We consider the constrained convex minimization problem: min_x∈C f(x) we will present gradient projection method which generates a sequence x^k according to the formula x^(k+1) = P_c(x^k − £\_k∇f(x^k)), k= 0, 1, ¡P ¡P ¡P , our ideal is rewritten the formula as a xed point algorithm: x^(k+1) = T_(£\k)x^k, k = 0, 1, ¡P ¡P ¡P is used to solve the minimization problem. In this paper, we present the gradient projection method(GPM) and different choices of the stepsize to discuss the convergence of gradient projection method which converge to a solution of the concerned problem.

Esta tese é um estudo acerca da análise de convergência de vários métodos numéricos de primeira e de segunda ordem para resolver problemas de programação matemática e as condições de otimalidade associadas. Nossas principais ferramentas são as condições sequenciais de otimalidade. As condições sequenciais de otimalidade oferecem um quadro teórico para a análise de convergência para várias famílias de métodos de primeira ordem sob condições de qualificações fracas. Nesta tese, apresentamos, para cada condição sequencial de otimalidade, a condição de qualificação mínima associada e mostramos as relações com outras condições de qualificação conhecidas. Este fato tem implicações práticas, uma vez que enfraquece as hipóteses requeridas para a convergência de vários métodos numéricos cujos critérios de paradas estão associados às condições sequenciais de otimalidade. Ainda mais, esse tipo de resultado não pode ser melhorado usando outras condições de qualificações. Nós estendemos a noção de condições sequenciais de otimalidade de primeira ordem, para incorporar informações de segunda ordem. Apresentamos, segundo nosso conhecimento, a primeira condição sequencial de otimalidade de segunda ordem, adequada para a análise de convergência de vários métodos numéricos com convergência a pontos estacionários de segunda ordem, como por exemplo métodos baseados no Lagrangeano aumentado, regiões de confiança e SQP regularizado. Associada com a nova condição sequencial de segunda ordem, temos uma nova condição de qualificação, mais fraca que as outras condições de qualificações utilizadas para a análise de convergência para métodos numéricos de segunda ordem. Nós situamos essa nova condição de qualificação com respeito a outras condições de qualificação usadas em análise de convergência. Finalmente apresentamos outra razão pela qual a condição fraca necessária de segunda ordem é a condição de segunda ordem adequada quando lidarmos com a convergência de algoritmos práticos / This thesis deals with the convergence analysis for several rst-and-second-order numerical methods used to solve mathematical programming problems. Our main tools are the sequential optimality conditions. First-order sequential optimality conditions oer a framework to the study of the convergence analysis of several families of rst-order methods, under weak constraint qualications. In this thesis, we will introduce, for each sequential optimality condition the minimal constraint qualications associated with it and we will show their relationships with other constraint qualications. This fact has a practical aspect, since, we improve the convergence analysis of practical methods with stopping criteria associated with sequential optimality conditions. This results can not be improved by using another weak constraint qualications. We will extend the notion of rst-order sequential optimality conditions to incorporate secondorder information. We will introduce, to the best of our knowledge, the rst second-order sequential optimality condition, suitable to the study of the convergence analysis of several second-order methods including methods based on the augmented lagrangian, trust-region and regularized SQP. Associated with the second-order sequential optimality condition, we have a new constraint qualication weaker than all constraint qualications used for the convergence analysis of second-order methods. We show the relationships of this new constraint qualications with other constraint qualications used for algorithmic purposes. We will also present a new reason why the weak secondorder necessary condition is the natural second-order condition when we are dealing with practical numerical methods

Esta tese é um estudo acerca da análise de convergência de vários métodos numéricos de primeira e de segunda ordem para resolver problemas de programação matemática e as condições de otimalidade associadas. Nossas principais ferramentas são as condições sequenciais de otimalidade. As condições sequenciais de otimalidade oferecem um quadro teórico para a análise de convergência para várias famílias de métodos de primeira ordem sob condições de qualificações fracas. Nesta tese, apresentamos, para cada condição sequencial de otimalidade, a condição de qualificação mínima associada e mostramos as relações com outras condições de qualificação conhecidas. Este fato tem implicações práticas, uma vez que enfraquece as hipóteses requeridas para a convergência de vários métodos numéricos cujos critérios de paradas estão associados às condições sequenciais de otimalidade. Ainda mais, esse tipo de resultado não pode ser melhorado usando outras condições de qualificações. Nós estendemos a noção de condições sequenciais de otimalidade de primeira ordem, para incorporar informações de segunda ordem. Apresentamos, segundo nosso conhecimento, a primeira condição sequencial de otimalidade de segunda ordem, adequada para a análise de convergência de vários métodos numéricos com convergência a pontos estacionários de segunda ordem, como por exemplo métodos baseados no Lagrangeano aumentado, regiões de confiança e SQP regularizado. Associada com a nova condição sequencial de segunda ordem, temos uma nova condição de qualificação, mais fraca que as outras condições de qualificações utilizadas para a análise de convergência para métodos numéricos de segunda ordem. Nós situamos essa nova condição de qualificação com respeito a outras condições de qualificação usadas em análise de convergência. Finalmente apresentamos outra razão pela qual a condição fraca necessária de segunda ordem é a condição de segunda ordem adequada quando lidarmos com a convergência de algoritmos práticos / This thesis deals with the convergence analysis for several rst-and-second-order numerical methods used to solve mathematical programming problems. Our main tools are the sequential optimality conditions. First-order sequential optimality conditions oer a framework to the study of the convergence analysis of several families of rst-order methods, under weak constraint qualications. In this thesis, we will introduce, for each sequential optimality condition the minimal constraint qualications associated with it and we will show their relationships with other constraint qualications. This fact has a practical aspect, since, we improve the convergence analysis of practical methods with stopping criteria associated with sequential optimality conditions. This results can not be improved by using another weak constraint qualications. We will extend the notion of rst-order sequential optimality conditions to incorporate secondorder information. We will introduce, to the best of our knowledge, the rst second-order sequential optimality condition, suitable to the study of the convergence analysis of several second-order methods including methods based on the augmented lagrangian, trust-region and regularized SQP. Associated with the second-order sequential optimality condition, we have a new constraint qualication weaker than all constraint qualications used for the convergence analysis of second-order methods. We show the relationships of this new constraint qualications with other constraint qualications used for algorithmic purposes. We will also present a new reason why the weak secondorder necessary condition is the natural second-order condition when we are dealing with practical numerical methods

>Magister Scientiae - MSc / In this thesis, we investigate the HIV/AIDS epidemic in a population which experiences a significant flow of immigrants. We derive and analyse a math- ematical model that describes the dynamics of HIV infection among the im- migrant youths and intervention that can minimize or prevent the spread of the disease in the population. In particular, we are interested in the effects of public-health education and of parental care.We consider existing models of public-health education in HIV/AIDS epidemi-ology, and provide some new insights on these. In this regard we focus atten-tion on the papers [b] and [c], expanding those researches by adding sensitivity analysis and optimal control problems with their solutions.Our main emphasis will be on the effect of parental care on HIV/AIDS epidemi-ology. In this regard we introduce a new model. Firstly, we analyse the model without parental care and investigate its stability and sensitivity behaviour.We conduct both qualitative and quantitative analyses. It is observed that in the absence of infected youths, disease-free equilibrium is achievable and is asymptotically stable. Further, we use optimal control methods to determine the necessary conditions for the optimality of intervention, and for disease eradication or control. Using Pontryagin’s Maximum Principle to check the effects of screening control and parental care on the spread of HIV/AIDS, we observe that parental care is more effective than screening control. However, the most efficient control strategy is in fact a combination of parental care and screening control. The results form the central theme of this thesis, and are included in the manuscript [a] which is now being reviewed for publication. Finally, numerical simulations are performed to illustrate the analytical results.

Dans cette thèse, tout d’abord, nous faisons l’Analyse Mathématique du modèle exact du chauffage radiatif d’un corps semi-transparent $\Omega$ par une source radiative noire qui l’entoure. Il s’agit donc d’étudier le couplage d’un système d’Equations de Transfert Radiatif avec condition au bord de réflectivité indépendantes avec une équation de conduction de la chaleur non linéaire avec condition limite non linéaire de type Robin. Nous prouvons l’existence et l’unicité de la solution et nous démontrons des bornes uniformes sur la solution et les intensités radiatives dans chaque bande de longueurs d’ondes pour laquelle le corps est semi-transparent, en fonction de bornes sur les données, Deuxièmement, nous considérons le problème du contrôle optimal de la température absolue à l’intérieur du corps semi-transparent $\Omega$ en agissant sur la température absolue de la source radiative noire qui l’entoure. À cet égard, nous introduisons la fonctionnelle coût appropriée et l’ensemble des contrôles admissibles $T_{S}$, pour lesquels nous prouvons l’existence de contrôles optimaux. En introduisant l’espace des états et l’équation d’état, une condition nécessaire de premier ordre pour qu’un contrôle $T_{S}$ : t ! $T_{S}$ (t) soit optimal, est alors dérivée sous la forme d’une inéquation variationnelle en utilisant le théorème des fonctions implicites et le problème adjoint. Ensuite, nous considérons le problème de l’existence et de l’unicité d’une solution faible des équations de la thermoviscoélasticité dans une formulation mixte de type Hellinger- Reissner, la nouveauté par rapport au travail de M.E. Rognes et R. Winther (M3AS, 2010) étant ici l’apparition de la viscosité dans certains coefficients de l’équation constitutive, viscosité qui dépend dans ce contexte de la température absolue T(x, t) et donc en particulier du temps t. Enfin, nous considérons dans ce cadre le problème du contrôle optimal de la déformation du corps semi-transparent $\Omega$, en agissant sur la température absolue de la source radiative noire qui l’entoure. Nous prouvons l’existence d’un contrôle optimal et nous calculons la dérivée Fréchet de la fonctionnelle coût réduite. / This thesis begins with a rigorous mathematical analysis of the radiative heating of a semi-transparent body made of glass, by a black radiative source surrounding it. This requires the study of the coupling between quasi-steady radiative transfer boundary value problems with nonhomogeneous reflectivity boundary conditions (one for each wavelength band in the semi-transparent electromagnetic spectrum of the glass) and a nonlinear heat conduction evolution equation with a nonlinear Robin boundary condition which takes into account those wavelengths for which the glass behaves like an opaque body. We prove existence and uniqueness of the solution, and give also uniform bounds on the solution i.e. on the absolute temperature distribution inside the body and on the radiative intensities. Now, we consider the temperature $T_{S}$ of the black radiative source S surrounding the semi-transparent body $\Omega$ as the control variable. We adjust the absolute temperature distribution (x, t) 7! T(x, t) inside the semi-transparent body near a desired temperature distribution Td(·, ·) during the time interval of radiative heating ]0, tf [ by acting on $T_{S}$. In this respect, we introduce the appropriate cost functional and the set of admissible controls $T_{S}$, for which we prove the existence of optimal controls. Introducing the State Space and the State Equation, a first order necessary condition for a control $T_{S}$ : t 7! $T_{S}$ (t) to be optimal is then derived in the form of a Variational Inequality by using the Implicit Function Theorem and the adjoint problem. We come now to the goal problem which is the deformation of the semi-transparent body $\Omega$ by heating it with a black radiative source surrounding it. We introduce a weak mixed formulation of this thermoviscoelasticity problem and study the existence and uniqueness of its solution, the novelty here with respect to the work of M.E. Rognes et R. Winther (M3AS, 2010) being the apparition of the viscosity in some of the coefficients of the constitutive equation, viscosity which depends on the absolute temperature T(x, t) and thus in particular on the time t. Finally, we state in this setting the related optimal control problem of the deformation of the semi-transparent body $\Omega$, by acting on the absolute temperature of the black radiative source surrounding it. We prove the existence of an optimal control and we compute the Fréchet derivative of the associated reduced cost functional.