Nous proposons dans cette thèse trois contributions principales aux méthodes d'apprentissage de dictionnaire. La première est un critère de parcimonie par groupes adapté à la NMF lorsque la mesure de distorsion choisie est la divergence d'Itakura-Saito. Dans la plupart des signaux de musique on peut trouver de longs intervalles où seulement une source est active (des soli). Le critère de parcimonie par groupe que nous proposons permet de trouver automatiquement de tels segments et d'apprendre un dictionnaire adapté à chaque source. Ces dictionnaires permettent ensuite d'effectuer la tâche de séparation dans les intervalles où les sources sont mélangés. Ces deux tâches d'identification et de séparation sont effectuées simultanément en une seule passe de l'algorithme que nous proposons. Notre deuxième contribution est un algorithme en ligne pour apprendre le dictionnaire à grande échelle, sur des signaux de plusieurs heures. L'espace mémoire requis par une NMF estimée en ligne est constant alors qu'il croit linéairement avec la taille des signaux fournis dans la version standard, ce qui est impraticable pour des signaux de plus d'une heure. Notre troisième contribution touche à l'interaction avec l'utilisateur. Pour des signaux courts, l'apprentissage aveugle est particulièrement dificile, et l'apport d'information spécifique au signal traité est indispensable. Notre contribution est similaire à l'inpainting et permet de prendre en compte des annotations temps-fréquences. Elle repose sur l'observation que la quasi-totalité du spectrogramme peut etre divisé en régions spécifiquement assignées à chaque source. Nous décrivons une extension de NMF pour prendre en compte cette information et discutons la possibilité d'inférer cette information automatiquement avec des outils d'apprentissage statistique simples. / In this thesis we provide three main contributions to blind source separation methods based on NMF. Our first contribution is a group-sparsity inducing penalty specifically tailored for Itakura-Saito NMF. In many music tracks, there are whole intervals where only one source is active at the same time. The group-sparsity penalty we propose allows to blindly indentify these intervals and learn source specific dictionaries. As a consequence, those learned dictionaries can be used to do source separation in other parts of the track were several sources are active. These two tasks of identification and separation are performed simultaneously in one run of group-sparsity Itakura-Saito NMF. Our second contribution is an online algorithm for Itakura-Saito NMF that allows to learn dictionaries on very large audio tracks. Indeed, the memory complexity of a batch implementation NMF grows linearly with the length of the recordings and becomes prohibitive for signals longer than an hour. In contrast, our online algorithm is able to learn NMF on arbitrarily long signals with limited memory usage. Our third contribution deals user informed NMF. In short mixed signals, blind learning becomes very hard and sparsity do not retrieve interpretable dictionaries. Our contribution is very similar in spirit to inpainting. It relies on the empirical fact that, when observing the spectrogram of a mixture signal, an overwhelming proportion of it consists in regions where only one source is active. We describe an extension of NMF to take into account time-frequency localized information on the absence/presence of each source. We also investigate inferring this information with tools from machine learning.

La séparation aveugle de source consiste à estimer les signaux de sources uniquement à partir des mélanges observés. Le problème peut être séparé en deux catégories en fonction du modèle de mélange: mélanges instantanés, où le retard et la réverbération (effet multi-chemin) ne sont pas pris en compte, et des mélanges convolutives qui sont plus généraux mais plus compliqués. De plus, le bruit additif au niveaux des capteurs et le réglage sous-déterminé, où il y a moins de capteurs que les sources, rendent le problème encore plus difficile.Dans cette thèse, tout d'abord, nous avons étudié le lien entre deux méthodes existantes pour les mélanges instantanés: analyse des composants indépendants (ICA) et analyse des composant parcimonieux (SCA). Nous avons ensuite proposé une nouveau formulation qui fonctionne dans les cas déterminés et sous-déterminés, avec et sans bruit. Les évaluations numériques montrent l'avantage des approches proposées.Deuxièmement, la formulation proposés est généralisés pour les mélanges convolutifs avec des signaux de parole. En intégrant un nouveau modèle d'approximation, les algorithmes proposés fonctionnent mieux que les méthodes existantes, en particulier dans des scénarios bruyant et / ou de forte réverbération.Ensuite, on prend en compte la technique de décomposition morphologique et l'utilisation de parcimonie structurée qui conduit à des algorithmes qui peuvent mieux exploiter les structures des signaux audio. De telles approches sont testées pour des mélanges convolutifs sous-déterminés dans un scénario non-aveugle.Enfin, en bénéficiant du modèle NMF (factorisation en matrice non-négative), nous avons combiné l'hypothèse de faible-rang et de parcimonie et proposé de nouvelles approches pour les mélanges convolutifs sous-déterminés. Les expériences illustrent la bonne performance des algorithmes proposés pour les signaux de musique, en particulier dans des scénarios de forte réverbération. / Blind source separation (BSS) consists of estimating the source signals only from the observed mixtures. The problem can be divided into two categories according to the mixing model: instantaneous mixtures, where delay and reverberation (multi-path effect) are not taken into account, and convolutive mixtures which are more general but more complicated. Moreover, the additive noise at the sensor level and the underdetermined setting, where there are fewer sensors than the sources, make the problem even more difficult.In this thesis, we first studied the link between two existing methods for instantaneous mixtures: independent component analysis (ICA) and sparse component analysis (SCA). We then proposed a new formulation that works in both determined and underdetermined cases, with and without noise. Numerical evaluations show the advantage of the proposed approaches.Secondly, the proposed formulation is generalized for convolutive mixtures with speech signals. By integrating a new approximation model, the proposed algorithms work better than existing methods, especially in noisy and/or high reverberation scenarios.Then, we take into account the technique of morphological decomposition and the use of structured sparsity which leads to algorithms that can better exploit the structures of audio signals. Such approaches are tested for underdetermined convolutive mixtures in a non-blind scenario.At last, being benefited from the NMF model, we combined the low-rank and sparsity assumption and proposed new approaches for under-determined convolutive mixtures. The experiments illustrate the good performance of the proposed algorithms for music signals, especially in strong reverberation scenarios.

Nous sommes intéressé par la reconstitution et la prédiction des séries temporelles multivariées à partir des données partiellement observées et/ou agrégées.La motivation du problème vient des applications dans la gestion du réseau électrique.Nous envisageons des outils capables de résoudre le problème d'estimation de plusieurs domaines.Après investiguer le krigeage, qui est une méthode de la litérature de la statistique spatio-temporelle, et une méthode hybride basée sur le clustering des individus, nous proposons un cadre général de reconstitution et de prédiction basé sur la factorisation de matrice nonnégative.Ce cadre prend en compte de manière intrinsèque la corrélation entre les séries temporelles pour réduire drastiquement la dimension de l'espace de paramètres.Une fois que le problématique est formalisé dans ce cadre, nous proposons deux extensions par rapport à l'approche standard.La première extension prend en compte l'autocorrélation temporelle des individus.Cette information supplémentaire permet d'améliorer la précision de la reconstitution.La deuxième extension ajoute une composante de régression dans la factorisation de matrice nonnégative.Celle-ci nous permet d'utiliser dans l'estimation du modèle des variables exogènes liées avec la consommation électrique, ainsi de produire des facteurs plus interprétatbles, et aussi améliorer la reconstitution.De plus, cette méthod nous donne la possibilité d'utiliser la factorisation de matrice nonnégative pour produire des prédictions.Sur le côté théorique, nous nous intéressons à l'identifiabilité du modèle, ainsi qu'à la propriété de la convergence des algorithmes que nous proposons.La performance des méthodes proposées en reconstitution et en prédiction est testé sur plusieurs jeux de données de consommation électrique à niveaux d'agrégation différents. / We are interested in the recovery and prediction of multiple time series from partially observed and/or aggregate data.Motivated by applications in electricity network management, we investigate tools from multiple fields that are able to deal with such data issues.After examining kriging from spatio-temporal statistics and a hybrid method based on the clustering of individuals, we propose a general framework based on nonnegative matrix factorization.This frameworks takes advantage of the intrisic correlation between the multivariate time series to greatly reduce the dimension of the parameter space.Once the estimation problem is formalized in the nonnegative matrix factorization framework, two extensions are proposed to improve the standard approach.The first extension takes into account the individual temporal autocorrelation of each of the time series.This increases the precision of the time series recovery.The second extension adds a regression layer into nonnegative matrix factorization.This allows exogenous variables that are known to be linked with electricity consumption to be used in estimation, hence makes the factors obtained by the method to be more interpretable, and also increases the recovery precision.Moreover, this method makes the method applicable to prediction.We produce a theoretical analysis on the framework which concerns the identifiability of the model and the convergence of the algorithms that are proposed.The performance of proposed methods to recover and forecast time series is tested on several multivariate electricity consumption datasets at different aggregation level.

The advances in data-collecting technologies provides great opportunities to access large sample-size data sets with high dimensionality. Variable selection is an important procedure to extract useful knowledge from such complex data. While in many real-data applications, appropriate selection of variables should facilitate the model interpretation and computation efficiency. It is thus important to incorporate domain knowledge of underlying data generation mechanism to select key variables for improving the model performance. However, general variable selection techniques, such as the best subset selection and the Lasso, often do not take the underlying data generation mechanism into considerations. This thesis proposal aims to develop statistical modeling methodologies with a focus on the structured variable selection towards better model interpretation and computation efficiency. Specifically, this thesis proposal consists of three parts: an additive heredity model with coefficients incorporating the multi-level data, a regularized dynamic generalized linear model with piecewise constant functional coefficients, and a structured variable selection method within the best subset selection framework. In Chapter 2, an additive heredity model is proposed for analyzing mixture-of-mixtures (MoM) experiments. The MoM experiment is different from the classical mixture experiment in that the mixture component in MoM experiments, known as the major component, is made up of sub-components, known as the minor components. The proposed model considers an additive structure to inherently connect the major components with the minor components. To enable a meaningful interpretation for the estimated model, we apply the hierarchical and heredity principles by using the nonnegative garrote technique for model selection. The performance of the additive heredity model was compared to several conventional methods in both unconstrained and constrained MoM experiments. The additive heredity model was then successfully applied in a real problem of optimizing the Pringlestextsuperscript{textregistered} potato crisp studied previously in the literature. In Chapter 3, we consider the dynamic effects of variables in the generalized linear model such as logistic regression. This work is motivated from the engineering problem with varying effects of process variables to product quality caused by equipment degradation. To address such challenge, we propose a penalized dynamic regression model which is flexible to estimate the dynamic coefficient structure. The proposed method considers modeling the functional coefficient parameter as piecewise constant functions. Specifically, under the penalized regression framework, the fused lasso penalty is adopted for detecting the changes in the dynamic coefficients. The group lasso penalty is applied to enable a sparse selection of variables. Moreover, an efficient parameter estimation algorithm is also developed based on alternating direction method of multipliers. The performance of the dynamic coefficient model is evaluated in numerical studies and three real-data examples. In Chapter 4, we develop a structured variable selection method within the best subset selection framework. In the literature, many techniques within the LASSO framework have been developed to address structured variable selection issues. However, less attention has been spent on structured best subset selection problems. In this work, we propose a sparse Ridge regression method to address structured variable selection issues. The key idea of the proposed method is to re-construct the regression matrix in the angle of experimental designs. We employ the estimation-maximization algorithm to formulate the best subset selection problem as an iterative linear integer optimization (LIO) problem. the mixed integer optimization algorithm as the selection step. We demonstrate the power of the proposed method in various structured variable selection problems. Moverover, the proposed method can be extended to the ridge penalized best subset selection problems. The performance of the proposed method is evaluated in numerical studies. / Doctor of Philosophy / The advances in data-collecting technologies provides great opportunities to access large sample-size data sets with high dimensionality. Variable selection is an important procedure to extract useful knowledge from such complex data. While in many real-data applications, appropriate selection of variables should facilitate the model interpretation and computation efficiency. It is thus important to incorporate domain knowledge of underlying data generation mechanism to select key variables for improving the model performance. However, general variable selection techniques often do not take the underlying data generation mechanism into considerations. This thesis proposal aims to develop statistical modeling methodologies with a focus on the structured variable selection towards better model interpretation and computation efficiency. The proposed approaches have been applied to real-world problems to demonstrate their model performance.

In Markov chain models in finance and healthcare a transition matrix over a certain time interval is needed but only a transition matrix over a longer time interval may be available. The problem arises of determining a stochastic $p$th root of astochastic matrix (the given transition matrix). By exploiting the theory of functions of matrices, we develop results on the existence and characterization of stochastic $p$th roots. Our contributions include characterization of when a real matrix hasa real $p$th root, a classification of $p$th roots of a possibly singular matrix,a sufficient condition for a $p$th root of a stochastic matrix to have unit row sums,and the identification of two classes of stochastic matrices that have stochastic $p$th roots for all $p$. We also delineate a wide variety of possible configurationsas regards existence, nature (primary or nonprimary), and number of stochastic roots,and develop a necessary condition for existence of a stochastic root in terms of the spectrum of the given matrix. On the computational side, we emphasize finding an approximate stochastic root: perturb the principal root $A^{1/p}$ or the principal logarithm $\log(A)$ to the nearest stochastic matrix or the nearest intensity matrix, respectively, if they are not valid ones;minimize the residual $\normF{X^p-A}$ over all stochastic matrices $X$ and also over stochastic matrices that are primary functions of $A$. For the first two nearness problems, the global minimizers are found in the Frobenius norm. For the last two nonlinear programming problems, we derive explicit formulae for the gradient and Hessian of the objective function $\normF{X^p-A}^2$ and investigate Newton's method, a spectral projected gradient method (SPGM) and the sequential quadratic programming method to solve the problem as well as various matrices to start the iteration. Numerical experiments show that SPGM starting with the perturbed $A^{1/p}$to minimize $\normF{X^p-A}$ over all stochastic matrices is method of choice.Finally, a new algorithm is developed for computing arbitrary real powers $A^\a$ of a matrix $A\in\mathbb{C}^{n\times n}$. The algorithm starts with a Schur decomposition,takes $k$ square roots of the triangular factor $T$, evaluates an $[m/m]$ Pad\'e approximant of $(1-x)^\a$ at $I - T^$, and squares the result $k$ times. The parameters $k$ and $m$ are chosen to minimize the cost subject to achieving double precision accuracy in the evaluation of the Pad\'e approximant, making use of a result that bounds the error in the matrix Pad\'e approximant by the error in the scalar Pad\'e approximant with argument the norm of the matrix. The Pad\'e approximant is evaluated from the continued fraction representation in bottom-up fashion, which is shown to be numerically stable. In the squaring phase the diagonal and first superdiagonal are computed from explicit formulae for $T^$, yielding increased accuracy. Since the basic algorithm is designed for $\a\in(-1,1)$, a criterion for reducing an arbitrary real $\a$ to this range is developed, making use of bounds for the condition number of the $A^\a$ problem. How best to compute $A^k$ for a negative integer $k$ is also investigated. In numerical experiments the new algorithm is found to be superior in accuracy and stability to several alternatives,including the use of an eigendecomposition, a method based on the Schur--Parlett\alg\ with our new algorithm applied to the diagonal blocks and approaches based on the formula $A^\a = \exp(\a\log(A))$.

This is a literature study, in linear algebra, about positive and nonnegative matrices and their special properties. We say that a matrix or a vector is positive/nonnegative if all of its entries are positive/nonnegative. First, we study some generalities and become acquainted with two types of nonnegative matrices; irreducible and reducible. After exploring their characteristics we investigate and prove the two main theorems of this subject, namely Perron's and Perron-Frobenius' theorem. In short Perron's theorem from 1907 tells us that the spectral radius of a positive matrix is a simple eigenvalue of the matrix and that its eigenvector can be taken to be positive. In 1912, Georg Frobenius generalized Perron's results also to irreducible nonnegative matrices. The two theorems have a wide range of applications in both pure mathematics and practical matters. In real world scenarios, many measurements are nonnegative (length, time, amount, etc.) and so their mathematical formulations often relate to Perron-Frobenius theory. The theory's importance to linear dynamical systems, such as Markov chains, cannot be overstated; it determines when, and to what, an iterative process will converge. This result is in turn the underlying theory for the page-ranking algorithm developed by Google in 1998. We will see examples of all these applications in chapters four and five where we will be particularly interested in different types of Markov chains.  The theory in this thesis can be found in many books. Here, most of the material is gathered from Horn-Johnson [5], Meyer [9] and Shapiro [10]. However, all of the theorems and proofs are formulated in my own way and the examples and illustrations are concocted by myself, unless otherwise noted. / Det här är en litteraturstudie, inom linjär algebra, om positiva och icke-negativa matriser och deras speciella egenskaper. Vi säger att en matris eller en vektor är positiv/icke-negativ om alla dess element är positiva/icke-negativa. Inledningsvis går vi igenom några grundläggande begrepp och bekanta oss med två typer av icke-negativa matriser; irreducibla och reducibla. Efter att vi utforskat deras egenskaper så studerar vi och bevisar ämnets två huvudsatser; Perrons och Perron-Frobenius sats. Kortfattat så säger Perrons sats, från 1907, att spektralradien för en positiv matris är ett simpelt egenvärde till matrisen och att dess egenvektor kan tas positiv. År 1912 så generaliserade Georg Frobenius Perrons resultat till att gälla också för irreducibla icke-negativa matriser.  De två satserna har både många teoretiska och praktiska tillämpningar. Många verkliga scenarios har icke-negativa mått (längd, tid, mängd o.s.v) och därför relaterar dess matematiska formulering till Perron-Frobenius teori. Teorin är betydande även för linjära dynamiska system, såsom Markov-kedjor, eftersom den avgör när, och till vad, en iterativ process konvergerar. Det resultatet är i sin tur den underliggande teorin bakom algoritmen PageRank som utvecklades av Google år 1998. Vi kommer se exempel på alla dessa tillämpningar i kapitel fyra och fem, där vi speciellt intresserar oss för olika typer av Markov-kedjor. Teorin i den här artikeln kan hittas i många böcker. Det mesta av materialet som presenteras här har hämtats från Horn-Johnson [5], Meyer [9] och Shapiro [10]. Däremot är alla satser och bevis formulerade på mitt eget sätt och alla exempel, samt illustrationer, har jag skapat själv, om inget annat sägs.

Factor models attribute stock returns to a linear combination of factors. A model with great explanatory power (R2) can be used to estimate the systematic risk of an investment. One of the most important factors is the industry which the company of the stock operates in. In commercial risk models this factor is often determined with a manually constructed stock classification scheme such as GICS. We present Natural Language Industry Scheme (NLIS), an automatic and multivalued classification scheme based on topic modeling. The topic modeling is performed on transcripts of company earnings calls and identifies a number of topics analogous to industries. We use non-negative matrix factorization (NMF) on a term-document matrix of the transcripts to perform the topic modeling. When set to explain returns of the MSCI USA index we find that NLIS consistently outperforms GICS, often by several hundred basis points. We attribute this to NLIS’ ability to assign a stock to multiple industries. We also suggest that the proportions of industry assignments for a given stock could correspond to expected future revenue sources rather than current revenue sources. This property could explain some of NLIS’ success since it closely relates to theoretical stock pricing. / Faktormodeller förklarar aktieprisrörelser med en linjär kombination av faktorer. En modell med hög förklaringsgrad (R2) kan användas föratt skatta en investerings systematiska risk. En av de viktigaste faktorerna är aktiebolagets industritillhörighet. I kommersiella risksystem bestäms industri oftast med ett aktieklassifikationsschema som GICS, publicerat av ett finansiellt institut. Vi presenterar Natural Language Industry Scheme (NLIS), ett automatiskt klassifikationsschema baserat på topic modeling. Vi utför topic modeling på transkript av aktiebolags investerarsamtal. Detta identifierar ämnen, eller topics, som är jämförbara med industrier. Topic modeling sker genom icke-negativmatrisfaktorisering (NMF) på en ord-dokumentmatris av transkripten. När NLIS används för att förklara prisrörelser hos MSCI USA-indexet finner vi att NLIS överträffar GICS, ofta med 2-3 procent. Detta tillskriver vi NLIS förmåga att ge flera industritillhörigheter åt samma aktie. Vi föreslår också att proportionerna hos industritillhörigheterna för en aktie kan motsvara förväntade inkomstkällor snarare än nuvarande inkomstkällor. Denna egenskap kan också vara en anledning till NLIS framgång då den nära relaterar till teoretisk aktieprissättning.

Cette thèse considère l'apprentissage de motifs récurrents dans la perception multimodale. Elle s'attache à développer des modèles robotiques de ces facultés telles qu'observées chez l'enfant, et elle s'inscrit en cela dans le domaine de la robotique développementale.Elle s'articule plus précisément autour de deux thèmes principaux qui sont d'une part la capacité d'enfants ou de robots à imiter et à comprendre le comportement d'humains, et d'autre part l'acquisition du langage. A leur intersection, nous examinons la question de la découverte par un agent en développement d'un répertoire de motifs primitifs dans son flux perceptuel. Nous spécifions ce problème et établissons son lien avec ceux de l'indétermination de la traduction décrit par Quine et de la séparation aveugle de source tels qu'étudiés en acoustique.Nous en étudions successivement quatre sous-problèmes et formulons une définition expérimentale de chacun. Des modèles d'agents résolvant ces problèmes sont également décrits et testés. Ils s'appuient particulièrement sur des techniques dites de sacs de mots, de factorisation de matrices et d'apprentissage par renforcement inverse. Nous approfondissons séparément les trois problèmes de l'apprentissage de sons élémentaires tels les phonèmes ou les mots, de mouvements basiques de danse et d'objectifs primaires composant des tâches motrices complexes. Pour finir nous étudions le problème de l'apprentissage d'éléments primitifs multimodaux, ce qui revient à résoudre simultanément plusieurs des problèmes précédents. Nous expliquons notamment en quoi cela fournit un modèle de l'ancrage de mots acoustiques / This thesis focuses on learning recurring patterns in multimodal perception. For that purpose it develops cognitive systems that model the mechanisms providing such capabilities to infants; a methodology that fits into thefield of developmental robotics.More precisely, this thesis revolves around two main topics that are, on the one hand the ability of infants or robots to imitate and understand human behaviors, and on the other the acquisition of language. At the crossing of these topics, we study the question of the how a developmental cognitive agent can discover a dictionary of primitive patterns from its multimodal perceptual flow. We specify this problem and formulate its links with Quine's indetermination of translation and blind source separation, as studied in acoustics.We sequentially study four sub-problems and provide an experimental formulation of each of them. We then describe and test computational models of agents solving these problems. They are particularly based on bag-of-words techniques, matrix factorization algorithms, and inverse reinforcement learning approaches. We first go in depth into the three separate problems of learning primitive sounds, such as phonemes or words, learning primitive dance motions, and learning primitive objective that compose complex tasks. Finally we study the problem of learning multimodal primitive patterns, which corresponds to solve simultaneously several of the aforementioned problems. We also details how the last problems models acoustic words grounding.

Modern complex dynamical systems typically possess a multiechelon hierarchical hybrid structure characterized by continuous-time dynamics at the lower-level units and logical decision-making units at the higher-level of hierarchy. Hybrid dynamical systems involve an interacting countable collection of dynamical systems defined on subregions of the partitioned state space. Thus, in addition to traditional control systems, hybrid control systems involve supervising controllers which serve to coordinate the (sometimes competing) actions of the lower-level controllers. A subclass of hybrid dynamical systems are impulsive dynamical systems which consist of three elements, namely, a continuous-time differential equation, a difference equation, and a criterion for determining when the states of the system are to be reset. One of the main topics of this dissertation is the development of stability analysis and control design for impulsive dynamical systems. Specifically, we generalize Poincare's theorem to dynamical systems possessing left-continuous flows to address the stability of limit cycles and periodic orbits of left-continuous, hybrid, and impulsive dynamical systems. For nonlinear impulsive dynamical systems, we present partial stability results, that is, stability with respect to part of the system's state. Furthermore, we develop adaptive control framework for general class of impulsive systems as well as energy-based control framework for hybrid port-controlled Hamiltonian systems. Extensions of stability theory for impulsive dynamical systems with respect to the nonnegative orthant of the state space are also addressed in this dissertation. Furthermore, we design optimal output feedback controllers for set-point regulation of linear nonnegative dynamical systems. Another main topic that has been addressed in this research is the stability analysis of large-scale dynamical systems. Specifically, we extend the theory of vector Lyapunov functions by constructing a generalized comparison system whose vector field can be a function of the comparison system states as well as the nonlinear dynamical system states. Furthermore, we present a generalized convergence result which, in the case of a scalar comparison system, specializes to the classical Krasovskii-LaSalle invariant set theorem. Moreover, we develop vector dissipativity theory for large-scale dynamical systems based on vector storage functions and vector supply rates. Finally, using a large-scale dynamical systems perspective, we develop a system-theoretic foundation for thermodynamics. Specifically, using compartmental dynamical system energy flow models, we place the universal energy conservation, energy equipartition, temperature equipartition, and entropy nonconservation laws of thermodynamics on a system-theoretic basis.

Basés sur les outils théoriques et algorithmiques de la programmation DC et DCA, les travaux de recherche dans cette thèse portent sur les approches locales et globales pour l'optimisation non convexe et l'optimisation globale en variables mixtes entières. La thèse comporte 5 chapitres. Le premier chapitre présente les fondements de la programmation DC et DCA, et techniques de Séparation et Evaluation (B&B) (utilisant la technique de relaxation DC pour le calcul des bornes inférieures de la valeur optimale) pour l'optimisation globale. Y figure aussi des résultats concernant la pénalisation exacte pour la programmation en variables mixtes entières. Le deuxième chapitre est consacré au développement d'une méthode DCA pour la résolution d'une classe NP-difficile des programmes non convexes non linéaires en variables mixtes entières. Ces problèmes d'optimisation non convexe sont tout d'abord reformulées comme des programmes DC via les techniques de pénalisation en programmation DC de manière que les programmes DC résultants soient efficacement résolus par DCA et B&B bien adaptés. Comme première application en optimisation financière, nous avons modélisé le problème de gestion de portefeuille sous le coût de transaction concave et appliqué DCA et B&B à sa résolution. Dans le chapitre suivant nous étudions la modélisation du problème de minimisation du coût de transaction non convexe discontinu en gestion de portefeuille sous deux formes : la première est un programme DC obtenu en approximant la fonction objectif du problème original par une fonction DC polyèdrale et la deuxième est un programme DC mixte 0-1 équivalent. Et nous présentons DCA, B&B, et l'algorithme combiné DCA-B&B pour leur résolution. Le chapitre 4 étudie la résolution exacte du problème multi-objectif en variables mixtes binaires et présente deux applications concrètes de la méthode proposée. Nous nous intéressons dans le dernier chapitre à ces deux problématiques challenging : le problème de moindres carrés linéaires en variables entières bornées et celui de factorisation en matrices non négatives (Nonnegative Matrix Factorization (NMF)). La méthode NMF est particulièrement importante de par ses nombreuses et diverses applications tandis que les applications importantes du premier se trouvent en télécommunication. Les simulations numériques montrent la robustesse, rapidité (donc scalabilité), performance et la globalité de DCA par rapport aux méthodes existantes. / Based on theoretical and algorithmic tools of DC programming and DCA, the research in this thesis focus on the local and global approaches for non convex optimization and global mixed integer optimization. The thesis consists of 5 chapters. The first chapter presents fundamentals of DC programming and DCA, and techniques of Branch and Bound method (B&B) for global optimization (using the DC relaxation technique for calculating lower bounds of the optimal value). It shall include results concerning the exact penalty technique in mixed integer programming. The second chapter is devoted of a DCA method for solving a class of NP-hard nonconvex nonlinear mixed integer programs. These nonconvex problems are firstly reformulated as DC programs via penalty techniques in DC programming so that the resulting DC programs are effectively solved by DCA and B&B well adapted. As a first application in financial optimization, we modeled the problem pf portfolio selection under concave transaction costs and applied DCA and B&B to its solutions. In the next chapter we study the modeling of the problem of minimization of nonconvex discontinuous transaction costs in portfolio selection in two forms: the first is a DC program obtained by approximating the objective function of the original problem by a DC polyhedral function and the second is an equivalent mixed 0-1 DC program. And we present DCA, B&B algorithm, and a combined DCA-B&B algorithm for their solutions. Chapter 4 studied the exact solution for the multi-objective mixed zero-one linear programming problem and presents two practical applications of proposed method. We are interested int the last chapter two challenging problems: the linear integer least squares problem and the Nonnegative Mattrix Factorization problem (NMF). The NMF method is particularly important because of its many various applications of the first are in telecommunications. The numerical simulations show the robustness, speed (thus scalability), performance, and the globality of DCA in comparison to existent methods.