This dissertation deals with the use of Shewhart control charts, modified to have variable sampling intervals, to simultaneously monitor a set of parameters. Fixed sampling interval control charts are modified to utilize sampling intervals that vary depending on what is being observed from the data. Two problems are emphasized, namely, the simultaneous monitoring of the mean and the variance and the simultaneous monitoring of several means. For each problem, two basic strategies are investigated. One strategy uses separate control charts for each parameter. A second strategy uses a single statistic which combines the information in the entire sample and is sensitive to shifts in any of the parameters. Several variations on these two basic strategies are studied. Numerical studies investigate the optimal number of sampling intervals and the length of the sampling intervals to be used. Each procedure is compared to corresponding fixed interval procedures in terms of time and the number of samples taken to signal. The effect of correlation on multiple means charts is studied through simulation. For both problems, it is seen that the variable sampling interval approach is substantially more efficient than fixed interval procedures, no matter which strategy is used. / Ph. D.

The main emphasis of this thesis is to study a posteriori error analysis of discontinuous Galerkin (DG) methods for the elliptic variational inequalities. The DG methods have become very pop-ular in the last two decades due to its nature of handling complex geometries, allowing irregular meshes with hanging nodes and different degrees of polynomial approximation on different ele-ments. Moreover they are high order accurate and stable methods. Adaptive algorithms refine the mesh locally in the region where the solution exhibits irregular behaviour and a posteriori error estimates are the main ingredients to steer the adaptive mesh refinement. The solution of linear elliptic problem exhibits singularities due to change in boundary con-ditions, irregularity of coefficients and reentrant corners in the domain. Apart from this, the solu-tion of variational inequality exhibits additional irregular behaviour due to occurrence of the free boundary (the part of the domain which is a priori unknown and must be found as a component of the solution). In the lack of full elliptic regularity of the solution, uniform refinement is inefficient and it does not yield optimal convergence rate. But adaptive refinement, which is based on the residuals ( or a posteriori error estimator) of the problem, enhance the efficiency by refining the mesh locally and provides the optimal convergence. In this thesis, we derive a posteriori error estimates of the DG methods for the elliptic variational inequalities of the first kind and the second kind. This thesis contains seven chapters including an introductory chapter and a concluding chap-ter. In the introductory chapter, we review some fundamental preliminary results which will be used in the subsequent analysis. In Chapter 2, a posteriori error estimates for a class of DG meth-ods have been derived for the second order elliptic obstacle problem, which is a prototype for elliptic variational inequalities of the first kind. The analysis of Chapter 2 is carried out for the general obstacle function therefore the error estimator obtained therein involves the min/max func-tion and hence the computation of the error estimator becomes a bit complicated. With a mild assumption on the trace of the obstacle, we have derived a significantly simple and easily com-putable error estimator in Chapter 3. Numerical experiments illustrates that this error estimator indeed behaves better than the error estimator derived in Chapter 2. In Chapter 4, we have carried out a posteriori analysis of DG methods for the Signorini problem which arises from the study of the frictionless contact problems. A nonlinear smoothing map from the DG finite element space to conforming finite element space has been constructed and used extensively, in the analysis of Chapter 2, Chapter 3 and Chapter 4. Also, a common property shared by all DG methods allows us to carry out the analysis in unified setting. In Chapter 5, we study the C0 interior penalty method for the plate frictional contact problem, which is a fourth order variational inequality of the second kind. In this chapter, we have also established the medius analysis along with a posteriori analy-sis. Numerical results have been presented at the end of every chapter to illustrate the theoretical results derived in respective chapters. We discuss the possible extension and future proposal of the work presented in the Chapter 6. In the last chapter, we have documented the FEM codes used in the numerical experiments.

This work aims to investigate some applications of the conjugate duality for scalar and vector optimization problems to the construction of gap functions for variational inequalities and equilibrium problems. The basic idea of the approach is to reformulate variational inequalities and equilibrium problems into optimization problems depending on a fixed variable, which allows us to apply duality results from optimization problems. Based on some perturbations, first we consider the conjugate duality for scalar optimization. As applications, duality investigations for the convex partially separable optimization problem are discussed. Afterwards, we concentrate our attention on some applications of conjugate duality for convex optimization problems in finite and infinite-dimensional spaces to the construction of a gap function for variational inequalities and equilibrium problems. To verify the properties in the definition of a gap function weak and strong duality are used. The remainder of this thesis deals with the extension of this approach to vector variational inequalities and vector equilibrium problems. By using the perturbation functions in analogy to the scalar case, different dual problems for vector optimization and duality assertions for these problems are derived. This study allows us to propose some set-valued gap functions for the vector variational inequality. Finally, by applying the Fenchel duality on the basis of weak orderings, some variational principles for vector equilibrium problems are investigated.

Dans une première partie, nous étudions la stabilité des solutions d'une inégalité variationnelle de la forme cône normal perturbé par une fonction. Pour ce faire, nous généralisons la méthode de S. Robinson, basée sur le degré topologique, aux espaces de Hilbert et à une classe de multi-applications non nécessairement convexes, appelées multi-applications prox-régulières.  Dans une deuxième partie, nous étudions des problèmes de contrôle optimal liés à la modélisation de problèmes de bio-procédés, et l'on s'intéresse à des contraintes périodiques sur l'état. Ainsi, nous étendons les résultats d'existence de solutions périodiques des EDOs de Kolmogorov au cadre du contrôle en rajoutant un paramètre contrôlé à ces équations. Ceci nous permet d'étudier par la suite un problème de commande optimale d'un chemostat sous forçage périodique, et d'en déduire la synthèse optimale pour ce problème. / In the first part, we study stability of solutions of a variational inequality of the form normal cone perturbed by a mapping. To do so, we generalize the method introduced by S. Robinson, based on the topological degree, to the general Hilbert setting on the class of non-necessarily convex set-valued mapping, called prox-regular set-valued mapping. In the second part, we study optimal control problems connected to the modelization of bio-processes and we consider periodic constraints on the state variable. We first extend the existence result of periodic solutions of Kolmogorov ODEs to the setting of control by adding a controlled parameter to those ODEs. This allows us to study an optimal control problem modeling a chemostat under a periodic forcing for which we give the optimal synthesis.

[pt] O mercado brasileiro de derivados de petróleo está sendo aberto para competição este ano, saindo de um ambiente de preços regulados pelo Governo Federal para um ambiente onde os preços são estabelecidos pelas leis de oferta e demanda. Neste contexto, existe a preocupação de como serão estes preços, e seus impactos sobre os consumidores e sobre os produtores locais. Esta Tese propõe alguns modelos matemáticos para estimar preços, níveis de produção, níveis de consumo (demanda), e importação e exportação de derivados de petróleo nas diversas regiões do mercado brasileiro. O fornecimento de derivados de petróleo não é considerado um mercado competitivo, e sim oligopolizado, principalmente no curto prazo, devido à capacidade instalada de refinarias e aos altos custos envolvidos na construção de novas refinarias. Estes modelos são multi- produto, considerando um fato importante na produção de derivados que é a impossibilidade de produzir apenas um derivado. Assim, existem restrições onde a oferta de um derivado é relacionada a oferta dos outros. O primeiro modelo considera um mercado de oligopólio fechado, com um número fixo de firmas. Tal modelo é formulado como um problema de equilíbrio a Nash. Um segundo modelo é apresentado expandindo o primeiro para o caso em que existem preços teto de demanda definidos politicamente. O terceiro modelo relaxa a suposição do mercado fechado, com número fixo de firmas, e considera a possibilidade de competição de novas firmas no mercado. Um quarto modelo é discutido, onde assume-se que existe uma firma líder no mercado, que consegue definir sua estratégia antes das demais firmas, semelhante ao problema econômico de Stackelberg. Todos os modelos foram formulados como problemas de inequações variacionais, sendo que o último modelo é ainda um problema de programação binível. Algoritmos de solução são propostos para os três primeiros modelos. Simulações sobre o mercado brasileiro de derivados são apresentadas. / [en] The Brazilian petroleum refined products market is being opened to competition this year, leaving an environment of regulated prices to another one where the prices are defined by the supply demand interactions. Considering this new scenario, there is a concern about how high the prices will be, and about their impact on the consumers and on the local producers. This thesis proposes some mathematical models to predict prices, production, consumption, and import and export levels of petroleum-refined products in all the sub-regions of the Brazilian market. Instead of a competitive market, the supply of refined products is considered an oligopoly market, especially in the short term, given the already installed refining capacity and the high costs involved in building new refineries. These models are multi-products, and they consider an important characteristic of the production of refined products, the impossibility of producing only one refined product. Hence, constraints where the production of one refined product is related to the production of the others are considered. The first model considers a closed oligopoly market, with a fixed number of firms. This problem is formulated as a Nash equilibrium problem. A second model is presented generalizing the first one to consider the possibility of ceiling demand prices politically defined. The third model relaxes the assumption of a fixed number of firms in the first model, and considers the possibility of competition by new entrants. A fourth model is discussed, where it is assumed that there is a leader firm in the market, which can define its strategy before the other firms, similar to the economic problem of Stackelberg. All the models are formulated as variational inequalities problems, and the last model is also a bi-level programming problem. Solution algorithms for the three first models are proposed. Some analyses of the Brazilian petroleum refined- products market are presented.

Incremental Sheet Forming (ISF) is a progressive metal forming process, where the deformation occurs locally around the point of contact between a tool and the metal sheet. The final work-piece is formed cumulatively by the movements of the tool, which is usually attached to a CNC milling machine. The ISF process is dieless in nature and capable of producing different parts of geometries with a universal tool. The tooling cost of ISF can be as low as 5–10% compared to the conventional sheet metal forming processes. On the laboratory scale, the accuracy of the parts created by ISF is between ±1.5 mm and ±3mm. However, in order for ISF to be competitive with a stamping process, an accuracy of below ±1.0 mm and more realistically below ±0.2 mm would be needed. In this work, we first studied the ISF deformation process by a simplified phenomenal linear model and employed a predictive controller to obtain an optimised tool trajectory in the sense of minimising the geometrical deviations between the targeted shape and the shape made by the ISF process. The algorithm is implemented at a rig in Cambridge University and the experimental results demonstrate the ability of the model predictive controller (MPC) strategy. We can achieve the deviation errors around ±0.2 mm for a number of simple geometrical shapes with our controller. The limitations of the underlying linear model for a highly nonlinear problem lead us to study the ISF process by a physics based model. We use the elastoplastic constitutive relation to model the material law and the contact mechanics with Signorini’s type of boundary conditions to model the process, resulting in an infinite dimensional system described by a partial differential equation. We further developed the computational method to solve the proposed mathematical model by using an augmented Lagrangian method in function space and discretising by finite element method. The preliminary results demonstrate the possibility of using this model for optimal controller design.

<p>This thesis consists of four papers and a summary. The common topic of the included papers are the pricing equations of path-dependent options. Various properties of barrier options and American options are studied, such as convexity of option prices, the size of the continuation region in American option pricing and pricing formulas for turbo warrants. In Paper I we study the effect of model misspecification on barrier option pricing. It turns out that, as in the case of ordinary European and American options, this is closely related to convexity properties of the option prices. We show that barrier option prices are convex under certain conditions on the contract function and on the relation between the risk-free rate of return and the dividend rate. In Paper II a new condition is given to ensure that the early exercise feature in American option pricing has a positive value. We give necessary and sufficient conditions for the American option price to coincide with the corresponding European option price in at least one diffusion model. In Paper III we study parabolic obstacle problems related to American option pricing and in particular the size of the non-coincidence set. The main result is that if the boundary of the set of points where the obstacle is a strict subsolution to the differential equation is C¹-Dini in space and Lipschitz in time, there is a positive distance, which is uniform in space, between the boundary of this set and the boundary of the non-coincidence set. In Paper IV we derive explicit pricing formulas for turbo warrants under the classical Black-Scholes assumptions.</p>

This thesis consists of four papers and a summary. The common topic of the included papers are the pricing equations of path-dependent options. Various properties of barrier options and American options are studied, such as convexity of option prices, the size of the continuation region in American option pricing and pricing formulas for turbo warrants. In Paper I we study the effect of model misspecification on barrier option pricing. It turns out that, as in the case of ordinary European and American options, this is closely related to convexity properties of the option prices. We show that barrier option prices are convex under certain conditions on the contract function and on the relation between the risk-free rate of return and the dividend rate. In Paper II a new condition is given to ensure that the early exercise feature in American option pricing has a positive value. We give necessary and sufficient conditions for the American option price to coincide with the corresponding European option price in at least one diffusion model. In Paper III we study parabolic obstacle problems related to American option pricing and in particular the size of the non-coincidence set. The main result is that if the boundary of the set of points where the obstacle is a strict subsolution to the differential equation is C1-Dini in space and Lipschitz in time, there is a positive distance, which is uniform in space, between the boundary of this set and the boundary of the non-coincidence set. In Paper IV we derive explicit pricing formulas for turbo warrants under the classical Black-Scholes assumptions.

Dans cette thèse, nous proposons des algorithmes et présentons des résultats numériques pour la résolution de jeux répétés stochastiques, à deux joueurs et somme nulle dont l'espace d'état est de grande taille. En particulier, nous considérons la classe de jeux en information complète et en horizon infini. Dans cette classe, nous distinguons d'une part le cas des jeux avec gain actualisé et d'autre part le cas des jeux avec gain moyen. Nos algorithmes, implémentés en C, sont principalement basés sur des algorithmes de type itérations sur les politiques et des méthodes multigrilles. Ces algorithmes sont appliqués soit à des équations de la programmation dynamique provenant de problèmes de jeux à deux joueurs à espace d'états fini, soit à des discrétisations d'équations de type Isaacs associées à des jeux stochastiques différentiels. Dans la première partie de cette thèse, nous proposons un algorithme qui combine l'algorithme des itérations sur les politiques pour les jeux avec gain actualisé à des méthodes de multigrilles algébriques utilisées pour la résolution des systèmes linéaires. Nous présentons des résultats numériques pour des équations d'Isaacs et des inéquations variationnelles. Nous présentons également un algorithme d'itérations sur les politiques avec raffinement de grilles dans le style de la méthode FMG. Des exemples sur des inéquations variationnelles montrent que cet algorithme améliore de façon non négligeable le temps de résolution de ces inéquations. Pour le cas des jeux avec gain moyen, nous proposons un algorithme d'itération sur les politiques pour les jeux à deux joueurs avec espaces d'états et d'actions finis, dans le cas général multichaine (c'est-à-dire sans hypothèse d'irréductibilité sur les chaînes de Markov associées aux stratégies des deux joueurs). Cet algorithme utilise une idée développée dans Cochet-Terrasson et Gaubert (2006). Cet algorithme est basé sur la notion de projecteur spectral non-linéaire d'opérateurs de la programmation dynamique de jeux à un joueur (lequel est monotone et convexe). Nous montrons que la suite des valeurs et valeurs relatives satisfont une propriété de monotonie lexicographique qui implique que l'algorithme termine en temps fini. Nous présentons des résultats numériques pour des jeux discrets provenant d'une variante des jeux de Richman et sur des problèmes de jeux de poursuite. Finalement, nous présentons de nouveaux algorithmes de multigrilles algébriques pour la résolution de systèmes linéaires singuliers particuliers. Ceux-ci apparaissent, par exemple, dans l'algorithme d'itérations sur les politiques pour les jeux stochastiques à deux joueurs et somme nulle avec gain moyen, décrit ci-dessus. Nous introduisons également une nouvelle méthode pour la recherche de mesures invariantes de chaînes de Markov irréductibles basée sur une approche de contrôle stochastique. Nous présentons un algorithme qui combine les itérations sur les politiques d'Howard et des itérations de multigrilles algébriques pour les systèmes linéaires singuliers.

Diese Arbeit untersucht Nicht-Glattheiten im Bereich der Formoptimierung aus zwei verschiedenen Perspektiven. Einerseits werden Formoptimierungsprobleme betrachtet, die durch Variationsungleichungen eingeschränkt sind. Häufig fallen diese unter die sogenannten Hindernis-Probleme. Diese Probleme haben zahlreiche Anwendungen, beispielsweise bei der Konstruktion von Formen, die Einschränkungen durch die Lösung von partiellen Differentialgleichungen unterliegen, welche wiederum von der optimierten Geometrie abhängen. Oft werden verschiedene Regularisierungsmethoden genutzt, um der Nicht-Glattheit und Nicht-Konvexität Herr zu werden. Über die Nicht-Glattheiten hinaus ergeben sich weitere Herausforderungen, welche durch die Variationsungleichheiten sowie durch Nicht-Konvexität und Unendlich-Dimensionalität auftreten. Diese Faktoren erschweren die Formulierung von Optimalitätsbedingungen und die Entwicklung effizienter Lösungsalgorithmen. In dieser Arbeit wird ein Ansatz vorgestellt, welcher es ermöglicht, Nicht-Glattheiten ohne Regularisierung zu behandeln. Dazu wird die Hadamard-Semiableitung verwendet. Auf der anderen Seite steht die Frage nach geeigneten Formräumen im Fokus und motiviert den zweiten Teil dieser Arbeit. Herkömmliche Formräume umfassen typischerweise glatte Verformungen der Kugel und sind mit einer glatten Struktur versehen. Dadurch kommt es zu einer Vernachlässigung von Formen mit Ecken und Kanten. Die Konstruktion eines Formraums, der nicht-glatte Formen beinhaltet, ist keine Herausforderung. Das Arbeiten mit einem solchen Raum hingegen schon. Typische glatte Strukturen, wie die der riemannschen Mannigfaltigkeit, gehen unter Umständen verloren. Diese Arbeit führt daher diffeologische Räume als natürliche Verallgemeinerung glatter Mannigfaltigkeiten ein. Die Erweiterung von Optimierungstechniken von glatten Mannigfaltigkeiten auf diffeologische Räume stellt eine Herausforderung da. Besondere Aspekte sind die Existenz nicht-äquivalenter Definitionen des Tangentialraums sowie die Notwendigkeit einer erweiterten riemannschen Struktur zur Herleitung von Gradienten. Diese Arbeit präsentiert eine Erweiterung der bereits bekannten riemannschen Optimierung und ihrer Objekte. Dazu gehören unter anderem Definitionen für einen geeigneten Tangentialraum, ein diffeologisches riemannsches Setting, einen diffeologischen Gradienten, eine diffeologische Retraktion und einen diffeologischen Levi-Civita-Zusammenhang. Dies resultiert in der Formulierung eines diffeologischen Gradientenverfahrens, das auf ein Optimierungsproblem angewendet wird.:Abstract Zusammenfassung Acknowledgments Preface 1 Introduction 1.1 Motivation,Aim,andScopeoftheThesis 1.2 StructureoftheThesis 2 Background Knowledge 2.1 DifferentialGeometry 2.2 ShapeOptimization 2.2.1 ABasicIntroduction 2.2.2 A Brief Overview of Variational Inequality Constraints 2.2.3 ShapeSpaces 3 A Hadamard Approach for Variational Inequality Constrains 3.1 HadamardSemiderivative 3.1.1 A Brief Introduction into Hadamard Semiderivatives 3.1.2 HadamardShapeDerivativeCalculus 3.2 HadamardOptimalitySystem 3.3 Application 4 Optimization on Diffeological Spaces 4.1 ABriefIntroductiontoDiffeologicalSpaces 4.2 Towards Optimization Algorithms on Diffeological Spaces 4.2.1 TangentSpace 4.2.2 Examples of Diffeological Tangent Spaces 4.2.3 DiffeologicalRiemannianSpace 4.2.4 Towards Updates of Iterates: Diffeological Levi-Civita ConnectionandDiffeologicalRetraction 4.3 Formulation of Diffeological Optimization Algorithms andTheirApplication 5 Conclusion 6 Notations / This thesis is concerned with non-smoothness from two different points of view regarding shape optimization problems. On one hand this thesis considers shape optimization problems that are constrained by variational inequalities of the first kind, often known as obstacle-type problems. These problems find numerous applications when constructing a shape that must adhere to constraints imposed on the solution of a partial differential equation dependent on the geometry being optimized. Since those problems are non-smooth and non-convex optimization problems, they are often handled using several regularization methods. Besides the non-smoothness there are complementary aspects due to the variational inequalities as well as non-linear, non-convex and infinite-dimensional aspects due to the shapes. This complicates setting up an optimality system, and thus developing an efficient solution algorithm. This thesis is presenting a way to deal with the non-smoothness without the requirement of regularizations. Therefore, the Hadamard semiderivative setting is considered. After introducing the Hadamard semiderivative and considering the Hadamard shape calculus, the Hadamard adjoint is introduced. This allows us to handle shape optimization problems that are constrained by variational inequalities of the first kind without using any kind of regularization. On the other hand this thesis is confronted with the question of suitable shape spaces. What is a suitable space that contains all important shapes? This question is the motivation of the second part of this thesis. A common shape space usually contains some kind of smooth deformations of the sphere. Often those shape spaces are equipped with a suitable smooth structure. This can results in the neglection of shapes that have kinks and corners and are non-smooth. The construction of a shape space that includes non-smooth shapes is not a major challenge but working with such a space is. How do you optimize if your space is not a manifold? On answering this question lies the main focus of the second part of this thesis. Therefore, this thesis introduces so-called diffeological spaces. Diffeological spaces, firstly introduced by J.M. Souriau in the 1980s, are a natural generalization of smooth manifolds. To date, optimization techniques have primarily been developed on manifolds. Extending these methods to diffeological spaces presents a significant challenge for several reasons. One prominent obstacle is the existence of different definitions for tangent spaces that do not coincide with one another. Furthermore, the expansion necessitates the creation of a broader concept of a Riemannian structure to establish gradients, which are essential components for optimization strategies. The first major step is a suitable definition of a tangent space in view of optimization methods. This definition is then used in order to present a diffeological Riemannian space and a diffeological gradient, which this thesis needs to formulate an optimization algorithm on diffeological spaces. Moreover, this thesis presents a diffeological retraction and the Levi-Civita connection on diffeological spaces. As a result a diffeological version of the steepest decent method is obtained. This thesis gives examples for the novel objects and apply the presented diffeological algorithm (an algorithm for diffeological spaces) to an optimization problem.:Abstract Zusammenfassung Acknowledgments Preface 1 Introduction 1.1 Motivation,Aim,andScopeoftheThesis 1.2 StructureoftheThesis 2 Background Knowledge 2.1 DifferentialGeometry 2.2 ShapeOptimization 2.2.1 ABasicIntroduction 2.2.2 A Brief Overview of Variational Inequality Constraints 2.2.3 ShapeSpaces 3 A Hadamard Approach for Variational Inequality Constrains 3.1 HadamardSemiderivative 3.1.1 A Brief Introduction into Hadamard Semiderivatives 3.1.2 HadamardShapeDerivativeCalculus 3.2 HadamardOptimalitySystem 3.3 Application 4 Optimization on Diffeological Spaces 4.1 ABriefIntroductiontoDiffeologicalSpaces 4.2 Towards Optimization Algorithms on Diffeological Spaces 4.2.1 TangentSpace 4.2.2 Examples of Diffeological Tangent Spaces 4.2.3 DiffeologicalRiemannianSpace 4.2.4 Towards Updates of Iterates: Diffeological Levi-Civita ConnectionandDiffeologicalRetraction 4.3 Formulation of Diffeological Optimization Algorithms andTheirApplication 5 Conclusion 6 Notations