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. / 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.

In this study two nonlinear analysis methods are proposed for investigation of nonlinear effects of high temperature superconductive(HTSC) thin film planar microwave circuits. The MoM-HB combination method is based on the combination formulation of the moment method(MoM) and the harmonic balance(HB) technique. It consists of linear and nonlinear solvers. The power series method treats the voltages at higher order frequencies as the excitations at the corresponding frequencies, and the higher order current distributions are then obtained by using the moment method again. The power series method is simple and fast for finding the output power at higher order frequencies. The MoM-HB combination method is suitable for strong nonlinearity, and it can be also used to find the fundamental current redistribution, conductor loss, and the scattering parameters variation at the fundamental frequency. These two proposed methods are efficient, accurate, and suitable for distributed-type HTSC nonlinearity. They can be easily incorporated into commercial EM CAD softwares to expand their capabilities. These two nonlinear analysis method are validated by analyzing a HTSC stripline filter and HTSC antenna dipole circuits. HTSC microstrip lines are then investigated for the nonlinear effects of HTSC material on the current density distribution over the cross section and the conductor loss as a function of the applied power. The HTSC microstrip patch filters are then studied to show that the HTSCinterconnecting line could dominate the behaviors of the circuits at high power. The variation of the transmission and reflection coefficients with the applied power and the third output power are calculated. The HTSC microstrip line structure with gilded edges is proposed for improving the power handling capability of HTSC thin film circuit based on a specified limit of harmonic generation and conductor loss. A general analysis approach suitable for any thickness of gilding layer is developed by integrating the multi-port network theory into aforementioned proposed nonlinear analysis methods. The conductor loss and harmonic generation of the gilded HTSC microstrip line are investigated.

We present results on smooth and nonsmooth variational properties of {it symmetric} functions of the eigenvalues of a real symmetric matrix argument, as well as {it absolutely symmetric} functions of the singular values of a real rectangular matrix. Such results underpin the theory of optimization problems involving such functions. We answer the question of when a symmetric function of the eigenvalues allows a quadratic expansion around a matrix, and then the stronger question of when it is twice differentiable. We develop simple formulae for the most important nonsmooth subdifferentials of functions depending on the singular values of a real rectangular matrix argument and give several examples. The analysis of the above two classes of functions may be generalized in various larger abstract frameworks. In particular, we investigate how functions depending on the eigenvalues or the singular values of a matrix argument may be viewed as the composition of symmetric functions with the roots of {it hyperbolic polynomials}. We extend the relationship between hyperbolic polynomials and {it self-concordant barriers} (an extremely important class of functions in contemporary interior point methods for convex optimization) by exhibiting a new class of self-concordant barriers obtainable from hyperbolic polynomials.

In this study two nonlinear analysis methods are proposed for investigation of nonlinear effects of high temperature superconductive(HTSC) thin film planar microwave circuits. The MoM-HB combination method is based on the combination formulation of the moment method(MoM) and the harmonic balance(HB) technique. It consists of linear and nonlinear solvers. The power series method treats the voltages at higher order frequencies as the excitations at the corresponding frequencies, and the higher order current distributions are then obtained by using the moment method again. The power series method is simple and fast for finding the output power at higher order frequencies. The MoM-HB combination method is suitable for strong nonlinearity, and it can be also used to find the fundamental current redistribution, conductor loss, and the scattering parameters variation at the fundamental frequency. These two proposed methods are efficient, accurate, and suitable for distributed-type HTSC nonlinearity. They can be easily incorporated into commercial EM CAD softwares to expand their capabilities. These two nonlinear analysis method are validated by analyzing a HTSC stripline filter and HTSC antenna dipole circuits. HTSC microstrip lines are then investigated for the nonlinear effects of HTSC material on the current density distribution over the cross section and the conductor loss as a function of the applied power. The HTSC microstrip patch filters are then studied to show that the HTSCinterconnecting line could dominate the behaviors of the circuits at high power. The variation of the transmission and reflection coefficients with the applied power and the third output power are calculated. The HTSC microstrip line structure with gilded edges is proposed for improving the power handling capability of HTSC thin film circuit based on a specified limit of harmonic generation and conductor loss. A general analysis approach suitable for any thickness of gilding layer is developed by integrating the multi-port network theory into aforementioned proposed nonlinear analysis methods. The conductor loss and harmonic generation of the gilded HTSC microstrip line are investigated.

We present results on smooth and nonsmooth variational properties of {it symmetric} functions of the eigenvalues of a real symmetric matrix argument, as well as {it absolutely symmetric} functions of the singular values of a real rectangular matrix. Such results underpin the theory of optimization problems involving such functions. We answer the question of when a symmetric function of the eigenvalues allows a quadratic expansion around a matrix, and then the stronger question of when it is twice differentiable. We develop simple formulae for the most important nonsmooth subdifferentials of functions depending on the singular values of a real rectangular matrix argument and give several examples. The analysis of the above two classes of functions may be generalized in various larger abstract frameworks. In particular, we investigate how functions depending on the eigenvalues or the singular values of a matrix argument may be viewed as the composition of symmetric functions with the roots of {it hyperbolic polynomials}. We extend the relationship between hyperbolic polynomials and {it self-concordant barriers} (an extremely important class of functions in contemporary interior point methods for convex optimization) by exhibiting a new class of self-concordant barriers obtainable from hyperbolic polynomials.

Many real-world problems can be modeled by generalized equations. The solution of the latter can be a challenging task, and typically requires the use of some efficient numerical procedures, whose convergence analysis often relies on stability properties of a solution in question, and on a suitable over-estimate for the distance of a given point to the solution set of the problem, called error bound. With this thesis, we aim at a unified approach to first- and second-order conditions for stability properties and error bounds for generalized equations. To this end, we study existing and develop new concepts for generalized first-order derivatives of set-valued mappings, and use them to formulate criteria for Lipschitzian stability properties and Lipschitzian error bound conditions. These criteria can all be regarded as the property that a suitable generalized least singular value of a generalized derivative is nonzero. By considering generalized least singular values as an extended real-valued function that depends on arguments of an underlying mapping, we will be able to obtain second-order conditions arising from generalized derivatives of this function to guarantee non-Lipschitzian stability properties and non-Lipschitzian error bound conditions. This allows us to extend the territory covered by some seminal monographs dealing with stability properties and error bounds for generalized equations under first-order conditions. Furthermore, we discuss some specializations of our findings, and work out relations to existing results. Finally, we also investigate correlations between stability properties and error bounds with respect to different problem-formulations of one and the same generalized equation.

We have considered the bilevel programming problem in the case where the lower-level problem admits more than one optimal solution. It is well-known in the literature that in such a situation, the problem is ill-posed from the view point of scalar objective optimization. Thus the optimistic and pessimistic approaches have been suggested earlier in the literature to deal with it in this case. In the thesis, we have developed a unified approach to derive necessary optimality conditions for both the optimistic and pessimistic bilevel programs, which is based on advanced tools from variational analysis. We have obtained various constraint qualifications and stationarity conditions depending on some constructive representations of the solution set-valued mapping of the follower’s problem. In the auxiliary developments, we have provided rules for the generalized differentiation and robust Lipschitzian properties for the lower-level solution setvalued map, which are of a fundamental interest for other areas of nonlinear and nonsmooth optimization. Some of the results of the aforementioned theory have then been applied to derive stationarity conditions for some well-known transportation problems having the bilevel structure.

We have considered the bilevel programming problem in the case where the lower-level problem admits more than one optimal solution. It is well-known in the literature that in such a situation, the problem is ill-posed from the view point of scalar objective optimization. Thus the optimistic and pessimistic approaches have been suggested earlier in the literature to deal with it in this case. In the thesis, we have developed a unified approach to derive necessary optimality conditions for both the optimistic and pessimistic bilevel programs, which is based on advanced tools from variational analysis. We have obtained various constraint qualifications and stationarity conditions depending on some constructive representations of the solution set-valued mapping of the follower’s problem. In the auxiliary developments, we have provided rules for the generalized differentiation and robust Lipschitzian properties for the lower-level solution setvalued map, which are of a fundamental interest for other areas of nonlinear and nonsmooth optimization. Some of the results of the aforementioned theory have then been applied to derive stationarity conditions for some well-known transportation problems having the bilevel structure.

Cette thèse est consacrée à la résolution d'un problème fondamental de l'analyse variationnelle qu'est la recherchede zéros d'opérateurs maximaux monotones dans un espace de Hilbert. Nous nous sommes tout d'abord intéressés au cas de l'opérateur somme étendue de deux opérateurs maximaux monotones; la recherche d'un zéro de cet opérateur est un problème dont la bibliographie est peu fournie: nous proposons une version modifiée de l'algorithme d'éclatement forward-backward utilisant à chaque itération, l'epsilon-élargissement d'un opérateur maximal monotone,afin de construire une solution. Nous avons ensuite étudié la convergence d'un nouvel algorithme de faisceaux pour construire ID zéro d'un opérateur maximal monotone quelconque en dimension finie. Cet algorithme fait intervenir une double approximation polyédrale de l'epsilon-élargissement de l'opérateur considéré / This thesis is devoted to solving a basic problem of variational analysis which is the search of zeros of maximal monotone operators in a Hilbert space. First of aIl, we concentrate on the case of the extended som of two maximal monotone operators; the search of a zero of this operator is a problem for which the bibliography is not abondant: we purpose a modified version of the forward-backward splitting algorithm using at each iteration, the epsilon-enlargement of a maximal monotone operator, in order to construet a solution. Secondly, we study the convergence of a new bondie algorithm to construet a zero of an arbitrary maximal monotone operator in a finite dimensional space. In this algorithm, intervenes a double polyhedral approximation of the epsilon-enlargement of the considered operator

Orientador: Roberto Andreani / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-22T06:15:13Z (GMT). No. of bitstreams: 1 BaygorreaCusihuallpa_Nancy_M.pdf: 1656614 bytes, checksum: 036b8eeb6a7f3e461c7ca051fed6fd3d (MD5) Previous issue date: 2013 / Resumo: Esta dissertação desenvolve um estudo detalhado da convergência local do método de ponto proximal para resolver o problema de encontrar zeros de operadores maximais sem a condição de monotonicidade. Em particular, é estudada a convergência dos métodos de multiplicadores proximais para resolver problemas de otimização não linear sem a condição de convexidade. Para obter os resultados desejados apresentaremos ferramentas de análise variacional para substituir a condição de monotonicidade maximal do operador como também, a teoria de dualidade generalizada para a aplicação do método de multiplicadores proximais. Apresentamos também uma aplicação do algoritmo do ponto proximal aos métodos dos multiplicadores para uma classe de problemas gerais baseados num esquema de dualidade generalizada / Abstract: In this dissertation we will develop a detailed study of local convergence of proximal point method for finding a root of maximal operators without monotonicity. In particular, it is studied the convergence for proximal method of multipliers by solving nonlinear optimization problems without convexity conditions. In order to obtain the desired results we will study some variational analysis tools to replace maximal monotonicity condition of operators as well as general duality theory which is t reacted to study an application to proximal method of multipliers. Also, we show an application of the proximal point algorithm to the multipliers methods for a class of problems which is based in general duality scheme / Mestrado / Matematica Aplicada / Mestra em Matemática Aplicada