Cette thèse est consacrée à la recherche des zéros d'un opérateur maximal monotone structuré, à l'aide de systèmes dynamiques dissipatifs continus et discrets. Les solutions sont obtenues comme limites des trajectoires lorsque le temps t tend vers l'infini. On s'intéressera principalement aux dynamiques obtenues par régularisation de type Levenberg-Marquardt de la méthode de Newton. On décrira aussi les approches basées sur des dynamiques voisines.Dans un cadre Hilbertien, on s'intéresse à la recherche des zéros de l'opérateur maximal monotone structuré M = A + B, où A est un opérateur maximal monotone général et B est un opérateur monotone Lipschitzien. Nous introduisons des dynamiques continues et discrètes de type Newton régularisé faisant intervenir d'une façon séparée les résolvantes de l'opérateur A (implicites), et des évaluations de B (explicites). A l'aide de la représentation de Minty de l'opérateur A comme une variété Lipschitzienne, nous reformulons ces dynamiques sous une forme relevant du théorème de Cauchy-Lipschitz. Nous nous intéressons au cas particulier où A est le sous différentiel d'une fonction convexe, semi-continue inférieurement, et propre, et B est le gradient d'une fonction convexe, différentiable. Nous étudions le comportement asymptotique des trajectoires. Lorsque le terme de régularisation ne tend pas trop vite vers zéro, et en s'appuyant sur une analyse asymptotique de type Lyapunov, nous montrons la convergence des trajectoires. Par ailleurs, nous montrons la dépendance Lipschitzienne des trajectoires par rapport au terme de régularisation.Puis nous élargissons notre étude en considérant différentes classes de systèmes dynamiques visant à résoudre les inclusions monotones gouvernées par un opérateur maximal monotone structuré M = $partialPhi$+ B, où $partialPhi$ désigne le sous différentiel d'une fonction convexe, semicontinue inférieurement, et propre, et B est un opérateur monotone cocoercif. En s'appuyant sur une analyse asymptotique de type Lyapunov, nous étudions le comportement asymptotique des trajectoires de ces systèmes. La discrétisation temporelle de ces dynamiques fournit desalgorithmes forward-backward (certains nouveaux ).Finalement, nous nous intéressons à l'étude du comportement asymptotique des trajectoires de systèmes dynamiques de type Newton régularisé, dans lesquels on introduit un terme supplémentaire de viscosité évanescente de type Tikhonov. On obtient ainsi la sélection asymptotique d'une solution de norme minimale. / This thesis is devoted to finding zeroes of structured maximal monotone operators, by using discrete and continuous dissipative dynamical systems. The solutions are obtained as the limits of trajectories when the time t tends towards infinity.We pay special attention to the dynamics that are obtained by Levenberg-Marquardt regularization of Newton's method. We also revisit the approaches based on some related dynamical systems.In a Hilbert framework, we are interested in finding zeroes of a structured maximal monotone operator M = A + B, where A is a general maximal monotone operator, and B is monotone and locally Lipschitz continuous. We introduce discrete and continuous dynamical systems which are linked to Newton's method. They involve separately B and the resolvents of A, and are designed to splitting methods. Based on the Minty representation of A as a Lipschitz manifold, we show that these dynamics can be formulated as differential systems, which are relevant to the Cauchy-Lipschitz theorem. We focus on the particular case where A is the subdifferential of a convex lower semicontinuous proper function, and B is the gradient of a convex, continuously differentiable function. We study the asymptotic behavior of trajectories. When the regularization parameter does not tend to zero too rapidly, and by using Lyapunov asymptotic analysis, we show the convergence of trajectories. Besides, we show the Lipschitz continuous dependence of the solution with respect to the regularization term.Then we extend our study by considering various classes of dynamical systems which aim at solving inclusions governed by structured monotone operators M = $partialPhi$+ B, where $partialPhi$ is the subdifferential of a convex lower semicontinuous function, and B is a monotone cocoercive operator. By a Lyapunov analysis, we show the convergence properties of the orbits of these systems. The time discretization of these dynamics gives various forward-backward splittingmethods (some new).Finally, we focus on the study of the asymptotic behavior of trajectories of the regularized Newton dynamics, in which we introduce an additional vanishing Tikhonov-like viscosity term.We thus obtain the asymptotic selection of the solution of minimal norm.

This thesis is concerned with the development of novel numerical methods for solving nondifferentiable convex optimization problems in real Hilbert spaces and with the investigation of their asymptotic behavior. To this end, we are also making use of monotone operator theory as some of the provided algorithms are originally designed to solve monotone inclusion problems. After introducing basic notations and preliminary results in convex analysis, we derive two numerical methods based on different smoothing strategies for solving nondifferentiable convex optimization problems. The first approach, known as the double smoothing technique, solves the optimization problem with some given a priori accuracy by applying two regularizations to its conjugate dual problem. A special fast gradient method then solves the regularized dual problem such that an approximate primal solution can be reconstructed from it. The second approach affects the primal optimization problem directly by applying a single regularization to it and is capable of using variable smoothing parameters which lead to a more accurate approximation of the original problem as the iteration counter increases. We then derive and investigate different primal-dual methods in real Hilbert spaces. In general, one considerable advantage of primal-dual algorithms is that they are providing a complete splitting philosophy in that the resolvents, which arise in the iterative process, are only taken separately from each maximally monotone operator occurring in the problem description. We firstly analyze the forward-backward-forward algorithm of Combettes and Pesquet in terms of its convergence rate for the objective of a nondifferentiable convex optimization problem. Additionally, we propose accelerations of this method under the additional assumption that certain monotone operators occurring in the problem formulation are strongly monotone. Subsequently, we derive two Douglas–Rachford type primal-dual methods for solving monotone inclusion problems involving finite sums of linearly composed parallel sum type monotone operators. To prove their asymptotic convergence, we use a common product Hilbert space strategy by reformulating the corresponding inclusion problem reasonably such that the Douglas–Rachford algorithm can be applied to it. Finally, we propose two primal-dual algorithms relying on forward-backward and forward-backward-forward approaches for solving monotone inclusion problems involving parallel sums of linearly composed monotone operators. The last part of this thesis deals with different numerical experiments where we intend to compare our methods against algorithms from the literature. The problems which arise in this part are manifold and they reflect the importance of this field of research as convex optimization problems appear in lots of applications of interest.

This thesis is concerned with the development of novel numerical methods for solving nondifferentiable convex optimization problems in real Hilbert spaces and with the investigation of their asymptotic behavior. To this end, we are also making use of monotone operator theory as some of the provided algorithms are originally designed to solve monotone inclusion problems. After introducing basic notations and preliminary results in convex analysis, we derive two numerical methods based on different smoothing strategies for solving nondifferentiable convex optimization problems. The first approach, known as the double smoothing technique, solves the optimization problem with some given a priori accuracy by applying two regularizations to its conjugate dual problem. A special fast gradient method then solves the regularized dual problem such that an approximate primal solution can be reconstructed from it. The second approach affects the primal optimization problem directly by applying a single regularization to it and is capable of using variable smoothing parameters which lead to a more accurate approximation of the original problem as the iteration counter increases. We then derive and investigate different primal-dual methods in real Hilbert spaces. In general, one considerable advantage of primal-dual algorithms is that they are providing a complete splitting philosophy in that the resolvents, which arise in the iterative process, are only taken separately from each maximally monotone operator occurring in the problem description. We firstly analyze the forward-backward-forward algorithm of Combettes and Pesquet in terms of its convergence rate for the objective of a nondifferentiable convex optimization problem. Additionally, we propose accelerations of this method under the additional assumption that certain monotone operators occurring in the problem formulation are strongly monotone. Subsequently, we derive two Douglas–Rachford type primal-dual methods for solving monotone inclusion problems involving finite sums of linearly composed parallel sum type monotone operators. To prove their asymptotic convergence, we use a common product Hilbert space strategy by reformulating the corresponding inclusion problem reasonably such that the Douglas–Rachford algorithm can be applied to it. Finally, we propose two primal-dual algorithms relying on forward-backward and forward-backward-forward approaches for solving monotone inclusion problems involving parallel sums of linearly composed monotone operators. The last part of this thesis deals with different numerical experiments where we intend to compare our methods against algorithms from the literature. The problems which arise in this part are manifold and they reflect the importance of this field of research as convex optimization problems appear in lots of applications of interest.