a reverberant room, we wish to estimate the acoustic responses (a.k.a. mixing filters) between the sources and the microphones. To solve this inverse problem one need to take into account additional hypotheses on the nature of the acoustic responses. Our approach consists in first identifying mathematically the necessary hypotheses on the acoustic responses for their estimation and then building cost functions and algorithms to effectively estimate them. First, we considered the case where the source signals are known. We developed a method to estimate the acoustic responses based on a convex regularization which exploits both the temporal sparsity of the filters and the exponentially decaying envelope. Real-world experiments confirmed the effectiveness of this method on real data. Then, we considered the case where the sources signal are unknown, but statistically independent. The mixing filters can be estimated up to a permutation and scaling ambiguity. We brought up an exhaustive study of the theoretical conditions under which we can solve the indeterminacy, when the multichannel filters are sparse in the temporal domain. Finally, we started to analyse the hypotheses under which this algorithm could be extended to the joint estimation of the sources and the filters, and showed a first unexpected results : in the context of blind deconvolution with sparse priors, for a quite large family of regularised cost functions, the global minimum is trivial. Additional constraints on the source signals and the filters are needed.

Thanks to its ability to evaluate metabolic functions in tissues from the temporal evolution of a previously injected radiotracer, dynamic positron emission tomography (PET) has become an ubiquitous analysis tool to quantify biological processes. Several quantification techniques from the PET imaging literature require a previous estimation of global time-activity curves (TACs) (herein called \textit{factors}) representing the concentration of tracer in a reference tissue or blood over time. To this end, factor analysis has often appeared as an unsupervised learning solution for the extraction of factors and their respective fractions in each voxel. Inspired by the hyperspectral unmixing literature, this manuscript addresses two main drawbacks of general factor analysis techniques applied to dynamic PET. The first one is the assumption that the elementary response of each tissue to tracer distribution is spatially homogeneous. Even though this homogeneity assumption has proven its effectiveness in several factor analysis studies, it may not always provide a sufficient description of the underlying data, in particular when abnormalities are present. To tackle this limitation, the models herein proposed introduce an additional degree of freedom to the factors related to specific binding. To this end, a spatially-variant perturbation affects a nominal and common TAC representative of the high-uptake tissue. This variation is spatially indexed and constrained with a dictionary that is either previously learned or explicitly modelled with convolutional nonlinearities affecting non-specific binding tissues. The second drawback is related to the noise distribution in PET images. Even though the positron decay process can be described by a Poisson distribution, the actual noise in reconstructed PET images is not expected to be simply described by Poisson or Gaussian distributions. Therefore, we propose to consider a popular and quite general loss function, called the $\beta$-divergence, that is able to generalize conventional loss functions such as the least-square distance, Kullback-Leibler and Itakura-Saito divergences, respectively corresponding to Gaussian, Poisson and Gamma distributions. This loss function is applied to three factor analysis models in order to evaluate its impact on dynamic PET images with different reconstruction characteristics.

The availability of kinetic data for the oxidative dehydrogenation (ODH) of n-butane from Téllez et al. (1999a and 1999b) and Assabumrungrat et al. (2002) presented an opportunity to submit a chemical process of industrial significance to Attainable Region (AR) analysis. The process thermodynamics for the ODH of n-butane and 1-butene have been reviewed. The addition of oxygen in less than the stoichiometric ratios was found to be essential to prevent deep oxidation of hydrocarbon products {Milne et al. (2004 and 2006c)}. The AR concept has been used to determine the maximum product yields from the ODH of n-butane and 1-butene under two control régimes, one where the partial pressure of oxygen along the length of the reactor was maintained at a constant level and the second where the oxygen partial pressure was allowed to wane. Theoretical maxima under the first régime were associated with very large and impractical residence times. The Recursive Convex Control policy {Seodigeng (2006)} and the second régime were applied to confirm these maxima {Milne et al. (2008)}. Lower and more practical residence times ensued. A differential side-stream reactor was the preferred reactor configuration as was postulated by Feinberg (2000a). Abstract A.D. Milne Page 4 of 430 The maximum yield of hydrocarbon product, the associated residence time and the required reactor configuration as functions of oxygen partial pressure were investigated for the series combinations of an inert porous membrane reactor and a fixed-bed reactor. The range of oxygen partial pressures was from 85 kPa to 0.25 kPa. The geometric profile for hydrocarbon reactant and product influences the residence times for the series reactors. The concept of a residence time ratio is introduced to identify the operating circumstances under which it becomes advantageous to select an inert membrane reactor in preference to a continuously stirred tank reactor and vice versa from the perspective of minimising the overall residence time for a reaction {Milne et al. (2006b)}. A two-dimensional graphical analytical technique is advocated to examine and balance the interplay between feed conditions, required product yields and residence times in the design of a reactor {Milne et al. (2006a)}.. A simple graphical technique is demonstrated to identify the point in a reaction at which the selectivity of the feed relative to a product is a maximum {Milne et al. (2006a)}. Literature Cited Assabumrungrat, S. Rienchalanusarn, T. Praserthdam, P. and Goto, S. (2002) Theoretical study of the application of porous membrane reactor to Abstract A.D. Milne Page 5 of 430 oxidative dehydrogenation of n-butane, Chemical Engineering Journal, vol. 85, pp. 69-79. Feinberg, M. (2000a) Optimal reactor design from a geometric viewpoint – Part II. Critical side stream reactors, Chemical Engineering Science, vol. 55, pp. 2455-2479. Milne, D., Glasser, D., Hildebrandt, D., Hausberger, B., (2004), Application of the Attainable Region Concept to the Oxidative Dehydrogenation of 1- Butene in Inert Porous Membrane Reactors, Industrial and. Engineering Chemistry Research, vol. 43, pp. 1827-1831 with corrections subsequently published in Industrial and Engineering Chemistry Research, vol. 43, p. 7208. Milne, D., Glasser, D., Hildebrandt, D., Hausberger, B., (2006a), Graphical Technique for Assessing a Reactor’s Characteristics, Chemical Engineering Progress, vol. 102, no. 3, pp. 46-51. Milne, D., Glasser, D., Hildebrandt, D., Hausberger, B., (2006b), Reactor Selection : Plug Flow or Continuously Stirred Tank?, Chemical Engineering Progress. vol. 102, no. 4, pp. 34-37. Milne, D., Glasser, D., Hildebrandt, D., Hausberger, B., (2006c), The Oxidative Dehydrogenation of n-Butane in a Fixed Bed Reactor and in an Inert Porous Membrane Reactor - Maximising the Production of Butenes and Butadiene, Industrial and Engineering Chemistry Research vol. 45, pp. 2661-2671. Abstract A.D. Milne Page 6 of 430 Milne, D., Seodigeng, T., Glasser, D., Hildebrandt, D., Hausberger, B., (2008), The Application of the Recursive Convex Control (RCC) policy to the Oxidative Dehydrogenation of n-Butane and 1-Butene, Industrial and Engineering Chemistry Research, (submitted for publication). Seodigeng, T.G. (2006), Numerical Formulations for Attainable Region Analysis, Ph.D. thesis, University of the Witwatersrand, Johannesburg, South Africa. Téllez, C. Menéndez, M. Santamaría, J. (1999a) Kinetic study of the oxidative dehydrogenation of butane on V/MgO catalysts, Journal of Catalysis, vol. 183, pp. 210-221. Téllez, C. Menéndez, M. Santamaría, J. (1999b) Simulation of an inert membrane reactor for the oxidative dehydrogenation of butane, Chemical Engineering Science, vol. 54, pp. 2917-2925. __________________________________

Anomalias ou outliers são exemplos ou grupo de exemplos que apresentam comportamento diferente do esperado. Na prática,esses exemplos podem representar doenças em um indivíduo ou em uma população, além de outros eventos como fraudes em operações bancárias e falhas em sistemas. Diversas técnicas existentes buscam identificar essas anomalias, incluindo adaptações de métodos de classificação e métodos estatísticos. Os principais desafios são o desbalanceamento do número de exemplos em cada uma das classes e a definição do comportamento normal associada à formalização de um modelo para esse comportamento. Nesta dissertação propõe-se a utilização de um novo espaço para realizar a detecção,esse espaço é chamado espaço de parâmetros. Um espaço de parâmetros é criado utilizando parâmetros estimados a partir da concatenação(encadeamento) de dois exemplos. Apresenta-se,então,um novo framework para realizar a detecção de anomalias através da fusão de detectores que utilizam fechos convexos em múltiplos espaços de parâmetros para realizar a detecção. O método é considerado um framework pois é possível escolher quais os espaços de parâmetros que serão utilizados pelo método de acordo como comportamento da base de dados alvo. Nesse trabalho utilizou-se,para experimentos,dois conjuntos de parâmetros(média e desvio padrão; média, variância, obliquidade e curtose) e os resultados obtidos foram comparados com alguns métodos comumente utilizados para detecção de anomalias. Os resultados atingidos foram comparáveis ou melhores aos obtidos pelos demais métodos. Além disso, acredita-se que a utilização de espaços de parâmetros cria uma grande flexibilidade do método proposto, já que o usuário pode escolher um espaço de parâmetros que se adeque a sua aplicação. Tanto a flexibilidade quanto a extensibilidade disponibilizada pelo espaço de parâmetros, em conjunto como bom desempenho do método proposto nos experimentos realizados, tornam atrativa a utilização de espaços de parâmetros e, mais especificamente, dos métodos apresentados na solução de problemas de detecção de anomalias. / Anomalies or outliers are examples or group of examples that have a behaviour different from the expected. These examples may represent diseases in individuals or populations,as well as other events such as fraud and failures in banking systems.Several existing techniques seek to identify these anomalies, including adaptations of classification methods, statistical methods and methods based on information theory. The main challenges are that the number of samples of each class is unbalanced, the cases when anomalies are disguised among normal samples and the definition of normal behaviour associated with the formalization of a model for this behaviour. In this dissertation,we propose the use of a new space to helpwith the detection task, this space is called parameter space. We also present a new framework to perform anomaly detection by using the fusion of convex hulls in multiple parameter spaces to perform the detection.The method is considered a framework because it is possible to choose which parameter spaces will be used by the method according to the behaviour of the target data set.For the experiments, two parameter spaces were used (mean and standard deviation; mean, variance, skewness and kurtosis) and the results were compared to some commonly used anomaly detection methods. The results achieved were comparable or better than those obtained by the other methods. Furthermore, we believe that a parameter space created great fexibility for the proposed method, since it allowed the user to choose a parameter space that best models the application. Both the flexibility and extensibility provided by the use of parameter spaces, together with the good performance achieved by the proposed method in the experiments, make parameter spaces and, more specifically, the proposed methods appealing when solving anomaly detection problems.

Apresentamos dois novos métodos para a solução de problemas de otimização convexa em dois níveis não necessariamente diferenciáveis, i.e., mostramos que as sequências geradas por ambos os métodos convergem para o conjunto ótimo de uma função não suave sujeito a um conjunto que também envolve a minimização de uma função não diferenciável. Ambos os algoritmos dispensam qualquer tipo de resolução de subproblemas ou busca linear durante suas iterações. Ao final, para demonstrar que os métodos são viáveis, resolvemos um problema de reconstrução de imagens tomográficas / We present two new methods for solving bilevel convex optimization problems, where both functions are not necessarily differentiable, i.e., we show that the sequences generated by those methods converge to the optimal set of a nonsmooth function subject to a set that also involves a function minimization. Both algorithms do not require any kind of subproblems resolution or linear search during the iterations. At the end, to prove that our methods are viable, we solve a problem of tomographic image reconstruction

Neste trabalho apresentamos um novo método ótimo para otimização de uma função convexa diferenciável sujeita a restrições convexas. Nosso método é baseado em ideias de Nesterov e Auslender e Teboulle. A proposta dos últimos autores usa uma distância de Bregman coerciva para garantir que os iterados permaneçam no interior do conjunto viável. Nosso método estende esses resultados para permitir o emprego da distância Euclidiana ao quadrado. Mostramos também como estimar a constante de Lipschitz para o gradiente da função objetivo, o que resulta em uma melhora na eficiência numérica do método. Finalmente, apresentamos experimentos numéricos para validar nossa proposta e comparar com o algoritmo de Nesterov. / In this work we introduce a new optimal method for constrained differentiable convex optimization which is based on previous ideas by Nesterov and Auslender and Teboulle. The method proposed by the last authors use a coercive Bregman distance to ensure that the iterates remain in the interior of the feasible set. Our results extend this method to allow the use of the squared Euclidean distance. We also show how to estimate the Lipschitz constant of the gradient of the objective function, improving the numerical behavior of the method. Finally, we present numerical experiments to validate our approach and compare it to Nesterov\'s algorithm.

Este trabalho apresenta uma arquitetura de controle de movimento entre duas posturas distintas para um robô móvel sob rodas com acionamento diferencial em um ambiente estruturado e livre de obstáculos. O conceito clássico de eficiência foi utilizado para a definição das estratégias de controle: um robô se movimenta de forma eficiente quando realiza a tarefa determinada no menor tempo e utilizando menor quantidade energética. A arquitetura proposta é um recorte do modelo de Controle Hierárquico Aninhado (NHC), composto por três níveis de abstração: (i) Planejamento de Caminho, (ii) Planejamento de Trajetória e (iii) Rastreamento de Trajetória. O Planejamento de Caminho proposto suaviza uma geodésica Dubins - o caminho mais eficiente - por uma Spline Grampeada para que este caminho seja definido por uma curva duplamente diferenciável. Uma transformação do espaço de configuração do robô é realizada. O Planejamento de Trajetória é um problema de otimização convexa na forma de Programação Cônica de Segunda Ordem, cujo objetivo é uma função ponderada entre tempo e energia. Como o tempo de percurso e a energia total consumida pelo robô possui uma relação hiperbólica, um algoritmo de sintonia do coeficiente de ponderação entre estas grandezas é proposta. Por fim, um Rastreador de Trajetória de dupla malha baseado em linearização entrada-saída e controle PID é proposto, e obteve resultados satisfatórios no rastreamento do caminho pelo robô. / This work presents a motion control architecture between two different positions for a differential driven wheeled mobile robot in a obstacles free structured environment. The classic concept of efficiency was used to define the control strategies: a robot moves efficiently when it accomplishes the determined task in the shortest time and using less amount of energy. The proposed architecture is a clipping of the Nested Hierarchical Controller (NHC) model, composed of three levels of abstraction: (i) Path Planning, (ii) Trajectory Planning and (iii) Trajectory Tracking. The proposed Path Planning smoothes a geodesic Dubins - the most efficient path - by a Clamped Spline as this path is defined by a twice differentiable curve. A transformation of the robot configuration space is performed. The Trajectory Planning is a convex optimization problem in the form of Second Order Cone Programming, whose objective is a weighted function between time and energy. As the travel time and the total energy consumed by the robot has a hyperbolic relation, a tuning algorithm to the weighting is proposed. Finnaly, a dual-loop Trajectory Tracker based on input-output feedback linearization and PID control is proposed, which obtained satisfactory results in tracking the path by the robot.

Lam Sze Wan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 93-95). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Background --- p.4 / Chapter 3 --- Review of Iri-Imai Algorithm for Convex Programming Prob- lems --- p.10 / Chapter 3.1 --- Iri-Imai Algorithm for Convex Programming --- p.11 / Chapter 3.2 --- Numerical Results --- p.14 / Chapter 3.2.1 --- Linear Programming Problems --- p.15 / Chapter 3.2.2 --- Convex Quadratic Programming Problems with Linear Inequality Constraints --- p.17 / Chapter 3.2.3 --- Convex Quadratic Programming Problems with Con- vex Quadratic Inequality Constraints --- p.18 / Chapter 3.2.4 --- Summary of Numerical Results --- p.21 / Chapter 3.3 --- Chapter Summary --- p.22 / Chapter 4 --- Value Estimation Approach to Iri-Imai Method for Con- strained Optimization --- p.23 / Chapter 4.1 --- Value Estimation Function Method --- p.24 / Chapter 4.1.1 --- Formulation and Properties --- p.24 / Chapter 4.1.2 --- Value Estimation Approach to Iri-Imai Method --- p.33 / Chapter 4.2 --- "A New Smooth Multiplicative Barrier Function Φθ+,u" --- p.35 / Chapter 4.2.1 --- Formulation and Properties --- p.35 / Chapter 4.2.2 --- "Value Estimation Approach to Iri-Imai Method by Us- ing Φθ+,u" --- p.41 / Chapter 4.3 --- Convergence Analysis --- p.43 / Chapter 4.4 --- Numerical Results --- p.46 / Chapter 4.4.1 --- Numerical Results Based on Algorithm 4.1 --- p.46 / Chapter 4.4.2 --- Numerical Results Based on Algorithm 4.2 --- p.50 / Chapter 4.4.3 --- Summary of Numerical Results --- p.59 / Chapter 4.5 --- Chapter Summary --- p.60 / Chapter 5 --- Extension of Value Estimation Approach to Iri-Imai Method for More General Constrained Optimization --- p.61 / Chapter 5.1 --- Extension of Iri-Imai Algorithm 3.1 for More General Con- strained Optimization --- p.62 / Chapter 5.1.1 --- Formulation and Properties --- p.62 / Chapter 5.1.2 --- Extension of Iri-Imai Algorithm 3.1 --- p.63 / Chapter 5.2 --- Extension of Value Estimation Approach to Iri-Imai Algo- rithm 4.1 for More General Constrained Optimization --- p.64 / Chapter 5.2.1 --- Formulation and Properties --- p.64 / Chapter 5.2.2 --- Value Estimation Approach to Iri-Imai Method --- p.67 / Chapter 5.3 --- Extension of Value Estimation Approach to Iri-Imai Algo- rithm 4.2 for More General Constrained Optimization --- p.69 / Chapter 5.3.1 --- Formulation and Properties --- p.69 / Chapter 5.3.2 --- Value Estimation Approach to Iri-Imai Method --- p.71 / Chapter 5.4 --- Numerical Results --- p.72 / Chapter 5.4.1 --- Numerical Results Based on Algorithm 5.1 --- p.73 / Chapter 5.4.2 --- Numerical Results Based on Algorithm 5.2 --- p.76 / Chapter 5.4.3 --- Numerical Results Based on Algorithm 5.3 --- p.78 / Chapter 5.4.4 --- Summary of Numerical Results --- p.86 / Chapter 5.5 --- Chapter Summary --- p.87 / Chapter 6 --- Conclusion --- p.88 / Bibliography --- p.93 / Chapter A --- Search Directions --- p.96 / Chapter A.1 --- Newton's Method --- p.97 / Chapter A.1.1 --- Golden Section Method --- p.99 / Chapter A.2 --- Gradients and Hessian Matrices --- p.100 / Chapter A.2.1 --- Gradient of Φθ(x) --- p.100 / Chapter A.2.2 --- Hessian Matrix of Φθ(x) --- p.101 / Chapter A.2.3 --- Gradient of Φθ(x) --- p.101 / Chapter A.2.4 --- Hessian Matrix of φθ (x) --- p.102 / Chapter A.2.5 --- Gradient and Hessian Matrix of Φθ(x) in Terms of ∇xφθ (x) and∇2xxφθ (x) --- p.102 / Chapter A.2.6 --- "Gradient of φθ+,u(x)" --- p.102 / Chapter A.2.7 --- "Hessian Matrix of φθ+,u(x)" --- p.103 / Chapter A.2.8 --- "Gradient and Hessian Matrix of Φθ+,u(x) in Terms of ∇xφθ+,u(x)and ∇2xxφθ+,u(x)" --- p.103 / Chapter A.3 --- Newton's Directions --- p.103 / Chapter A.3.1 --- Newton Direction of Φθ (x) in Terms of ∇xφθ (x) and ∇2xxφθ(x) --- p.104 / Chapter A.3.2 --- "Newton Direction of Φθ+,u(x) in Terms of ∇xφθ+,u(x) and ∇2xxφθ,u(x)" --- p.104 / Chapter A.4 --- Feasible Descent Directions for the Minimization Problems (Pθ) and (Pθ+) --- p.105 / Chapter A.4.1 --- Feasible Descent Direction for the Minimization Prob- lems (Pθ) --- p.105 / Chapter A.4.2 --- Feasible Descent Direction for the Minimization Prob- lems (Pθ+) --- p.107 / Chapter B --- Randomly Generated Test Problems for Positive Definite Quadratic Programming --- p.109 / Chapter B.l --- Convex Quadratic Programming Problems with Linear Con- straints --- p.110 / Chapter B.l.1 --- General Description of Test Problems --- p.110 / Chapter B.l.2 --- The Objective Function --- p.112 / Chapter B.l.3 --- The Linear Constraints --- p.113 / Chapter B.2 --- Convex Quadratic Programming Problems with Quadratic In- equality Constraints --- p.116 / Chapter B.2.1 --- The Quadratic Constraints --- p.117