Quelques thèmes en l'analyse variationnelle et optimisation

In this thesis, we first study the theory of [gamma]-limits. Besides some basic properties of [gamma]-limits,expressions of sequential [gamma]-limits generalizing classical results of Greco are presented. These limits also give us a clue to a unified classification of derivatives and tangent cones. Next, we develop an approach to generalized differentiation theory. This allows us to deal with several generalized derivatives of set-valued maps defined directly in primal spaces, such as variational sets, radial sets, radial derivatives, Studniarski derivatives. Finally, we study calculus rules of these derivatives and applications related to optimality conditions and sensitivity analysis.Dans cette thèse, j’étudie d’abord la théorie des [gamma]-limites. En dehors de quelques propriétés fondamentales des [gamma]-limites, les expressions de [gamma]-limites séquentielles généralisant des résultats de Greco sont présentées. En outre, ces limites nous donnent aussi une idée d’une classification unifiée de la tangence et la différentiation généralisée. Ensuite, je développe une approche des théories de la différentiation généralisée. Cela permet de traiter plusieurs dérivées généralisées des multi-applications définies directement dans l’espace primal, tels que des ensembles variationnels,des ensembles radiaux, des dérivées radiales, des dérivées de Studniarski. Finalement, j’étudie les règles de calcul de ces dérivées et les applications liées aux conditions d’optimalité et à l’analyse de sensibilité

Hybridization of multi-objective deterministic particle swarm with derivative-free local searches

The paper presents a multi-objective derivative-free and deterministic global/local hybrid algorithm for the efficient and effective solution of simulation-based design optimization (SBDO) problems. The objective is to show how the hybridization of two multi-objective derivative-free global and local algorithms achieves better performance than the separate use of the two algorithms in solving specific SBDO problems for hull-form design. The proposed method belongs to the class of memetic algorithms, where the global exploration capability of multi-objective deterministic particle swarm optimization is enriched by exploiting the local search accuracy of a derivative-free multi-objective line-search method. To the authors best knowledge, studies are still limited on memetic, multi-objective, deterministic, derivative-free, and evolutionary algorithms for an effective and efficient solution of SBDO for hull-form design. The proposed formulation manages global and local searches based on the hypervolume metric. The hybridization scheme uses two parameters to control the local search activation and the number of function calls used by the local algorithm. The most promising values of these parameters were identified using forty analytical tests representative of the SBDO problem of interest. The resulting hybrid algorithm was finally applied to two SBDO problems for hull-form design. For both analytical tests and SBDO problems, the hybrid method achieves better performance than its global and local counterparts

A radial basis function method for solving optimal control problems.

This work presents two direct methods based on the radial basis function (RBF) interpolation and arbitrary discretization for solving continuous-time optimal control problems: RBF Collocation Method and RBF-Galerkin Method. Both methods take advantage of choosing any global RBF as the interpolant function and any arbitrary points (meshless or on a mesh) as the discretization points. The first approach is called the RBF collocation method, in which states and controls are parameterized using a global RBF, and constraints are satisfied at arbitrary discrete nodes (collocation points) to convert the continuous-time optimal control problem to a nonlinear programming (NLP) problem. The resulted NLP is quite sparse and can be efficiently solved by well-developed sparse solvers. The second proposed method is a hybrid approach combining RBF interpolation with Galerkin error projection for solving optimal control problems. The proposed solution, called the RBF-Galerkin method, applies a Galerkin projection to the residuals of the optimal control problem that make them orthogonal to every member of the RBF basis functions. Also, RBF-Galerkin costate mapping theorem will be developed describing an exact equivalency between the Karush–Kuhn–Tucker (KKT) conditions of the NLP problem resulted from the RBF-Galerkin method and discretized form of the first-order necessary conditions of the optimal control problem, if a set of conditions holds. Several examples are provided to verify the feasibility and viability of the RBF method and the RBF-Galerkin approach as means of finding accurate solutions to general optimal control problems. Then, the RBF-Galerkin method is applied to a very important drug dosing application: anemia management in chronic kidney disease. A multiple receding horizon control (MRHC) approach based on the RBF-Galerkin method is developed for individualized dosing of an anemia drug for hemodialysis patients. Simulation results are compared with a population-oriented clinical protocol as well as an individual-based control method for anemia management to investigate the efficacy of the proposed method

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Neural networks, surrogate models and black box algorithms: theory and applications

In this Ph. D. Thesis we will analyze some of the most used surrogate models, together with a particular type of line search black box strategy. After introducing these powerful tools, we will present the Canonical Duality Theory, the potentiality it has to improve these tools, and some of their applications. The principal contributes of this Thesis are the reformulation of the Radial Basis Neural Network problem in its canonical dual form in Section 2.2 and the application of the surrogate models and black box algorithms presented in this Thesis on various real world problems reported in Chapter 3

Neural networks, surrogate models and black box algorithms: theory and applications

In this Ph. D. Thesis we will analyze some of the most used surrogate models, together with a particular type of line search black box strategy. After introducing these powerful tools, we will present the Canonical Duality Theory, the potentiality it has to improve these tools, and some of their applications. The principal contributes of this Thesis are the reformulation of the Radial Basis Neural Network problem in its canonical dual form in Section 2.2 and the application of the surrogate models and black box algorithms presented in this Thesis on various real world problems reported in Chapter 3

OPTIMIZATION OF ALGORITHMS WITH THE OPAL FRAMEWORK

RÉSUMÉ La question d'identifier de bons paramètres a été étudiée depuis longtemps et on peut compter un grand nombre de recherches qui se concentrent sur ce sujet. Certaines de ces recherches manquent de généralité et surtout de re-utilisabilité. Une première raison est que ces projets visent des systèmes spécifiques. En plus, la plupart de ces projets ne se concentrent pas sur les questions fondamentales de l'identification de bons paramètres. Et enfin, il n'y avait pas un outil puissant capable de surmonter des difficulté dans ce domaine. En conséquence, malgré un grand nombre de projets, les utilisateurs n'ont pas trop de possibilité à appliquer les résultats antérieurs à leurs problèmes. Cette thèse propose le cadre OPAL pour identifier de bons paramètres algorithmiques avec des éléments essentiels, indispensables. Les étapes de l'élaboration du cadre de travail ainsi que les résultats principaux sont présentés dans trois articles correspondant aux trois chapitres 4, 5 et 6 de la thèse. Le premier article introduit le cadre par l'intermédiaire d'exemples fondamentaux. En outre, dans ce cadre, la question d'identifier de bons paramètres est modélisée comme un problème d'optimisation non-lisse qui est ensuite résolu par un algorithme de recherche directe sur treillis adaptatifs. Cela réduit l'effort des utilisateurs pour accomplir la tâche d'identifier de bons paramètres. Le deuxième article décrit une extension visant à améliorer la performance du cadre OPAL. L'utilisation efficace de ressources informatiques dans ce cadre se fait par l'étude de plusieurs stratégies d'utilisation du parallélisme et par l'intermédiaire d'une fonctionnalité particulière appelée l'interruption des tâches inutiles. Le troisième article est une description complète du cadre et de son implémentation en Python. En plus de rappeler les caractéristiques principales présentées dans des travaux antérieurs, l'intégration est présentée comme une nouvelle fonctionnalité par une démonstration de la coopération avec un outil de classification. Plus précisément, le travail illustre une coopération de OPAL et un outil de classification pour résoudre un problème d'optimisation des paramètres dont l'ensemble de problèmes tests est trop grand et une seule évaluation peut prendre une journée.----------ABSTRACT The task of parameter tuning question has been around for a long time, spread over most domains and there have been many attempts to address it. Research on this question often lacks in generality and re-utilisability. A first reason is that these projects aim at specific systems. Moreover, some approaches do not concentrate on the fundamental questions of parameter tuning. And finally, there was not a powerful tool that is able to take over the difficulties in this domain. As a result, the number of projects continues to grow, while users are not able to apply the previous achievements to their own problem. The present work systematically approaches parameter tuning by figuring out the fundamental issues and identifying the basic elements for a general system. This provides the base for developing a general and flexible framework called OPAL, which stands for OPtimization of ALgorithms. The milestones in developing the framework as well as the main achievements are presented through three papers corresponding to the three chapters 4, 5 and 6 of this thesis. The first paper introduces the framework by describing the crucial basic elements through some very simple examples. To this end, the paper considers three questions in constructing an automated parameter tuning framework. By answering these questions, we propose OPAL, consisting of indispensable components of a parameter tuning framework. OPAL models the parameter tuning task as a blackbox optimization problem. This reduces the effort of users in launching a tuning session. The second paper shows one of the opportunities to extend the framework. To take advantage of the situations where multiple processors are available, we study various ways of embedding parallelism and develop a feature called ''interruption of unnecessary tasks'' in order to improve performance of the framework. The third paper is a full description of the framework and a release of its Python} implementation. In addition to the confirmations on the methodology and the main features presented in previous works, the integrability is introduced as a new feature of this release through an example of the cooperation with a classification tool. More specifically, the work illustrates a cooperation of OPAL and a classification tool to solve a parameter optimization problem of which the test problem set is too large and an assessment can take a day

Geometric Numerical Integration

The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods