In this thesis we consider constrained systems of equations. The focus is on local Newton-type methods for the solution of constrained systems which converge locally quadratically under mild assumptions implying neither local uniqueness of solutions nor differentiability of the equation function at solutions. The first aim of this thesis is to improve existing local convergence results of the constrained Levenberg-Marquardt method. To this end, we describe a general Newton-type algorithm. Then we prove local quadratic convergence of this general algorithm under the same four assumptions which were recently used for the local convergence analysis of the LP-Newton method. Afterwards, we show that, besides the LP-Newton method, the constrained Levenberg-Marquardt method can be regarded as a special realization of the general Newton-type algorithm and therefore enjoys the same local convergence properties. Thus, local quadratic convergence of a nonsmooth constrained Levenberg-Marquardt method is proved without requiring conditions implying the local uniqueness of solutions. As already mentioned, we use four assumptions for the local convergence analysis of the general Newton-type algorithm. The second aim of this thesis is a detailed discussion of these convergence assumptions for the case that the equation function of the constrained system is piecewise continuously differentiable. Some of the convergence assumptions seem quite technical and difficult to check. Therefore, we look for sufficient conditions which are still mild but which seem to be more familiar. We will particularly prove that the whole set of the convergence assumptions holds if some set of local error bound conditions is satisfied and in addition the feasible set of the constrained system excludes those zeros of the selection functions which are not zeros of the equation function itself, at least in a sufficiently small neighborhood of some fixed solution. We apply our results to constrained systems arising from complementarity systems, i.e., systems of equations and inequalities which contain complementarity constraints. Our new conditions are discussed for a suitable reformulation of the complementarity system as constrained system of equations by means of the minimum function. In particular, it will turn out that the whole set of the convergence assumptions is actually implied by some set of local error bound conditions. In addition, we provide a new constant rank condition implying the whole set of the convergence assumptions. Particularly, we provide adapted formulations of our new conditions for special classes of complementarity systems. We consider Karush-Kuhn-Tucker (KKT) systems arising from optimization problems, variational inequalities, or generalized Nash equilibrium problems (GNEPs) and Fritz-John (FJ) systems arising from GNEPs. Thus, we obtain for each problem class conditions which guarantee local quadratic convergence of the general Newton-type algorithm and its special realizations to a solution of the particular problem. Moreover, we prove for FJ systems of GNEPs that generically some full row rank condition is satisfied at any solution of the FJ system of a GNEP. The latter condition implies the whole set of the convergence assumptions if the functions which characterize the GNEP are sufficiently smooth. Finally, we describe an idea for a possible globalization of our Newton-type methods, at least for the case that the constrained system arises from a certain smooth reformulation of the KKT system of a GNEP. More precisely, a hybrid method is presented whose local part is the LP-Newton method. The hybrid method turns out to be, under appropriate conditions, both globally and locally quadratically convergent.

Submitted by Jaqueline Silva (jtas29@gmail.com) on 2016-12-09T17:10:49Z No. of bitstreams: 2 Tese - Edvaldo Elias de Almeida Batista - 2016.pdf: 1198471 bytes, checksum: 88d7db305f0cfe6be9b62496a226217f (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Jaqueline Silva (jtas29@gmail.com) on 2016-12-09T17:11:03Z (GMT) No. of bitstreams: 2 Tese - Edvaldo Elias de Almeida Batista - 2016.pdf: 1198471 bytes, checksum: 88d7db305f0cfe6be9b62496a226217f (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2016-12-09T17:11:03Z (GMT). No. of bitstreams: 2 Tese - Edvaldo Elias de Almeida Batista - 2016.pdf: 1198471 bytes, checksum: 88d7db305f0cfe6be9b62496a226217f (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-10-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this thesis, we study variational inequalities and generalized vector equilibrium problems. In Chapter 1, several results and basic definitions of Riemannian geometry are listed; we present the concept of the monotone vector field in Hadamard manifolds and many of their properties, besides, we introduce the concept of enlargement of a monotone vector field, and we display its properties in a Riemannian context. In Chapter 2, an inexact proximal point method for variational inequalities in Hadamard manifolds is introduced, and its convergence properties are studied; see [7]. To present our method, we generalize the concept of enlargement of monotone operators, from a linear setting to the Riemannian context. As an application, an inexact proximal point method for constrained optimization problems is obtained. In Chapter 3, we present an extragradient algorithm for variational inequality associated with the point-to-set vector field in Hadamard manifolds and study its convergence properties; see [8]. In order to present our method, the concept of enlargement of maximal monotone vector fields is used and its lower-semicontinuity is established to obtain the convergence of the method in this new context. In Chapter 4, we present a sufficient condition for the existence of a solution to the generalized vector equilibrium problem on Hadamard manifolds using a version of the KnasterKuratowski-Mazurkiewicz Lemma; see [6]. In particular, the existence of solutions to optimization, vector optimization, Nash equilibria, complementarity, and variational inequality is a special case of the existence result for the generalized vector equilibrium problem. / Nesta tese, estudamos desigualdades variacionais e o problema de equilíbrio vetorial generalizado. No Capítulo 1, vários resultados e definições elementares sobre geometria Riemanniana são enunciados; apresentamos o conceito de campo vetorial monótono e muitas de suas propriedades, além de introduzir o conceito de alargamento de um campo vetorial monótono e exibir suas propriedades em um contexto Riemanniano. No Capítulo 2, um método de ponto proximal inexato para desigualdades variacionais em variedades de Hadamard é introduzido e suas propriedades de convergência são estudadas; veja [7]. Para apresentar o nosso método, generalizamos o conceito de alargamento de operadores monótonos, do contexto linear ao contexto de Riemanniano. Como aplicação, é obtido um método de ponto proximal inexato para problemas de otimização com restrições. No Capítulo 3, apresentamos um algoritmo extragradiente para desigualdades variacionais associado a um campo vetorial ponto-conjunto em variedades de Hadamard e estudamos suas propriedades de convergência; veja [8]. A fim de apresentar nosso método, o conceito de alargamento de campos vetoriais monótonos é utilizado e sua semicontinuidade inferior é estabelecida, a fim de obter a convergência do método neste novo contexto. No Capítulo 4, apresentamos uma condição suficiente para a existência de soluções para o problema de equilíbrio vetorial generalizado em variedades de Hadamard usando uma versão do Lema Knaster-Kuratowski-Mazurkiewicz; veja [6]. Em particular, a existência de soluções para problemas de otimização, otimização vetorial, equilíbrio de Nash, complementaridade e desigualdades variacionais são casos especiais do resultado de existência do problema de equilíbrio vetorial generalizado.

In this thesis we consider constrained systems of equations. The focus is on local Newton-type methods for the solution of constrained systems which converge locally quadratically under mild assumptions implying neither local uniqueness of solutions nor differentiability of the equation function at solutions. The first aim of this thesis is to improve existing local convergence results of the constrained Levenberg-Marquardt method. To this end, we describe a general Newton-type algorithm. Then we prove local quadratic convergence of this general algorithm under the same four assumptions which were recently used for the local convergence analysis of the LP-Newton method. Afterwards, we show that, besides the LP-Newton method, the constrained Levenberg-Marquardt method can be regarded as a special realization of the general Newton-type algorithm and therefore enjoys the same local convergence properties. Thus, local quadratic convergence of a nonsmooth constrained Levenberg-Marquardt method is proved without requiring conditions implying the local uniqueness of solutions. As already mentioned, we use four assumptions for the local convergence analysis of the general Newton-type algorithm. The second aim of this thesis is a detailed discussion of these convergence assumptions for the case that the equation function of the constrained system is piecewise continuously differentiable. Some of the convergence assumptions seem quite technical and difficult to check. Therefore, we look for sufficient conditions which are still mild but which seem to be more familiar. We will particularly prove that the whole set of the convergence assumptions holds if some set of local error bound conditions is satisfied and in addition the feasible set of the constrained system excludes those zeros of the selection functions which are not zeros of the equation function itself, at least in a sufficiently small neighborhood of some fixed solution. We apply our results to constrained systems arising from complementarity systems, i.e., systems of equations and inequalities which contain complementarity constraints. Our new conditions are discussed for a suitable reformulation of the complementarity system as constrained system of equations by means of the minimum function. In particular, it will turn out that the whole set of the convergence assumptions is actually implied by some set of local error bound conditions. In addition, we provide a new constant rank condition implying the whole set of the convergence assumptions. Particularly, we provide adapted formulations of our new conditions for special classes of complementarity systems. We consider Karush-Kuhn-Tucker (KKT) systems arising from optimization problems, variational inequalities, or generalized Nash equilibrium problems (GNEPs) and Fritz-John (FJ) systems arising from GNEPs. Thus, we obtain for each problem class conditions which guarantee local quadratic convergence of the general Newton-type algorithm and its special realizations to a solution of the particular problem. Moreover, we prove for FJ systems of GNEPs that generically some full row rank condition is satisfied at any solution of the FJ system of a GNEP. The latter condition implies the whole set of the convergence assumptions if the functions which characterize the GNEP are sufficiently smooth. Finally, we describe an idea for a possible globalization of our Newton-type methods, at least for the case that the constrained system arises from a certain smooth reformulation of the KKT system of a GNEP. More precisely, a hybrid method is presented whose local part is the LP-Newton method. The hybrid method turns out to be, under appropriate conditions, both globally and locally quadratically convergent.

L’instabilité posturale est fréquente chez les séniors et peut entrainer la chute. La chute chez les séniors est un problème majeur de santé publique. Les chiffres épidémiologiques sont éloquents : une personne sur trois âgées de plus de 70 ans fera une chute dans l’année. Les causes sont multifactorielles : ostéo-articulaire, visuelle, cognitive, vestibulaire…. Dans cette étude, nous nous sommes intéressés à l’évolution de la fonction des récepteurs vestibulaires périphériques avec l’âge et à la perception de rotation à partir des entrées canalaires horizontales (système vestibulaire central et projections vestibulaires corticales). Notre but est d’essayer de comprendre l’implication du vieillissement du système vestibulaire dans l’instabilité posturale des séniors. Au niveau périphérique, nous avons quantifié la fonction des canaux semi-circulaires horizontaux par le test calorique et le vidéo-head impulse test. La fonction des récepteurs otolithiques (utriculaire et sacculaire) a été évaluée par les potentiels évoqués myogéniques recueillis au niveau cervical (voies sacculo-spinales) et oculaire (voies utriculo-oculaires). Au niveau central, la perception de l’entrée vestibulaire canalaire horizontale a été appréciée après irrigation à l’eau chaude du conduit auditif externe en appliquant un score de perception (présence ou absence de sensation rotatoire). Finalement, l’équilibre a été quantifié grâce au test d’organisation sensorielle sur l’Equitest et grâce à un système que nous avons récemment mis au point en collaboration avec le Professeur Curthoys à Sydney, comprenant une Wii Balance Board, un tapis mousse et un masque de réalité virtuelle (Oculus Rift). Les résultats ont montré une diminution des réponses oculaires au test calorique après 70 ans mais une absence de baisse du gain du réflexe vestibulo-oculaire horizontal au vidéo-head impulse test. La fonction otolithique, sacculaire et utriculaire, est altérée avec l’âge quelle que soit la stimulation utilisée (aérienne ou osseuse). La perception de l’entrée vestibulaire canalaire horizontale induite par une stimulation calorique nous a permis de montrer pour la première fois que certains séniors ne percevaient pas la sensation de rotation malgré une réponse oculaire normale (vitesse maximale de la phase lente du nystagmus oculaire supérieure à 15°/s). Dans notre population, nous avons pu ainsi définir deux types de séniors : un groupe présentant une perception de vertige rotatoire et un groupe « négligeant » ne pouvant pas reconstruire une sensation rotatoire à partir des entrées vestibulaires canalaires horizontales. La comparaison de ces deux groupes de séniors appariés sur l’âge ne montre aucune différence de la fonction canalaire horizontale ni de la fonction otolithique sacculaire et utriculaire. Néanmoins, les séniors négligents présentent en majorité des performances anormales (chute ou score diminué) à l’Equitest notamment en conditions 5 et 6. De plus, leur score au DHI est plus élevé relevant ainsi le handicape ressenti par ces séniors à cause de leur instabilité. En conclusion, les troubles de l’équilibre chez certains seniors pourraient résulter en partie d’une dysfonction vestibulaire centrale. Des études ultérieures permettront de déterminer si l’augmentation du seuil de perception rotatoire est un bon facteur prédictif du risque de chute. / Postural instability is common in seniors and can lead to falls which seniors are a major problem for Public Health. Epidemiological studies clearly show the magnitude of this problem: one in three people aged than more 70 years will fall in a year. This is caused by multiple factors including: musculoskeletal, visual, cognition, vestibular… The present study concerns the effect of age on the vestibular peripheral receptors function and on the perception of rotation from horizontal canal inputs (central vestibular processing and vestibular cortical projection). The aim is to try to understand the vestibular mechanisms involved in postural instability and mobility with age. At the peripheral level, the horizontal canal function was assessed using caloric test and video-Head Impulse Test. Otolith function (saccular and utricular) was assessed using vestibular evoked myogenic potentials recorded at cervical level (sacculo-spinal pathways) and at ocular level (utriculo-ocular pathways). At the central level, perception of motion from vestibular horizontal canal inputs was studied after caloric stimulation with warm water using a subjective perceptual score (presence or absence of rotatory vertigo). Finally, postural equilibrium was assessed with the Sensory Organization Test on the Equitest machine and also with a new system developed in collaboration with Prof. Curthoys (Sydney) using a Wii Balance Board, a foam rubber pad and a virtual reality headset (Oculus Rift DK2). Results showed decreased ocular responses induced by caloric stimulation after 70 years of age but healthy horizontal gain of the vestibulo-ocular reflex assessed by video-head impulse testing. The otolithic (saccular and utricular) function is impaired with age for all the stimuli used (air or bone conducted). Perception of motion induced by caloric stimulation (vestibular horizontal canal inputs) allowed us to show for the first time that some seniors are unable to feel the induced rotatory vertigo even with normal ocular responses (peak of the slow phase eye velocity higher than 15°/s). We defined two types of seniors: one senior group having a normal feeling of vertigo and one senior ‘neglect’ group who did not feel any sensation of rotation from horizontal canal inputs. The comparison of these two age-matched groups showed no difference in horizontal canal function, or otolithic function. The majority of the ‘neglect’ seniors with an absence of perception exhibited falls or a decreased score in conditions 5 and 6 during the Equitest. Moreover, their DHI scores were higher, showing the handicap induced by postural instability in these seniors. In conclusion, postural instability and falls in seniors may result from central vestibular impairment (inadequate central processing). A prospective study is needed to determine whether the increase perceptual threshold of rotation could be a good predictor of fall risk in seniors.