Optimization is a scientific discipline that lies at the boundary<br />between pure and applied mathematics. Indeed, while on the one hand<br />some of its developments involve rather theoretical concepts, its<br />most successful algorithms are on the other hand heavily used by<br />numerous companies to solve scheduling and design problems on a<br />daily basis.<br /><br />Our research started with the study of the conic formulation for<br />convex optimization problems. This approach was already studied in<br />the seventies but has recently gained a lot of interest due to<br />development of a new class of algorithms called interior-point<br />methods. This setting is able to exploit the two most important<br />characteristics of convexity:<br /><br />- a very rich duality theory (existence of a dual problem that is<br />strongly related to the primal problem, with a very symmetric<br />formulation),<br />- the ability to solve these problems efficiently,<br />both from the theoretical (polynomial algorithmic complexity) and<br />practical (implementations allowing the resolution of large-scale<br />problems) points of view.<br /><br />Most of the research in this area involved so-called self-dual<br />cones, where the dual problem has exactly the same structure as the<br />primal: the most famous classes of convex optimization problems<br />(linear optimization, convex quadratic optimization and semidefinite<br />optimization) belong to this category. We brought some contributions <br />in this field:<br />- a survey of interior-point methods for linear optimization, with <br />an emphasis on the fundamental principles that lie behind the design <br />of these algorithms,<br />- a computational study of a method of linear approximation of convex <br />quadratic optimization (more precisely, the second-order cone that <br />can be used in the formulation of quadratic problems is replaced by a <br />polyhedral approximation whose accuracy can be guaranteed a priori),<br />- an application of semidefinite optimization to classification, <br />whose principle consists in separating different classes of patterns <br />using ellipsoids defined in the feature space (this approach was <br />successfully applied to the prediction of student grades).<br /><br />However, our research focussed on a much less studied category of<br />convex problems which does not rely on self-dual cones, i.e.<br />structured problems whose dual is formulated very differently from<br />the primal. We studied in particular<br />- geometric optimization, developed in the late sixties, which<br />possesses numerous application in the field of engineering<br />(entropy optimization, used in information theory, also belongs to<br />this class of problems)<br />- l_p-norm optimization, a generalization of linear and convex<br />quadratic optimization, which allows the formulation of constraints<br />built around expressions of the form |ax+b|^p (where p is a fixed<br />exponent strictly greater than 1).<br /><br />For each of these classes of problems, we introduced a new type of<br />convex cone that made their formulation as standard conic problems<br />possible. This allowed us to derive very simplified proofs of the<br />classical duality results pertaining to these problems, notably weak<br />duality (a mere consequence of convexity) and the absence of a<br />duality gap (strong duality property without any constraint<br />qualification, which does not hold in the general convex case). We<br />also uncovered a very surprising result that stipulates that<br />geometric optimization can be viewed as a limit case of l_p-norm<br />optimization. Encouraged by the similarities we observed, we<br />developed a general framework that encompasses these two classes of<br />problems and unifies all the previously obtained conic formulations.<br /><br />We also brought our attention to the design of interior-point<br />methods to solve these problems. The theory of polynomial algorithms<br />for convex optimization developed by Nesterov and Nemirovski asserts<br />that the main ingredient for these methods is a computable<br />self-concordant barrier function for the corresponding cones. We<br />were able to define such a barrier function in the case of<br />l_p-norm optimization (whose parameter, which is the main<br />determining factor in the algorithmic complexity of the method, is<br />proportional to the number of variables in the formulation and<br />independent from p) as well as in the case of the general<br />framework mentioned above.<br /><br />Finally, we contributed a survey of the self-concordancy property,<br />improving some useful results about the value of the complexity<br />parameter for certain categories of barrier functions and providing<br />some insight on the reason why the most commonly adopted definition<br />for self-concordant functions is the best possible.

Résumé Nous présentons en premier lieu dans ce travail les principales notions de la théorie de l'Analyse Limite (AL) — ou théorie des charges limites — en mécanique. Puis nous proposons une méthode de point intérieur destinée à résoudre des problèmes de programmation convexe posés par la méthode statique de l'AL, en vue d'obtenir des bornes inférieures de la charge limite (ou de ruine) d'un système mécanique. Les principales caractéristiques de cette méthode de point intérieur sont exposées en détail, et particulièrement son itération type. En second lieu, nous exposons l'application de cet algorithme sur un problème concret d'analyse limite, sur une large gamme de tailles numériques, et nous comparons pour validation les résultats obtenus avec ceux déjà existants ainsi qu'avec ceux calculés à partir de versions linéarisées du problème statique. Nous analysons également les résultats obtenus pour des problèmes classiques avec matériaux de Gurson, pour lesquels la linéarisation ou la programmation conique ne s'applique pas. La deuxième partie de cet ouvrage a trait à la méthode cinématique de l'analyse limite, qui, elle, s'occupe de fournir des bornes supérieures des charges limites. En premier lieu, nous traitons de l'équivalence entre la méthode cinématique classique et la méthode cinématique mixe, en partant d'une l'approche variationnelle fournie précédemment par Radenkovic et Nguyen. Ensuite, prenant en compte les exigences particulières aux formulations numériques, nous présentons une méthode mixte originale, parfaitement cinématique, utilisant aussi bien des champs de vitesses linéaires que quadratiques, continus ou discontinus. Son modus operandi pratique est tiré de l'analyse des conditions d'optimalité de Karush, Kuhn et Tucker, fournissant par là un exemple significatif d'interaction fructueuse entre la mécanique et la programmation mathématique. La méthode est testée sur des problèmes classiques avec les critères de plasticité de von Mises/Tresca et Gurson. Ces test démontrent l'efficacité remarquable de cette méthode mixte — qui par ailleurs n'utilise que le critère de plasticité comme information sur le matériau — et sa robustesse, laquelle s'avère même supérieure à celle de codes commerciaux récents de programmation conique. Enfin, nous présentons une approche de décomposition, elle aussi originale, des problèmes de bornes supérieures en analyse limite. Cette approche est basée à la fois sur la méthode cinématique mixte et l'algorithme de point intérieur précédents, et elle est conçue pour utiliser jusqu'à des champs de vitesse quadratiques discontinus. Détaillée dans le cas de la déformation plane, cette approche apparaît très rapidement convergente, ainsi que nous le vérifions sur le problème du barreau comprimé de von Mises/Tresca dans le cas de champs de vitesse linéaires continus. Puis elle est appliquée, dans le cas de champs quadratiques discontinus, au problème classique de la stabilité du talus vertical de Tresca, avec des résultats particulièrement remarquables puisqu'ils améliorent nettement les solutions cinématiques connues jusqu'à présent dans la littérature sur le sujet. Cette caractéristique de forte convergence qualifie particulièrement cette méthode de décomposition comme algorithme de base pour une parallélisation directe— ou récursive — de l'approche par éléments finis de l'analyse limite. Abstract Firstly, the main notions of the theory of Limit analysis (LA) in Mechanics —or collapse load theory – is presented. Then is proposed an Interior Point method to solve convex programming problems raised by the static method of LA, in order to obtain lower bounds to the collapse (or limit) load of a mechanical system. We explain the main features of this Interior Point method, describing in particular its typical iteration. Secondly, we show and analyze the results of its application to a practical Limit Analysis problem, for a wide range of sizes, and we compare them for validation with existing results and with those of linearized versions of the static problem. Classical problems are also analyzed for Gurson materials to which linearization or conic programming does not apply. The second part of this work focuses on the kinematical method of Limit Analysis, aiming this time to provide upper bounds on collapse loads. In a first step, we detail the equivalence between the classical an general mixed approaches, starting from an earlier variational approach of Radenkovic and Nguyen. In a second step, keeping in mind numerical formulation requirements, an original purely kinematical mixed method—using linear or quadratic, continuous or discontinuous velocity fields as virtual variables—is proposed. Its practical modus operandi is deduced from the Karush-Kuhn-Tucker optimality conditions, providing an example of crossfertilization between mechanics and mathematical programming. The method is tested on classical problems for von Mises/tresca and Gurson plasticity criteria. Using only the yield criterion as material data, it appears very efficient and robust, even more reliable than recent conic commercial codes. Furthermore, both static and kinematic present approaches give rise to the first solutions of problem for homogeneous Gurson materials. Finally, an original decomposition approach of the upper bound method of limit analysis is proposed. It is based on both previous kinematical approach and interior point solver, using up to discontinuous quadratic velocity. Detailed in plane strain, this method appears very rapidly convergent, as verified in the von Mises/Tresca compressed bar problem in the linear continuous velocity case. Then the method is applied, using discontinuous quadratic velocity fields, to the classical problem of the stability of a Tresca vertical cut, with very significant results as they notably improved the kinematical solutions of the literature. Moreover its strong convergence qualifies this decomposition scheme as a suitable algorithm for a direct—or recursive—parallelization of the LA finite element approach.

Optimization is a scientific discipline that lies at the boundary between pure and applied mathematics. Indeed, while on the one hand some of its developments involve rather theoretical concepts, its most successful algorithms are on the other hand heavily used by numerous companies to solve scheduling and design problems on a daily basis. Our research started with the study of the conic formulation for convex optimization problems. This approach was already studied in the seventies but has recently gained a lot of interest due to development of a new class of algorithms called interior-point methods. This setting is able to exploit the two most important characteristics of convexity: - a very rich duality theory (existence of a dual problem that is strongly related to the primal problem, with a very symmetric formulation), - the ability to solve these problems efficiently, both from the theoretical (polynomial algorithmic complexity) and practical (implementations allowing the resolution of large-scale problems) point of views. Most of the research in this area involved so-called self-dual cones, where the dual problem has exactly the same structure as the primal: the most famous classes of convex optimization problems (linear optimization, convex quadratic optimization and semidefinite optimization) belong to this category. We brought some contributions in this field: - a survey of interior-point methods for linear optimization, with an emphasis on the fundamental principles that lie behind the design of these algorithms, - a computational study of a method of linear approximation of convex quadratic optimization (more precisely, the second-order cone that can be used in the formulation of quadratic problems is replaced by a polyhedral approximation whose accuracy that can be guaranteed a priori), - an application of semidefinite optimization to classification, whose principle consists in separating different classes of patterns using ellipsoids defined in the feature space (this approach was successfully applied to the prediction of student grades). However, our research focussed on a much less studied category of convex problems which does not rely on self-dual cones, i.e. structured problems whose dual is formulated very differently from the primal. We studied in particular - geometric optimization, developed in the late sixties, which possesses numerous application in the field of engineering (entropy optimization, used in information theory, also belongs to this class of problems) - l_p-norm optimization, a generalization of linear and convex quadratic optimization, which allows the formulation of constraints built around expressions of the form |ax+b|^p (where p is a fixed exponent strictly greater than 1). For each of these classes of problems, we introduced a new type of convex cone that made their formulation as standard conic problems possible. This allowed us to derive very simplified proofs of the classical duality results pertaining to these problems, notably weak duality (a mere consequence of convexity) and the absence of a duality gap (strong duality property without any constraint qualification, which does not hold in the general convex case). We also uncovered a very surprising result that stipulates that geometric optimization can be viewed as a limit case of l_p-norm optimization. Encouraged by the similarities we observed, we developed a general framework that encompasses these two classes of problems and unifies all the previously obtained conic formulations. We also brought our attention to the design of interior-point methods to solve these problems. The theory of polynomial algorithms for convex optimization developed by Nesterov and Nemirovsky asserts that the main ingredient for these methods is a computable self-concordant barrier function for the corresponding cones. We were able to define such a barrier function in the case of l_p-norm optimization (whose parameter, which is the main determining factor in the algorithmic complexity of the method, is proportional to the number of variables in the formulation and independent from p) as well as in the case of the general framework mentioned above. Finally, we contributed a survey of the self-concordancy property, improving some useful results about the value of the complexity parameter for certain categories of barrier functions and providing some insight on the reason why the most commonly adopted definition for self-concordant functions is the best possible.