Tesis sobre el tema "Analisi nonsmooth"

La funzione valore è un nodo centrale del controllo ottimo. `E noto che la funzione valore può essere irregolare anche per sistemi molto regolari. Pertanto l’analisi non liscia diviene un importante strumento per studiarne le proprietà, anche grazie alle numerose connessioni con la semiconcavità. Sotto opportune ipotesi, la funzione valore è localmente semiconcava. Questa proprietà è connessa anche con la proprietà di sfera interna dei suoi insiemi di livello e dei loro perimetri. In questa tesi introduciamo l’analisi non-liscia e le sue connessioni con funzioni semiconcave ed insiemi di perimetro finito. Descriviamo i sistemi di controllo ed introduciamo le proprietà basilari della funzione tempo minimo T(x) e della funzione valore V (x). Usando il principio del massimo, estendiamo alcuni risultati noti di sfera interna per gli insiemi raggiungibili A(T), al caso non-autonomo ed ai sistemi con costo corrente non costante. Questa proprietà ci permette di ottenere delle stime sui perimetri per alcuni sistemi di controllo. Infine queste proprietà degli insiemi raggiungibili possono essere estese agli insiemi di livello della funzione valore, e, sotto alcune ipotesi di controllabilità otteniamo anche semiconcavità locale per V (x). Inoltre studiamo anche sistemi di controllo vincolati. Nei sistemi vincolati la funzione valore perde regolarità. Infatti, quando una traiettoria tocca il bordo del vincolo Ω, si presentano delle singolarità. Questi effetti sono evidenziati anche dal principio del massimo, che produce un termine aggiuntivo di misura(eventualmente discontinuo), quando una traiettoria tocca il bordo ∂Ω. E la funzione valore perde la semiconcavità, anche per sistemi particolarmente semplici. Ma siamo in grado di recuperare lipschitzianità per il tempo minimo, ed enunciare il principio del massimo esplicitando il termine di bordo. In questo modo otteniamo delle particolari proprietà di sfera interna, e quindi anche stime sui perimetri, per gli insiemi raggiungibili.
The value function is a focal point in optimal control theory. It is a known fact that the value function can be nonsmooth even with very smooth data. So, nonsmooth analysis is a useful tool to study its regularity. Semiconcavity is a regularity property, with some fine connection with nonsmooth analysis. Under appropriate assumptions, the value function is locally semiconcave. This property is connected with the interior sphere property of its level sets and their perimeters. In this thesis we introduce basic concepts of nonsmooth analysis and their connections with semiconcave functions, and sets of finite perimeter. We describe control systems, and we introduce the basic properties of the minimum time function T(x) and of the value function V (x). Then, using maximum principle, we extend some known results of interior sphere property for the attainable setsA(t), to the nonautonomous case and to systems with nonconstant running cost L. This property allow us to obtain some fine perimeter estimates for some class of control systems. Finally these regularity properties of the attainable sets can be extended to the level sets of the value function, and, with some controllability assumption, we also obtain a local semiconcavity for V (x). Moreoverwestudycontrolsystemswithstateconstraints. Inconstrained systems we loose many of regularity properties related to the value function. In fact, when a trajectory of control system touches the boundary of the constraint set Ω, some singularity effect occurs. This effect is clear even in the statement of the maximum principle. Indeed, due to the times in which a trajectory stays on ∂Ω, a measure boundary term (possibly, discontinuous) appears. So, we have no more semiconcavity for the value function, even for very simple control systems. But we recover Lipschitz continuity for the minimum time and we rewrite the constrained maximum principle with an explicit boundary term. We also obtain a kind of interior sphere property, and perimeter estimates for the attainable sets for some class of control systems.

Several regularity results on the minimum time function are proved, together with regularity properties of a class of continuous functions whose hypograph satisfies an external sphere condition.
Si dimostrano risultati di regolarita' per la funzione tempo minimo, mediante particolari proprieta' di una classe di funzioni continue il cui ipografico soddisfa una condizione di sfera esterna.

In many applications nonsmooth nonconvex energy functions, which are Lipschitz continuous, appear quite naturally. Contact mechanics with friction is a classic example. A second example is the 1-Laplace operator and its eigenfunctions. In this work we will give an algorithm such that for every locally Lipschitz continuous function f and every sequence produced by this algorithm it holds that every accumulation point of the sequence is a critical point of f in the sense of Clarke. Here f is defined on a reflexive Banach space X, such that X and its dual space X' are strictly convex and Clarkson's inequalities hold. (E.g. Sobolev spaces and every closed subspace equipped with the Sobolev norm satisfy these assumptions for p>1.) This algorithm is designed primarily to solve variational problems or their high dimensional discretizations, but can be applied to a variety of locally Lipschitz functions. In elastic contact mechanics the strain energy is often smooth and nonconvex on a suitable domain, while the contact and the friction energy are nonsmooth and have a support on a subspace which has a substantially smaller dimension than the strain energy, since all points in the interior of the bodies only have effect on the strain energy. For such elastic contact problems we suggest a specialization of our algorithm, which treats the smooth part with Newton like methods. In the case that the gradient of the entire energy function is semismooth close to the minimizer, we can even prove superlinear convergence of this specialization of our algorithm. We test the algorithm and its specialization with a couple of benchmark problems. Moreover, we apply the algorithm to the 1-Laplace minimization problem restricted to finitely dimensional subspaces of piecewise affine, continuous functions. The algorithm developed here uses ideas of the bundle trust region method by Schramm, and a new generalization of the concept of gradients on a set. The basic idea behind this gradients on sets is that we want to find a stable descent direction, which is a descent direction on an entire neighborhood of an iteration point. This way we avoid oscillations of the gradients and very small descent steps (in the smooth and in the nonsmooth case). It turns out, that the norm smallest element of the gradient on a set provides a stable descent direction. The algorithm we present here is the first algorithm which can treat locally Lipschitz continuous functions in this generality, up to our knowledge. In particular, large finitely dimensional Banach spaces haven't been studied for nonsmooth nonconvex functions so far. We will show that the algorithm is very robust and often faster than common algorithms. Furthermore, we will see that with this algorithm it is possible to compute reliably the first eigenfunctions of the 1-Laplace operator up to disretization errors, for the first time
In vielen Anwendungen tauchen nichtglatte, nichtkonvexe, Lipschitz-stetige Energie Funktionen in natuerlicher Weise auf. Ein klassische Beispiel bildet die Kontaktmechanik mit Reibung. Ein weiteres Beispiel ist der $1$-Laplace Operator und seine Eigenfunktionen. In dieser Dissertation werden wir ein Abstiegsverfahren angeben, so dass fuer jede lokal Lipschitz-stetige Funktion f jeder Haeufungspunkt einer durch dieses Verfahren erzeugten Folge ein kritischer Punkt von f im Sinne von Clarke ist. Hier ist f auf einem einem reflexiver, strikt konvexem Banachraum definierert, fuer den der Dualraum ebenfalls strikt konvex ist und die Clarkeson Ungleichungen gelten. (Z.B. Sobolevraeume und jeder abgeschlossene Unterraum mit der Sobolevnorm versehen, erfuellt diese Bedingung fuer p>1.) Dieser Algorithmus ist primaer entwickelt worden um Variationsprobleme, bzw. deren hochdimensionalen Diskretisierungen zu loesen. Er kann aber auch fuer eine Vielzahl anderer lokal Lipschitz stetige Funktionen eingesetzt werden. In der elastischen Kontaktmechanik ist die Spannungsenergie oft glatt und nichtkonvex auf einem geeignetem Definitionsbereich, waehrend der Kontakt und die Reibung durch nicht glatte Funktionen modelliert werden, deren Traeger ein Unterraum mit wesentlich kleineren Dimension ist, da alle Punkte im Inneren des Koerpers nur die Spannungsenergie beeinflussen. Fuer solche elastischen Kontaktprobleme schlagen wir eine Spezialisierung unseres Algorithmuses vor, der den glatten Teil mit Newton aehnlichen Methoden behandelt. Falls der Gradient der gesamten Energiefunktion semiglatt in der Naehe der Minimalstelle ist, koennen wir sogar beweisen, dass der Algorithmus superlinear konvergiert. Wir testen den Algorithmus und seine Spezialisierung an mehreren Benchmark Problemen. Ausserdem wenden wir den Algorithmus auf 1-Laplace Minimierungsproblem eingeschraenkt auf eine endlich dimensionalen Unterraum der stueckweise affinen, stetigen Funktionen an. Der hier entwickelte Algorithmus verwendet Ideen des Bundle-Trust-Region-Verfahrens von Schramm, und einen neu entwickelten Verallgemeinerung von Gradienten auf Mengen. Die zentrale Idee hinter den Gradienten auf Mengen ist die, dass wir stabile Abstiegsrichtungen auf einer ganzen Umgebung der Iterationspunkte finden wollen. Auf diese Weise vermeiden wir das Oszillieren der Gradienten und sehr kleine Abstiegsschritte (im glatten, wie im nichtglatten Fall.) Es stellt sich heraus, dass das normkleinste Element dieses Gradienten auf der Umgebung eine stabil Abstiegsrichtung bestimmt. So weit es uns bekannt ist, koennen die hier entwickelten Algorithmen zum ersten Mal lokal Lipschitz-stetige Funktionen in dieser Allgemeinheit behandeln. Insbesondere wurden nichtglatte, nichtkonvexe Funktionen auf derart hochdimensionale Banachraeume bis jetzt nicht behandelt. Wir werden zeigen, dass unser Algorithmus sehr robust und oft schneller als uebliche Algorithmen ist. Des Weiteren, werden wir sehen, dass es mit diesem Algorithmus das erste mal moeglich ist, zuverlaessig die erste Eigenfunktion des 1-Laplace Operators bis auf Diskretisierungsfehler zu bestimmen

Data mining is about solving problems by analyzing data that present in databases. Supervised and unsupervised data classification (clustering) are among the most important techniques in data mining. Regression analysis is the process of fitting a function (often linear) to the data to discover how one or more variables vary as a function of another. The aim of clusterwise regression is to combine both of these techniques, to discover trends within data, when more than one trend is likely to exist. Clusterwise regression has applications for instance in market segmentation, where it allows one to gather information on customer behaviors for several unknown groups of customers. There exist different methods for solving clusterwise linear regression problems. In spite of that, the development of efficient algorithms for solving clusterwise linear regression problems is still an important research topic. In this thesis our aim is to develop new algorithms for solving clusterwise linear regression problems in large data sets based on incremental and nonsmooth optimization approaches. Three new methods for solving clusterwise linear regression problems are developed and numerically tested on publicly available data sets for regression analysis. The first method is a new algorithm for solving the clusterwise linear regression problems based on their nonsmooth nonconvex formulation. This is an incremental algorithm. The second method is a nonsmooth optimization algorithm for solving clusterwise linear regression problems. Nonsmooth optimization techniques are proposed to use instead of the Sp¨ath algorithm to solve optimization problems at each iteration of the incremental algorithm. The discrete gradient method is used to solve nonsmooth optimization problems at each iteration of the incremental algorithm. This approach allows one to reduce the CPU time and the number of regression problems solved in comparison with the first incremental algorithm. The third algorithm is an algorithm based on an incremental approach and on the smoothing techniques for solving clusterwise linear regression problems. The use of smoothing techniques allows one to apply powerful methods of smooth nonlinear programming to solve clusterwise linear regression problems. Numerical results are presented for all three algorithms using small to large data sets. The new algorithms are also compared with multi-start Sp¨ath algorithm for clusterwise linear regression.
Doctor of Philosophy

Cluster analysis deals with the problem of organization of a collection of patterns into clusters based on a similarity measure. Various distance functions can be used to define this measure. Clustering problems with the similarity measure defined by the squared Euclidean distance have been studied extensively over the last five decades. However, problems with other Minkowski norms have attracted significantly less attention. The use of different similarity measures may help to identify different cluster structures of a data set. This in turn may help to significantly improve the decision making process. High dimensional data visualization is another important task in the field of data mining and pattern recognition. To date, the principal component analysis and the self-organizing maps techniques have been used to solve such problems. In this thesis we develop algorithms for solving clustering problems in large data sets using various similarity measures. Such similarity measures are based on the squared L
Doctor of Philosophy

Desirability Functions continue to attract attention of scientists and researchers working in the area of multi-response optimization. There are many versions of such functions, differing mainly in formulations of individual and overall desirability functions. Derringer and Suich&rsquo
s desirability functions being used throughout this thesis are still the most preferred ones in practice and many other versions are derived from these. On the other hand, they have a drawback of containing nondifferentiable points and, hence, being nonsmooth. Current approaches to their optimization, which are based on derivative-free search techniques and modification of the functions by higher-degree polynomials, need to be diversified considering opportunities offered by modern nonlinear (global) optimization techniques and related softwares. A first motivation of this work is to develop a new efficient solution strategy for the maximization of overall desirability functions which comes out to be a nonsmooth composite constrained optimization problem by nonsmooth optimization methods. We observe that individual desirability functions used in practical computations are of mintype, a subclass of continuous selection functions. To reveal the mechanism that gives rise to a variation in the piecewise structure of desirability functions used in practice, we concentrate on a component-wise and generically piecewise min-type functions and, later on, max-type functions. It is our second motivation to analyze the structural and topological properties of desirability functions via piecewise max-type functions. In this thesis, we introduce adjusted desirability functions based on a reformulation of the individual desirability functions by a binary integer variable in order to deal with their piecewise definition. We define a constraint on the binary variable to obtain a continuous optimization problem of a nonlinear objective function including nondifferentiable points with the constraints of bounds for factors and responses. After describing the adjusted desirability functions on two well-known problems from the literature, we implement modified subgradient algorithm (MSG) in GAMS incorporating to CONOPT solver of GAMS software for solving the corresponding optimization problems. Moreover, BARON solver of GAMS is used to solve these optimization problems including adjusted desirability functions. Numerical applications with BARON show that this is a more efficient alternative solution strategy than the current desirability maximization approaches. We apply negative logarithm to the desirability functions and consider the properties of the resulting functions when they include more than one nondifferentiable point. With this approach we reveal the structure of the functions and employ the piecewise max-type functions as generalized desirability functions (GDFs). We introduce a suitable finite partitioning procedure of the individual functions over their compact and connected interval that yield our so-called GDFs. Hence, we construct GDFs with piecewise max-type functions which have efficient structural and topological properties. We present the structural stability, optimality and constraint qualification properties of GDFs using that of max-type functions. As a by-product of our GDF study, we develop a new method called two-stage (bilevel) approach for multi-objective optimization problems, based on a separation of the parameters: in y-space (optimization) and in x-space (representation). This approach is about calculating the factor variables corresponding to the ideal solutions of each individual functions in y, and then finding a set of compromised solutions in x by considering the convex hull of the ideal factors. This is an early attempt of a new multi-objective optimization method. Our first results show that global optimum of the overall problem may not be an element of the set of compromised solution. The overall problem in both x and y is extended to a new refined (disjunctive) generalized semi-infinite problem, herewith analyzing the stability and robustness properties of the objective function. In this course, we introduce the so-called robust optimization of desirability functions for the cases when response models contain uncertainty. Throughout this thesis, we give several modifications and extensions of the optimization problem of overall desirability functions.

This thesis tackles the traditional transport engineering problem of urban traffic demand estimation by using Bluetooth data and advanced signal processing algorithms. It proposes a method to recover vehicles trajectories from Bluetooth detectors and combining vehicle trajectories with traditional traffic datasets, traffic is estimated at a city level using signal processing algorithms. Involving new technologies in traffic demand estimation gave an opportunity to rethink traditional approaches and to come up with new method to jointly estimate origin-destinations flows and route flows. The whole methodology has been applied and evaluated with real Brisbane traffic data.

Optimierungsprobleme und Variationsungleichungen über Banach-Räumen stellen Themen von substantiellem Interesse dar, da beide Problemklassen einen abstrakten Rahmen für zahlreiche Anwendungen aus verschiedenen Fachgebieten stellen. Nach einer Einführung in Teil I werden im zweiten Teil allgemeine Approximationsmethoden, einschließlich verschiedener Diskretisierungs- und Regularisierungsansätze, zur Lösung von nichtglatten Variationsungleichungen und Optimierungsproblemen unter konvexen Restriktionen vorgestellt. In diesem allgemeinen Rahmen stellen sich gewisse Dichtheitseigenschaften der konvexen zulässigen Menge als wichtige Voraussetzungen für die Konsistenz einer abstrakten Klasse von Störungen heraus. Im Folgenden behandeln wir vor allem Restriktionsmengen in Sobolev-Räumen, die durch eine punktweise Beschränkung an den Funktionswert definiert werden. Für diesen Restriktionstyp werden verschiedene Dichtheitsresultate bewiesen. In Teil III widmen wir uns einem quasi-statischen Kontaktproblem der Elastoplastizität mit Härtung. Das entsprechende zeit-diskretisierte Problem kann als nichtglattes, restringiertes Minimierungsproblem betrachtet werden. Zur Lösung wird eine Pfadverfolgungsmethode auf Basis des verallgemeinerten Newton-Verfahrens entwickelt, dessen Teilprobleme lokal superlinear und gitterunabhängig lösbar sind. Teil III schließt mit verschiedenen numerischen Beispielen. Der letzte Teil der Arbeit ist der quasi-statischen, perfekten Plastizität gewidmet. Auf Basis des primalen Problems der perfekten Plastizität leiten wir eine reduzierte Formulierung her, die es erlaubt, das primale Problem als Fenchel-dualisierte Form des klassischen zeit-diskretisierten Spannungsproblems zu verstehen. Auf diese Weise werden auch neue Optimalitätsbedingungen hergeleitet. Zur Lösung des Problems stellen wir eine modifizierte Form der viskoplastischen Regularisierung vor und beweisen die Konvergenz dieses neuen Regularisierungsverfahrens.
Optimization problems and variational inequalities over Banach spaces are subjects of paramount interest since these mathematical problem classes serve as abstract frameworks for numerous applications. Solutions to these problems usually cannot be determined directly. Following an introduction, part II presents several approximation methods for convex-constrained nonsmooth variational inequality and optimization problems, including discretization and regularization approaches. We prove the consistency of a general class of perturbations under certain density requirements with respect to the convex constraint set. We proceed with the study of pointwise constraint sets in Sobolev spaces, and several density results are proven. The quasi-static contact problem of associative elasto-plasticity with hardening at small strains is considered in part III. The corresponding time-incremental problem can be equivalently formulated as a nonsmooth, constrained minimization problem, or, as a mixed variational inequality problem over the convex constraint. We propose an infinite-dimensional path-following semismooth Newton method for the solution of the time-discrete plastic contact problem, where each path-problem can be solved locally at a superlinear rate of convergence with contraction rates independent of the discretization. Several numerical examples support the theoretical results. The last part is devoted to the quasi-static problem of perfect (Prandtl-Reuss) plasticity. Building upon recent developments in the study of the (incremental) primal problem, we establish a reduced formulation which is shown to be a Fenchel predual problem of the corresponding stress problem. This allows to derive new primal-dual optimality conditions. In order to solve the time-discrete problem, a modified visco-plastic regularization is proposed, and we prove the convergence of this new approximation scheme.

L’estimation des matrices origine-destination (OD) est un sujet de recherche important depuis les années 1950. En effet, ces tableaux à deux entrées recensent la demande de transport d'une zone géographique donnée et sont de ce fait un élément clé de l'ingénierie du trafic. Historiquement, les seules données disponibles pour leur estimation par les statistiques étaient les comptages de véhicules par les boucles magnétiques. Ce travail s'inscrit alors dans le contexte de l'installation à Brisbane de plus de 600 détecteurs Bluetooth qui ont la capacité de détecter et d'identifier les appareils électroniques équipés de cette technologie.Dans un premier temps, il explore la possibilité offerte par ces détecteurs pour les applications en ingénierie du transport en caractérisant ces données et leurs bruits. Ce projet aboutit, à l'issue de cette étude, à une méthode de reconstruction des trajectoires des véhicules équipés du Bluetooth à partir de ces seules données. Dans un second temps, en partant de l'hypothèse que l'accès à des échantillons importants de trajectoires va se démocratiser, cette thèse propose d'étendre la notion de matrice OD à celle de matrice OD par lien afin de combiner la description de la demande avec celle de l'utilisation du réseau. Reposant sur les derniers outils méthodologies développés en optimisation convexe, nous proposons une méthode d'estimation de ces matrices à partir des trajectoires inférées par Bluetooth et des comptages routiers.A partir de peu d'hypothèses, il est possible d'inférer ces nouvelles matrices pour l'ensemble des utilisateurs d'un réseau routier (indépendamment de leur équipement en nouvelles technologies). Ce travail se distingue ainsi des méthodes traditionnelles d'estimation qui reposaient sur des étapes successives et indépendantes d'inférence et de modélisation
Origin Destination matrix estimation is a critical problem of the Transportation field since the fifties. OD matrix is a two-entry table taking census of the zone-to-zone traffic of a geographic area. This traffic description tools is therefore paramount for traffic engineering applications. Traditionally, the OD matrix estimation has solely been based on traffic counts collected by networks of magnetic loops. This thesis takes place in a context with over 600 Bluetooth detectors installed in the City of Brisbane. These detectors permit in-car Bluetooth device detection and thus vehicle identification.This manuscript explores first, the potentialities of Bluetooth detectors for Transport Engineering applications by characterising the data, their noises and biases. This leads to propose a new methodology for Bluetooth equipped vehicle trajectory reconstruction. In a second step, based on the idea that probe trajectories will become more and more available by means of new technologies, this thesis proposes to extend the concept of OD matrix to the one of link dependent origin destination matrix that describes simultaneously both the traffic demand and the usage of the network. The problem of LOD matrix estimation is formulated as a minimisation problem based on probe trajectories and traffic counts and is then solved thanks to the latest advances in nonsmooth convex optimisation.This thesis demonstrates that, with few hypothesis, it is possible to retrieve the LOD matrix for the whole set of users in a road network. It is thus different from traditional OD matrix estimation approaches that relied on successive steps of modelling and of statistical inferences

Orientador: Geraldo Nunes Silva
Banca: Vilma Alves de Oliveira
Banca: Masayoshi Tsuchida
Resumo: Estamos interessados em estudar uma generalização do Princípio do Máximo de Pontryagin para problema de controle ótimo com restrições mistas envolvendo funções nãodiferenciáveis, pois este princípio não se aplica para todos os tipos de problemas. O principal objetivo deste trabalho é apresentar as condições necessárias de otimalidade na forma do princípio do máximo que serão aplicadas para o problema de controle ótimo com restrições mistas envolvendo funções não-diferenciáveis. Para alcançar este objetivo apresentamos estudos sobre cones normais e cones tangentes os quais são utilizados no desenvolvimento da teoria de subdiferenciais. Após esse embasamento formulamos o problema de controle ótimo envolvendo funções não-diferenciáveis, e apresentamos as condições necessárias de otimalidade.
Abstract: We are interested in study a generalization of the Pontryagin Maximum Principle for optimal control problems with mixed constraints involving nondi erentiable functions, because this principle can not be applied for all the types of problems. The main objective of this work is to present the necessary conditions of optimality in the form of the maximum principle that will be applied for the optimal control problem with mixed constraints involving nondi erentiable functions. To achieve this objective we present studies above normal cones and tangent cones which are used in the development of the theory of subdi erentials. After this foundation we formulate the optimal control problem involving nondi erentiable functions, and we present the necessary conditions of optimality.
Mestre

Discontinuous dynamical systems and multiagent systems are encountered in numerous engineering applications. This dissertation develops stability and dissipativity of nonlinear dynamical systems with discontinuous right-hand sides, optimality of discontinuous feed-back controllers for Filippov dynamical systems, almost consensus protocols for multiagent systems with innaccurate sensor measurements, and adaptive estimation algorithms using multiagent network identifiers. In particular, we present stability results for discontinuous dynamical systems using nonsmooth Lyapunov theory. Then, we develop a constructive feedback control law for discontinuous dynamical systems based on the existence of a nonsmooth control Lyapunov function de fined in the sense of generalized Clarke gradients and set-valued Lie derivatives. Furthermore, we develop dissipativity notions and extended Kalman-Yakubovich-Popov conditions and apply these results to develop feedback interconnection stability results for discontinuous systems. In addition, we derive guaranteed gain, sector, and disk margins for nonlinear optimal and inverse optimal discontinuous feedback regulators that minimize a nonlinear-nonquadratic performance functional for Filippov dynamical systems. Then, we provide connections between dissipativity and optimality of nonlinear discontinuous controllers for Filippov dynamical systems. Furthermore, we address the consensus problem for a group of agent robots with uncertain interagent measurement data, and show that the agents reach an almost consensus state and converge to a set centered at the centroid of agents initial locations. Finally, we develop an adaptive estimation framework predicated on multiagent network identifiers with undirected and directed graph topologies that identifies the system state and plant parameters online.

La motivation industrielle et mécanique du problème sera présentée pour un problème continu: élasticité linéaire avec une contrainte unilatérale. Un système masse-ressort avec un contact unilatéral en découle par discrétisation. Le but de cette thèse est d'étudier ces systèmes à vibro-impact de N degrés de liberté avec un contact unilatéral. Le système résultant est linéaire en l'absence de contact; Il est régi par une loi d'impact autrement. L'auteur identifie les modes non linéaires qui présentent une phase de contact collant pour un modèle à deux degrés de liberté en présence d'un obstacle rigide. L'application de premier retour de Poincaré est un outil fondamental pour étudier la dynamique près de solutions périodiques. Étant donné que la section de Poincaré est un sous-ensemble de l'interface de contact dans l'espace des phases, elle peut être tangente aux orbites pour les contacts rasants et conduire à une singularité en « racine carrée » déjà connue en Mécanique. Cette singularité est revisitée dans un cadre mathématique rigoureux. Elle implique la discontinuité du temps de premier retour. Enfin, l’instabilité des modes linéaire rasants est abordée
The mechanical motivation is presented for a PDE with a constraint. The purpose of this thesis is to study N degree-of-freedom vibro-impact systems with an unilateral contact. The resulting system is linear in the absence of contact; it is governed by an impact law otherwise. The author identifies some nonlinear modes that display a sticking phase. The First Return Map is a fundamental tool to explore periodic solutions. Since the Poincaré section is a subset of the contact interface in the phase-space, it can be tangent to orbits which yields the well-known square-root singularity. This singularity is here revisited in a rigorous mathematical framework. Moreover, the study of this singularity implies a more important singularity: the discontinuity of the first return time. Finally, the square-root dynamics near the linear grazing modes which may lead to the instability of these linear grazing modes is studied

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Estamos interessados em estudar uma generalização do Princípio do Máximo de Pontryagin para problema de controle ótimo com restrições mistas envolvendo funções nãodiferenciáveis, pois este princípio não se aplica para todos os tipos de problemas. O principal objetivo deste trabalho é apresentar as condições necessárias de otimalidade na forma do princípio do máximo que serão aplicadas para o problema de controle ótimo com restrições mistas envolvendo funções não-diferenciáveis. Para alcançar este objetivo apresentamos estudos sobre cones normais e cones tangentes os quais são utilizados no desenvolvimento da teoria de subdiferenciais. Após esse embasamento formulamos o problema de controle ótimo envolvendo funções não-diferenciáveis, e apresentamos as condições necessárias de otimalidade.
We are interested in study a generalization of the Pontryagin Maximum Principle for optimal control problems with mixed constraints involving nondi erentiable functions, because this principle can not be applied for all the types of problems. The main objective of this work is to present the necessary conditions of optimality in the form of the maximum principle that will be applied for the optimal control problem with mixed constraints involving nondi erentiable functions. To achieve this objective we present studies above normal cones and tangent cones which are used in the development of the theory of subdi erentials. After this foundation we formulate the optimal control problem involving nondi erentiable functions, and we present the necessary conditions of optimality.

Le projet de cette thèse est double. Le premier concerne l’extension des résultats précédents sur les conditions nécessaires d’optimalité pour des problèmes avec contraintes d’état, dans le cadre du contrôle optimal ainsi que dans le cadre de calcul des variations. Le deuxième objectif consiste à travailler sur deux nouveaux aspects de recherche : dériver des résultats de viabilité pour une classe de systèmes de contrôle avec des contraintes d’état dans lesquels les conditions dites ‘standard inward pointing conditions’ sont violées; et établir les conditions nécessaires d’optimalité pour des problèmes de minimisation de coût moyen éventuellement perturbés par des paramètres inconnus.Dans la première partie, nous examinons les conditions nécessaires d’optimalité qui jouent un rôle important dans la recherche de candidats pour être des solutions optimales parmi toutes les solutions admissibles. Cependant, dans les problèmes d’optimisation dynamique avec contraintes d’état, certaines situations pathologiques pourraient survenir. Par exemple, il se peut que le multiplicateur associé à la fonction objective (à minimiser) disparaisse. Dans ce cas, la fonction objective à minimiser n’intervient pas dans les conditions nécessaires de premier ordre: il s’agit du cas dit anormal. Un phénomène pire, appelé le cas dégénéré montre que, dans certaines circonstances, l’ensemble des trajectoires admissibles coïncide avec l’ensemble des candidats minimiseurs. Par conséquent, les conditions nécessaires ne donnent aucune information sur les minimiseurs possibles.Pour surmonter ces difficultés, de nouvelles hypothèses supplémentaires doivent être imposées, appelées les qualifications de la contrainte. Nous étudions ces deux problèmes (normalité et non dégénérescence) pour des problèmes de contrôle optimal impliquant des contraintes dynamiques exprimées en termes d’inclusion différentielle, lorsque le minimiseur a son point de départ dans une région où la contrainte d’état est non lisse. Nous prouvons que sous une information supplémentaire impliquant principalement le cône tangent de Clarke, les conditions nécessaires sous la forme dite ‘Extended Euler-Lagrange condition’ sont satisfaites en forme normale et non dégénérée pour deux classes de problèmes de contrôle optimal avec contrainte d’état. Le résultat sur la normalité est également appliqué pour le problème de calcul des variations avec contrainte d’état.Dans la deuxième partie de la thèse, nous considérons d’abord une classe de systèmes de contrôle avec contrainte d’état pour lesquels les qualifications de la contrainte standard du ‘premier ordre’ ne sont pas satisfaites, mais une qualification de la contrainte d’ordre supérieure (ordre 2) est satisfaite.Nous proposons une nouvelle construction des trajectoires admissibles (dit un résultat de viabilité) et nous étudions des exemples (tels que l’intégrateur non holonomique de Brockett) fournissant en plus un résultat d’estimation non linéaire. L’autre sujet de la deuxième partie de la thèse concerne l’étude d’une classe de problèmes de contrôle optimal dans lesquels des incertitudes apparaissent dans les données en termes de paramètres inconnus. En tenant compte d’un critère de performance sous la forme de coût moyen, une question cruciale est clairement de pouvoir caractériser les contrôles optimaux indépendamment de l’action du paramètre inconnu: cela permet de trouver une sorte de ‘meilleur compromis’ parmi toutes les réalisations possibles du système de contrôle tant que le paramètre varie. Pour ce type de problèmes, nous obtenons des conditions nécessaires d’optimalité sous la forme du Principe du Maximum (éventuellement pour le cas non lisse)
The project of this thesis is twofold. The first concerns the extension of previous results on necessary optimality conditions for state constrained problems in optimal control and in calculus of variations. The second aim consists in working along two new research lines: derive viability results for a class of control systems with state constraints in which ‘standard inward pointing conditions’ are violated; and establish necessary optimality conditions for average cost minimization problems possibly perturbed by unknown parameters.In the first part, we examine necessary optimality conditions which play an important role in finding candidates to be optimal solutions among all admissible solutions. However, in dynamic optimization problems with state constraints, some pathological situations might arise. For instance, it might occur that the multiplier associated with the objective function (to minimize) vanishes. In this case, the objective function to minimize does not intervene in first order necessary conditions: this is referred to as the abnormal case. A worse phenomenon, called the degenerate case shows that in some circumstances the set of admissible trajectories coincides with the set of candidates to be minimizers. Therefore the necessary conditions give no information on the possible minimizers.To overcome these difficulties, new additional hypotheses have to be imposed, known as constraint qualifications. We investigate these two issues (normality and non-degeneracy) for optimal control problems involving state constraints and dynamics expressed as a differential inclusion, when the minimizer has its left end-point in a region where the state constraint set in nonsmooth. We prove that under an additional information involving mainly the Clarke tangent cone, necessary conditions in the form of the Extended Euler-Lagrange condition are derived in the normal and non-degenerate form for two different classes of state constrained optimal control problems. Application of the normality result is shown also for the calculus of variations problem subject to a state constraint.In the second part of the thesis, we consider first a class of state constrained control systems for which standard ‘first order’ constraint qualifications are not satisfied, but a higher (second) order constraint qualification is satisfied. We propose a new construction for feasible trajectories (a viability result) and we investigate examples (such as the Brockett nonholonomic integrator) providing in addition a non-linear stimate result. The other topic of the second part of the thesis concerns the study of a class of optimal control problems in which uncertainties appear in the data in terms of unknown parameters. Taking into consideration an average cost criterion, a crucial issue is clearly to be able to characterize optimal controls independently of the unknown parameter action: this allows to find a sort of ‘best compromise’ among all the possible realizations of the control system as the parameter varies. For this type of problems, we derive necessary optimality conditions in the form of Maximum Principle (possibly nonsmooth)

Zusammenfassung Diese Arbeit ist ein Versuch, verschiedene klassische und neuere Methodender glatten bzw. nichtglatten Optimierung zu verallgemeinern und in ihrem Zusammenhang darzustellen. Als Hauptinstrument erweist sich dabei die sogenannte verallgemeinerte Kojima-Funktion. Neben reichlichen Beispielen setzen wir einen besonderen Akzent auf die Betrachtung von Variationsungleichungen, Komplementaritaetsaufgaben und der Standartaufgabeder mathematischen Programmierung. Unter natuerlichen Voraussetzungen an diese Probleme kann man u.a. Barriere-, Straf- und SQP-Typ-Methoden, die auf Newton-Verfahrenbasieren, aber auch Modelle, die sogenannte NCP-Funktionen benutzen, mittelsspezieller Stoerungen der Kojima-Funktion exakt modellieren. Daneben werdendurch explizite und natuerliche Wahl der Stoerungsparameter auch neue Methoden dieser Arten vorgeschlagen. Die Vorteile solcher Modellierungsind ueberzeugend vor allem wegen der direkt moeglichen (auf Stabilitaetseigenschaften der Kojima-Gleichung beruhendenden)Loesungsabschaetzungen und weil die entsprechenden Nullstellen ziemlich einfach als Loesungen bekannter Ersatzprobleme interpretiert werden koennen. Ein weiterer Aspekt der Arbeit besteht in der genaueren Untersuchungder "nichtglatten Faelle". Hier wird die Theorie von verschiedenen verallgemeinerten Ableitungen und dadurch entstehenden verallgemeinerten Newton-Verfahren, die im Buch "Nonsmooth Equations in Optimization" von B. Kummer und D. Klatte vorgeschlagen und untersucht wurde, intensiv benutzt. Entscheidend ist dabei, dass die benutzten verallgemeinerten Ableitungen auch praktisch angewandt werden koennen, da man sie exakt ausrechnen kann.
This work attempts to generalize various classical and new methods of smooth or nonsmooth optimization and to show them in their interrelation. The main tool for doing this is the so-called generalized Kojima-function. In addition to numerous examples we specialy emphasize the consideration of variational inequalities, complementarity problems and the standard problem of mathematical programming. Under natural assumptions on these problems we can model e.g. barrier-, penalty-, and SQP-Type-methods basing on Newton methods, and also methods using the so-called NCP-function exactly by means of special perturbations of the Kojima-function. Furthermore, by the explicit and natural choice of the perturbation parameters new methods of these kinds are introduced. The benefit of such a modelling is obvious, first of all due to the direct solution estimation (basing on stability properties of the Kojima-equation) and because the corresponding zeros can easily be interpreted as solutions of known subproblems. A further aspect considered in this paper is the detailed investigation of "nonsmooth cases". The theory of various generalized derivatives and resulting generalized Newton methods, which is introduced and investigated in the book "Nonsmooth Equations in Optimization" of B. Kummer and D. Klatte, is intensely used here. The crucial point is the applicability of the used generalized derivatives in practice, since they can be calculated exactly.

Les problèmes de complémentarité interviennent dans de nombreux domaines scientifiques : économie, mécanique des solides, mécanique des fluides. Ce n'est que récemment qu'ils ont commencé d'intéresser les chercheurs étudiant les écoulements et le transport en milieu poreux. Les problèmes de complémentarité sont un cas particulier des inéquations variationnelles. Dans cette thèse, on offre plusieurs contributions aux méthodes numériques pour résoudre les problèmes de complémentarité. Dans la première partie de cette thèse, on étudie les problèmes de complémentarité linéaires 0 6 x ⊥ (Mx+q) > 0 où, x l'inconnue est dans Rn et où les données sont q, un vecteur de Rn, et M, une matrice d'ordre n. L'existence et l'unicité de ce problème est obtenue quand la matrice M est une P-matrice. Une méthode très efficace pour résoudre les problèmes de complémentarité est la méthode de Newton-min, une extension de la méthode de Newton aux problèmes non lisses.Dans cette thèse on montre d'abord, en construisant deux familles de contre-exemples, que la méthode de Newton-min ne converge pas pour la classe des P-matrices, sauf si n= 1 ou 2. Ensuite on caractérise algorithmiquement la classe des P-matrices : c'est la classe des matrices qui sont telles que quel que, soit le vecteur q, l'algorithme de Newton-min ne fait pas de cycle de deux points. Enfin ces résultats de non-convergence nous ont conduit à construire une méthode de globalisation de l'algorithme de Newton-min dont nous avons démontré la convergence globale pour les P-matrices. Des résultats numériques montrent l'efficacité de cet algorithme et sa convergence polynomiale pour les cas considérés. Dans la deuxième partie de cette thèse, nous nous sommes intéressés à un exemple de problème de complémentarité non linéaire concernant les écoulements en milieu poreux. Il s'agit d'un écoulement liquide-gaz à deux composants eau-hydrogène que l'on rencontre dans le cadre de l'étude du stockage des déchets radioactifs en milieu géologique. Nous présentons un modèle mathématique utilisant des conditions de complémentarité non linéaires décrivant ces écoulements. D'une part, nous proposons une méthode de résolution et un solveur pour ce problème. D'autre part, nous présentons les résultats numériques que nous avons obtenus suite à la simulation des cas-tests proposés par l'ANDRA (Agence Nationale pour la gestion des Déchets Radioactifs) et le GNR MoMaS. En particulier, ces résultats montrent l'efficacité de l'algorithme proposé et sa convergence quadratique pour ces cas-tests

L’objectif de cette thèse est d’obtenir des conditions nécessaires d’optimalité du premier ordre sous la forme d’un principe du maximum de Pontryagin (en abrégé PMP) pour des problèmes de contrôle échantillonné optimal avec temps d’échantillonnage libres, contraintes d’état et coûts de Mayer non lisses. Le Chapitre 1 est consacré aux notations et espaces fonctionnels utiles pour décrire les problèmes de contrôle échantillonné optimal qui seront rencontrés dans le manuscrit. Dans le Chapitre 2, nous obtenons une condition nécessaire d’optimalité lorsque les temps d’échantillonnage peuvent être choisis librement qui est appelée condition de continuité de la fonction Hamiltonienne. Rappelons que la fonction Hamiltonienne qui décrit l’évolution du Hamiltonien avec les valeurs de la trajectoire optimale et du contrôle échantillonné optimal est, en général, discontinue quand les temps d’échantillonnage sont fixés. Notre résultat démontre que la continuité de la fonction Hamiltonienne est retrouvée pour les contrôles échantillonnés optimaux avec temps d’échantillonnage optimaux. Pour terminer, nous implémentons une méthode de tir basée sur la condition de continuité de la fonction Hamiltonienne pour déterminer numériquement les temps d’échantillonnage optimaux dans deux exemples linéaires-quadratiques. Dans le Chapitre 3, nous obtenons un PMP pour des problèmes de contrôle échantillonné optimal avec contraintes d’état. Nous obtenons que les vecteurs adjoints sont solutions de problèmes de Cauchy-Stieltjes définis par des mesures de Borel associées à des fonctions à variation bornée. De plus, nous trouvons que, sous quelques hypothèses assez générales, toute trajectoire admissible (associée à un contrôle échantillonné) rebondit nécessairement sur les contraintes d’état. Nous exploitons ce phénomène de trajectoires rebondissantes pour implémenter une méthode indirecte qu’on utilise pour résoudre numériquement quelques exemples simples de problèmes de contrôle échantillonné optimal avec contraintes d’état. Dans le Chapitre 4, nous obtenons un PMP pour des problèmes de contrôle échantillonné optimal avec coûts de Mayer non lisses. Notre preuve est uniquement basée sur les outils de l’analyse non lisse et n’utilise aucune technique de régularisation. Nous déterminons l’existence d’une sélection dans le sous-différentiel de la fonction de coût de Mayer non lisse en établissant un résultat plus général sur l’existence d’un vecteur séparant universel pour les ensembles convexes compacts. En appliquant ce résultat, appelé théorème de vecteur séparant universel, nous obtenons un PMP pour des problèmes de contrôle échantillonné optimal avec coûts de Mayer non lisses où la condition de transversalité sur le vecteur adjoint est donnée par une inclusion dans le sous-différentiel de la fonction de coût de Mayer non lisse. Pour obtenir les conditions d’optimalité sous la forme d’un PMP, nous utilisons différentes techniques de perturbation sur le contrôle optimal. Pour traiter les contraintes d’état, nous pénalisons la distance à ces contraintes dans une fonctionnelle et nous appliquons le principe variationnel d’Ekeland. En particulier, nous invoquons des résultats sur la renormalisation des espaces de Banach pour assurer la régularité de la fonction distance dans les contextes de dimension infinie. Enfin nous utilisons des notions standards de l’analyse non lisse, telles que les dérivées directionnelles généralisées de Clarke et le sous-différentiel de Clarke, pour étudier les problèmes de contrôle échantillonné optimal avec coûts de Mayer non lisses
This dissertation is concerned with first-order necessary optimality conditions in the form of a Pontryagin maximum principle (in short, PMP) for optimal sampled-data control problems with free sampling times, running inequality state constraints and nonsmooth Mayer cost functions.Chapter 1 is devoted to notations and basic framework needed to describe the optimal sampled-data control problems to be encountered in the manuscript. In Chapter 2, considering that the sampling times can be freely chosen, we obtain an additional necessary optimality condition in the PMP called the Hamiltonian continuity condition. Recall that the Hamiltonian function, which describes the evolution of the Hamiltonian taking values of the optimal trajectory and of theoptimal sampled-data control, is in general discontinuous when the sampling times are fixed. Our result proves that the continuity of the Hamiltonian function is recovered in the case of optimal sampled-data controls with optimal sampling times. Finally we implement a shooting method based on the Hamiltonian continuity condition in order to numerically determine the optimal sampling times in two linear-quadratic examples.In Chapter 3, we obtain a PMP for optimal sampled-data control problems with running inequality state constraints. In particular we obtain that the adjoint vectors are solutions to Cauchy-Stieltjes problems defined by Borel measures associated to functions of bounded variation. Moreover, we find that, under certain general hypotheses, any admissible trajectory (associated to a sampled-data control) necessarily bounces on the runningine quality state constraints. Taking advantage of this bouncing trajectory phenomen on, we are able to use thePMP to implement an indirect numerical method which we use to numerically solve some simple examples of optimal sampled-data control problems with running inequality state constraints. In Chapter 4, we obtain a PMP for optimal sampled-data control problems with nonsmooth Mayer cost functions. Our proof directly follows from the tools of nonsmooth analysis and does not involve any regularization technique. We determine the existence of a selection in the subdifferential of the nonsmooth Mayer cost function by establishing a more general result asserting the existence a universal separating vector for a given compact convex set. From the application of this result, which is called universal separating vector theorem, we obtain a PMP for optimal sampled-data control problems with nonsmooth Mayer cost functions where the transversality conditon on the adjoint vector is given by an inclusion in the subdifferential of the nonsmooth Mayer cost function.To obtain the optimality conditions in the form of a PMP, we use different techniques of perturbations of theoptimal control. In order to handle the state constraints, we penalize the distance to them in a corresponding cost functional and then apply the Ekeland variational principle. In particular, we invoke some results on renorming Banach spaces in order to ensure the regularity of distance functions in the infinite-dimensional context. Finally we use standard notions in nonsmooth analysis such as the Clarke generalized directional derivative and theClarke subdifferential to study optimal sampled-data control problems with nonsmooth Mayer cost functions

La simulation numérique des systèmes multicorps en présence d'interactions complexes, dont le contact frottant, pose de nombreux défis, tant en terme de modélisation que de temps de calcul. Dans ce manuscrit de thèse, nous étudions deux familles de décomposition de domaine adaptées au formalisme de la dynamique non régulière des contacts (NSCD). Cette méthode d'intégration implicite en temps de l'évolution d'une collection de corps en interaction a pour caractéristique de prendre en compte le caractère discret et non régulier d'un tel milieu. Les techniques de décomposition de domaine classiques ne peuvent de ce fait être directement transposées. Deux méthodes de décomposition de domaine, proches des formalismes des méthodes de Schwarz et de complément de Schur sont présentées. Ces méthodes se révèlent être de puissants outils pour la parallélisation en mémoire distribuée des simulations granulaires 2D et 3D sur un centre de calcul haute performance. Le comportement de structure des milieux granulaires denses est de plus exploité afin de propager rapidement l'information sur l'ensemble des sous domaines via un schéma semi-implicite d'intégration en temps.

Huang Liren.
Thesis (Ph.D.)--Chinese University of Hong Kong, 1993.
Includes bibliographical references (leaves 96).
Abstract --- p.1
Introduction --- p.2
References --- p.5
Chapter Chapter 1. --- Some elementary results in nonsmooth analysis and optimization --- p.6
Chapter 1. --- "Some properties for ""lim sup"" and ""lim inf""" --- p.6
Chapter 2. --- The directional derivative of the sup-type function --- p.8
Chapter 3. --- Some results in nonsmooth analysis and optimization --- p.12
References --- p.19
Chapter Chapter 2. --- On generalized second-order derivatives and Taylor expansions in nonsmooth optimization --- p.20
Chapter 1. --- Introduction --- p.20
Chapter 2. --- "Dini-directional derivatives, Clark's directional derivatives and generalized second-order directional derivatives" --- p.20
Chapter 3. --- On Cominetti and Correa's conjecture --- p.28
Chapter 4. --- Generalized second-order Taylor expansion --- p.36
Chapter 5. --- Detailed proof of Theorem 2.4.2 --- p.40
Chapter 6. --- Corollaries of Theorem 2.4.2 and Theorem 2.4.3 --- p.43
Chapter 7. --- Some applications in optimization --- p.46
Ref erences --- p.51
Chapter Chapter 3. --- Second-order necessary and sufficient conditions in nonsmooth optimization --- p.53
Chapter 1. --- Introduction --- p.53
Chapter 2. --- Second-order necessary and sufficient conditions without constraint --- p.56
Chapter 3. --- Second-order necessary conditions with constrains --- p.66
Chapter 4. --- Sufficient conditions theorem with constraints --- p.77
References --- p.87
Appendix --- p.89
References --- p.96

In many applications nonsmooth nonconvex energy functions, which are Lipschitz continuous, appear quite naturally. Contact mechanics with friction is a classic example. A second example is the 1-Laplace operator and its eigenfunctions. In this work we will give an algorithm such that for every locally Lipschitz continuous function f and every sequence produced by this algorithm it holds that every accumulation point of the sequence is a critical point of f in the sense of Clarke. Here f is defined on a reflexive Banach space X, such that X and its dual space X' are strictly convex and Clarkson's inequalities hold. (E.g. Sobolev spaces and every closed subspace equipped with the Sobolev norm satisfy these assumptions for p>1.) This algorithm is designed primarily to solve variational problems or their high dimensional discretizations, but can be applied to a variety of locally Lipschitz functions. In elastic contact mechanics the strain energy is often smooth and nonconvex on a suitable domain, while the contact and the friction energy are nonsmooth and have a support on a subspace which has a substantially smaller dimension than the strain energy, since all points in the interior of the bodies only have effect on the strain energy. For such elastic contact problems we suggest a specialization of our algorithm, which treats the smooth part with Newton like methods. In the case that the gradient of the entire energy function is semismooth close to the minimizer, we can even prove superlinear convergence of this specialization of our algorithm. We test the algorithm and its specialization with a couple of benchmark problems. Moreover, we apply the algorithm to the 1-Laplace minimization problem restricted to finitely dimensional subspaces of piecewise affine, continuous functions. The algorithm developed here uses ideas of the bundle trust region method by Schramm, and a new generalization of the concept of gradients on a set. The basic idea behind this gradients on sets is that we want to find a stable descent direction, which is a descent direction on an entire neighborhood of an iteration point. This way we avoid oscillations of the gradients and very small descent steps (in the smooth and in the nonsmooth case). It turns out, that the norm smallest element of the gradient on a set provides a stable descent direction. The algorithm we present here is the first algorithm which can treat locally Lipschitz continuous functions in this generality, up to our knowledge. In particular, large finitely dimensional Banach spaces haven't been studied for nonsmooth nonconvex functions so far. We will show that the algorithm is very robust and often faster than common algorithms. Furthermore, we will see that with this algorithm it is possible to compute reliably the first eigenfunctions of the 1-Laplace operator up to disretization errors, for the first time.
In vielen Anwendungen tauchen nichtglatte, nichtkonvexe, Lipschitz-stetige Energie Funktionen in natuerlicher Weise auf. Ein klassische Beispiel bildet die Kontaktmechanik mit Reibung. Ein weiteres Beispiel ist der $1$-Laplace Operator und seine Eigenfunktionen. In dieser Dissertation werden wir ein Abstiegsverfahren angeben, so dass fuer jede lokal Lipschitz-stetige Funktion f jeder Haeufungspunkt einer durch dieses Verfahren erzeugten Folge ein kritischer Punkt von f im Sinne von Clarke ist. Hier ist f auf einem einem reflexiver, strikt konvexem Banachraum definierert, fuer den der Dualraum ebenfalls strikt konvex ist und die Clarkeson Ungleichungen gelten. (Z.B. Sobolevraeume und jeder abgeschlossene Unterraum mit der Sobolevnorm versehen, erfuellt diese Bedingung fuer p>1.) Dieser Algorithmus ist primaer entwickelt worden um Variationsprobleme, bzw. deren hochdimensionalen Diskretisierungen zu loesen. Er kann aber auch fuer eine Vielzahl anderer lokal Lipschitz stetige Funktionen eingesetzt werden. In der elastischen Kontaktmechanik ist die Spannungsenergie oft glatt und nichtkonvex auf einem geeignetem Definitionsbereich, waehrend der Kontakt und die Reibung durch nicht glatte Funktionen modelliert werden, deren Traeger ein Unterraum mit wesentlich kleineren Dimension ist, da alle Punkte im Inneren des Koerpers nur die Spannungsenergie beeinflussen. Fuer solche elastischen Kontaktprobleme schlagen wir eine Spezialisierung unseres Algorithmuses vor, der den glatten Teil mit Newton aehnlichen Methoden behandelt. Falls der Gradient der gesamten Energiefunktion semiglatt in der Naehe der Minimalstelle ist, koennen wir sogar beweisen, dass der Algorithmus superlinear konvergiert. Wir testen den Algorithmus und seine Spezialisierung an mehreren Benchmark Problemen. Ausserdem wenden wir den Algorithmus auf 1-Laplace Minimierungsproblem eingeschraenkt auf eine endlich dimensionalen Unterraum der stueckweise affinen, stetigen Funktionen an. Der hier entwickelte Algorithmus verwendet Ideen des Bundle-Trust-Region-Verfahrens von Schramm, und einen neu entwickelten Verallgemeinerung von Gradienten auf Mengen. Die zentrale Idee hinter den Gradienten auf Mengen ist die, dass wir stabile Abstiegsrichtungen auf einer ganzen Umgebung der Iterationspunkte finden wollen. Auf diese Weise vermeiden wir das Oszillieren der Gradienten und sehr kleine Abstiegsschritte (im glatten, wie im nichtglatten Fall.) Es stellt sich heraus, dass das normkleinste Element dieses Gradienten auf der Umgebung eine stabil Abstiegsrichtung bestimmt. So weit es uns bekannt ist, koennen die hier entwickelten Algorithmen zum ersten Mal lokal Lipschitz-stetige Funktionen in dieser Allgemeinheit behandeln. Insbesondere wurden nichtglatte, nichtkonvexe Funktionen auf derart hochdimensionale Banachraeume bis jetzt nicht behandelt. Wir werden zeigen, dass unser Algorithmus sehr robust und oft schneller als uebliche Algorithmen ist. Des Weiteren, werden wir sehen, dass es mit diesem Algorithmus das erste mal moeglich ist, zuverlaessig die erste Eigenfunktion des 1-Laplace Operators bis auf Diskretisierungsfehler zu bestimmen.

The main purpose of this dissertation is to investigate necessary optimality conditions for a class of very general nonsmooth optimization problems called the mathematical program with geometric constraints (MPGC). The geometric constraint means that the image of certain mapping is included in a nonempty and closed set. We first study the conventional nonlinear program with equality, inequality and abstract set constraints as a special case of MPGC. We derive the enhanced Fritz John condition and from which, we obtain the enhanced Karush-Kuhn-Tucker (KKT) condition and introduce the associated pseudonormality and quasinormality condition. We prove that either pseudonormality or quasinormality with regularity implies the existence of a local error bound. We also give a tighter upper estimate for the Fr\'chet subdifferential and the limiting subdifferential of the value function in terms of quasinormal multipliers which is usually a smaller set than the set of classical normal multipliers. We then consider a more general MPGC where the image of the mapping from a Banach space is included in a nonempty and closed subset of a finite dimensional space. We obtain the enhanced Fritz John necessary optimality conditions in terms of the approximate subdifferential. One of the technical difficulties in obtaining such a result in an infinite dimensional space is that no compactness result can be used to show the existence of local minimizers of a perturbed problem. We employ the celebrated Ekeland's variational principle to obtain the results instead. We then apply our results to the study of exact penalty and sensitivity analysis. We also study a special class of MPCG named mathematical programs with equilibrium constraints (MPECs). We argue that the MPEC-linear independence constraint qualification is not a constraint qualification for the strong (S-) stationary condition when the objective function is nonsmooth. We derive the enhanced Fritz John Mordukhovich (M-) stationary condition for MPECs. From this enhanced Fritz John M-stationary condition we introduce the associated MPEC generalized pseudonormality and quasinormality condition and build the relations between them and some other widely used MPEC constraint qualifications. We give upper estimates for the subdifferential of the value function in terms of the enhanced M- and C-multipliers respectively. Besides, we focus on some new constraint qualifications introduced for nonlinear extremum problems in the recent literature. We show that, if the constraint functions are continuously differentiable, the relaxed Mangasarian-Fromovitz constraint qualification (or, equivalently, the constant rank of the subspace component condition) implies the existence of local error bounds. We further extend the new result to the MPECs.
Graduate
0405

First, we study the associated regularized gap functions and the D-gap functions and compute their Clarke-Rockafellar directional derivatives and the Clarke generalized gradients. Second, using these tools and extending the works of Fukushima and Pang (who studied the case when F is smooth), we present results on the relationship between minimizing sequences and stationary sequences of the D-gap functions, regardless the existence of solutions of (VIP). Finally, as another application, we show that, under the strongly monotonicity assumption, the regularized gap functions have fractional exponent error bounds, and thereby we provide an algorithm of Armijo type to solve the (VIP).
In this thesis, we investigate a nonsmooth variational inequality problem (VIP) defined by a locally Lipschitz function F which is not necessarily differentiable or monotone on its domain which is a closed convex set in an Euclidean space.
Tan Lulin.
"December 2005."
Adviser: Kung Fu Ng.
Source: Dissertation Abstracts International, Volume: 67-11, Section: B, page: 6444.
Thesis (Ph.D.)--Chinese University of Hong Kong, 2005.
Includes bibliographical references (p. 79-84) and index.
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Abstracts in English and Chinese.
School code: 1307.