The first chapter provides some basic definitions and results from the theory of convex analysis and nonlinear mappings related to our work. Some sufficient conditions for the existence of a solution of problem (GVIP) are also recalled. In the second chapter, we first illustrate the scope of the auxiliary problem procedure designed to solve problems like (GVIP) by examining some well-known methods included in that framework. Then, we review the most representative convergence results for that class of methods that can be found in the literature in the case where F is singlevalued as well as in the multivalued case. Finally, we somewhat discuss the particular case of projection methods to solve affine variational inequalities. The third chapter introduces the variational convergence notion of Mosco and combines it with the auxiliary problem principle. Then, we recall the convergence conditions existing for the resulting perturbed scheme before our own contribution and we comment them. Finally, we introduce and illustrate the rate of convergence condition that we impose on the perturbations to obtain better convergence results. Chapter 4 presents global and local convergence results for the family of perturbed methods in the case where F is singlevalued. We also discuss how our results extend or improve the previous ones. Chapter 5 studies the multivalued case. First, we present convergence results generalizing those obtained when there is no perturbations. Then, we relax the scheme by means of a notion of enlargement of an operator and we provide convergence conditions for this inexact scheme. In Chapter 6, we build a bundle algorithm to solve problem (GVIP) and we study its convergence.

Uma maneira de resolver problemas de programação linear de grande escala é explorar a relaxação lagrangeana das restrições \"difíceis\'\' e utilizar métodos de subgradientes. Populares por fornecerem rapidamente boas aproximações de soluções duais, eles não produzem diretamente as soluções primais. Para obtê-las com custo computacional adequado, pode-se construir seqüências ergódicas ou utilizar uma técnica proposta recentemente, denominada algoritmo do volume. As propriedades teóricas de convergência não foram bem estabelecidas nesse algoritmo, mas pequenas modificações permitem a demonstração da convergência dual. Destacam-se como adaptações o algoritmo do volume revisado, um método de feixes específico, e o algoritmo do volume incorporado ao método de variação do alvo. Este trabalho foi baseado no estudo desses algoritmos e de todos os conceitos envolvidos, em especial, análise convexa e otimização não diferenciável. Estudamos as principais diferenças teóricas desses métodos e realizamos comparações numéricas com problemas lineares e lineares inteiros, em particular, o corte máximo em grafos. / One way to solve large-scale linear programming problems is to exploit the Lagrangian relaxation of the difficult constraints and use subgradient methods. Such methods are popular as they give good approximations of dual solutions. Unfortunately, they do not directly yield primal solutions. Two alternatives to obtain primal solutions under reasonable computational cost are the construction of ergodic sequences and the use of the recently developed volume algorithm. While the convergence of ergodic sequences is well understood, the convergence properties of the volume algorithm is not well established in the original paper. This lead to some modifications of the original method to ease the proof of dual convergence. Three alternatives are the revised volume algorithm, a special case of the bundle method, and the volume algorithm incorporated by the variable target value method. The aim of this work is to study such algorithms and all related concepts, especially convex analysis and nondifferentiable optimization. We analysed the main theoretical differences among the methods and performed numerical experiments with linear and integer problems, in particular, the maximum cut problem on graphs.

Uma maneira de resolver problemas de programação linear de grande escala é explorar a relaxação lagrangeana das restrições \"difíceis\'\' e utilizar métodos de subgradientes. Populares por fornecerem rapidamente boas aproximações de soluções duais, eles não produzem diretamente as soluções primais. Para obtê-las com custo computacional adequado, pode-se construir seqüências ergódicas ou utilizar uma técnica proposta recentemente, denominada algoritmo do volume. As propriedades teóricas de convergência não foram bem estabelecidas nesse algoritmo, mas pequenas modificações permitem a demonstração da convergência dual. Destacam-se como adaptações o algoritmo do volume revisado, um método de feixes específico, e o algoritmo do volume incorporado ao método de variação do alvo. Este trabalho foi baseado no estudo desses algoritmos e de todos os conceitos envolvidos, em especial, análise convexa e otimização não diferenciável. Estudamos as principais diferenças teóricas desses métodos e realizamos comparações numéricas com problemas lineares e lineares inteiros, em particular, o corte máximo em grafos. / One way to solve large-scale linear programming problems is to exploit the Lagrangian relaxation of the difficult constraints and use subgradient methods. Such methods are popular as they give good approximations of dual solutions. Unfortunately, they do not directly yield primal solutions. Two alternatives to obtain primal solutions under reasonable computational cost are the construction of ergodic sequences and the use of the recently developed volume algorithm. While the convergence of ergodic sequences is well understood, the convergence properties of the volume algorithm is not well established in the original paper. This lead to some modifications of the original method to ease the proof of dual convergence. Three alternatives are the revised volume algorithm, a special case of the bundle method, and the volume algorithm incorporated by the variable target value method. The aim of this work is to study such algorithms and all related concepts, especially convex analysis and nondifferentiable optimization. We analysed the main theoretical differences among the methods and performed numerical experiments with linear and integer problems, in particular, the maximum cut problem on graphs.

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Cette thèse développe une méthode de faisceau non convexe pour la minimisation de fonctions localement lipschitziennes lower C1 puis l’applique à des problèmes de synthèse de lois de commande structurées issus de l’industrie aéronautique. Ici loi de commande structurée fait référence à une architecture de contrôle, qui se compose d’éléments comme les PIDs, combinés avec des filtres variés, et comprenant beaucoup moins de paramètres de réglage qu’un contrôleur d’ordre plein. Ce type de problème peut se formuler dans le cadre théorique et général de la programmation non convexe et non lisse. Parmi les techniques numériques efficaces pour résoudre ces problèmes non lisses, nous avons dans ce travail, opté pour les méthodes de faisceau, convenablement étendues au cas non convexe. Celles-ci utilisent un oracle qui, en chaque itéré x, retourne la valeur de la fonction et un sous-gradient de Clarke arbitraire. Afin de générer un pas de descente satisfaisant à partir de l’itéré sérieux courant, ces techniques stockent et accumulent de l’information, dans ce que l’on appelle le faisceau, obtenu à partir d’évaluations successives de l’oracle à chaque pas d’essai insatisfaisant. Dans cette thèse, on propose de construire le faisceau en décalant vers le bas une tangente de l’objectif en un pas d’essai ne constituant pas un pas de descente satisfaisant. Le décalage est indispensable dans le cas non convexe pour préserver la consistance, on dit encore l’exactitude, du modèle vis à vis de l’objectif. L’algorithme développé est validé sur un problème de synthèse conjointe du pilote automatique et de la loi des commandes de vol d’un avion civil en un point de vol donné et sur un problème de synthèse de loi de commande par séquencement de gain pour le contrôle longitudinal dans une enveloppe de vol. / This thesis develops a non convex bundle method for the minimization of lower C1 locally Lipschitz functions which it then applies to the synthesis of structured control laws for problems arising in aerospace control. Here a structured control law refers to a control architecture preferred by practitioners, which consist of elements like PIDs, combined with various filters, featuring significantly less tunable parameters than a full-order controller. This type of problem can be formulated under the theoretical and general framework of non convex and non smooth programming. Among the efficient numerical techniques to solve such non smooth problems, we have in this work opted for bundle methods, suitably extended to address non-convex optimization programs. Bundle methods use oracles which at every iterate x return the function value and one unspecified Clarke subgradient. In order to generate descent steps away from a current serious iterate, these techniques hinge on storing and accumulating information, called the bundle, obtained from successive evaluations of the oracle along the unsuccessful trial steps. In this thesis, we propose to build the bundle by shifting down a tangent of the objective at a trial step which is not a satisfactory descent step. The shift is essential in the non convex case in order to preserve the consistency, named also the exactitude, of the model with regard to the objective. The developed algorithm is validated on a synthesis problem combining the automatic pilot and the flight control law of a civil aircraft at a given flying point ; and a gain scheduled control law synthesis for the longitudinal control in a flight envelope.

Le problème d’ordre linéaire (LOP) a reçu beaucoup d’attention dans différents domaines d’application, allant de l’archéologie à l’ordonnancement en passant par l’économie et même de la psychologie mathématique. Ce problème est aussi connu pour être parmi les problèmes NP-difficiles. Nous considérons dans cette thèse une variante de (LOP) sous contrainte de cardinalité. Nous cherchons donc un ordre linéaire d’un sous-ensemble de sommets du graphe de préférences de cardinalité fixée et de poids maximum. Ce problème, appelé (FCLOP) pour ’fixed-cardinality linear ordering problem’, n’a pas été étudié en tant que tel dans la littérature scientifique même si plusieurs applications dans les domaines de macro-économie, de classification dominante ou de transport maritime existent concrètement. On retrouve en fait ses caractéristiques dans les modèles étendus de sous-graphes acycliques. Le problème d’ordre linéaire est déjà connu comme un problème NP-difficile et il a donné lieu à de nombreuses études, tant théoriques sur la structure polyédrale de l’ensemble des solutions réalisables en variables 0-1 que numériques grâce à des techniques de relaxation et de séparation progressive. Cependant on voit qu’il existe de nombreux cas dans la littérature, dans lesquelles des solveurs de Programmation Linéaire en nombres entiers comme CPLEX peuvent en résoudre certaines instances en moins de 10 secondes, mais une fois que la cardinalité est limitée, ces mêmes instances deviennent très difficiles à résoudre. Sur les aspects polyédraux, nous avons étudié le polytope de FCLOP, défini plusieurs classes d’inégalités valides et identifié la dimension ainsi que certaines inégalités qui définissent des facettes pour le polytope de FCLOP. Nous avons introduit un algorithme Relax-and-Cut basé sur ces résultats pour résoudre les instances du problème. Dans cette étude, nous nous sommes également concentrés sur la relaxation Lagrangienne pour résoudre ces cas difficiles. Nous avons étudié différentes stratégies de relaxation et nous avons comparé les bornes duales par rapport à la consolidation obtenue à partir de chaque stratégie de relâcher les contraintes afin de détecter le sous-ensemble des contraintes le plus approprié. Les résultats numériques montrent que nous pouvons trouver des bornes duales de très haute qualité. Nous avons également mis en place une méthode de décomposition Lagrangienne. Dans ce but, nous avons décomposé le modèle de FCLOP en trois sous-problèmes (au lieu de seulement deux) associés aux contraintes de ’tournoi’, de ’graphes sans circuits’ et de ’cardinalité’. Les résultats numériques montrent une amélioration significative de la qualité des bornes duales pour plusieurs cas. Nous avons aussi mis en oeuvre une méthode de plans sécants (cutting plane algorithm) basée sur la relaxation pure des contraintes de circuits. Dans cette méthode, on a relâché une partie des contraintes et on les a ajoutées au modèle au cas où il y a des de/des violations. Les résultats numériques montrent des performances prometteuses quant à la réduction du temps de calcul et à la résolution d’instances difficiles hors d’atteinte des solveurs classiques en PLNE. / Linear Ordering Problem (LOP) has receive significant attention in different areas of application, ranging from transportation and scheduling to economics and even archeology and mathematical psychology. It is classified as a NP-hard problem. Assume a complete weighted directed graph on V n , |V n |= n. A permutation of the elements of this finite set of vertices is a linear order. Now let p be a given fixed integer number, 0 ≤ p ≤ n. The p-Fixed Cardinality Linear Ordering Problem (FCLOP) is looking for a subset of vertices containing p nodes and a linear order on the nodes in S. Graphically, there exists exactly one directed arc between every pair of vertices in an LOP feasible solution, which is also a complete cycle-free digraph and the objective is to maximize the sum of the weights of all the arcs in a feasible solution. In the FCLOP, we are looking for a subset S ⊆ V n such that |S|= p and an LOP on these S nodes. Hence the objective is to find the best subset of the nodes and an LOP over these p nodes that maximize the sum of the weights of all the arcs in the solution. Graphically, a feasible solution of the FCLOP is a complete cycle-free digraph on S plus a set of n − p vertices that are not connected to any of the other vertices. There are several studies available in the literature focused on polyhedral aspects of the linear ordering problem as well as various exact and heuristic solution methods. The fixed cardinality linear ordering problem is presented for the first time in this PhD study, so as far as we know, there is no other study in the literature that has studied this problem. The linear ordering problem is already known as a NP-hard problem. However one sees that there exist many instances in the literature that can be solved by CPLEX in less than 10 seconds (when p = n), but once the cardinality number is limited to p (p < n), the instance is not anymore solvable due to the memory issue. We have studied the polytope corresponding to the FCLOP for different cardinality values. We have identified dimension of the polytope, proposed several classes of valid inequalities and showed that among these sets of valid inequalities, some of them are defining facets for the FCLOP polytope for different cardinality values. We have then introduced a Relax-and-Cut algorithm based on these results to solve instances of the FCLOP. To solve the instances of the problem, in the beginning, we have applied the Lagrangian relaxation algorithm. We have studied different relaxation strategies and compared the dual bound obtained from each case to detect the most suitable subproblem. Numerical results show that some of the relaxation strategies result better dual bound and some other contribute more in reducing the computational time and provide a relatively good dual bound in a shorter time. We have also implemented a Lagrangian decomposition algorithm, decom-6 posing the FCLOP model to three subproblems (instead of only two subproblems). The interest of decomposing the FCLOP model to three subproblems comes mostly from the nature of the three subproblems, which are relatively quite easier to solve compared to the initial FCLOP model. Numerical results show a significant improvement in the quality of dual bounds for several instances. We could also obtain relatively quite better dual bounds in a shorter time comparing to the other relaxation strategies. We have proposed a cutting plane algorithm based on the pure relaxation strategy. In this algorithm, we firstly relax a subset of constraints that due to the problem structure, a very few number of them are active. Then in the course of the branch-and-bound tree we verify if there exist any violated constraint among the relaxed constraints or. Then the characterized violated constraints will be globally added to the model. (...)