Thesis (Ph. D.)--University of Missouri-Columbia, 1998. / Typescript. Vita. Includes bibliographical references (leaves 42-44). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1998. / Typescript. Vita. Includes bibliographical references (leaves 42-44). Also available on the Internet.

Nesta dissertação, definimos a geometria hiperbólica usando o disco de Poincaré (D2) e o semiplano superior (H2) com as respectivas propriedades. Além disso, apresentamos algumas funções e relações importantes da geometria hiperbólica; conceituamos as superfícies de Riemann, analisando suas propriedades e representações; estudamos o espaço de Teichmüller com a devida decomposição em calças. Esses temas são ferramentas necessárias para atingir o objetivo da dissertação: definir as coordenadas de Fenchel Nielsen como um sistema de coordenadas locais do espaço de Teichmüller Tg. / In this dissertation, we defined the hyperbolic geometry using the Poincares disk (D2) and upper half-plane (H2) with its properties. Besides, we presented some functions and important relations of the hyperbolic geometry; we conceptualize the Riemann surfaces, analyzing its properties and representations; we studied the Teichmüller Space with proper decomposition pants. These themes are essential tools to reach the goal of the work: The definition of the Fenchel Nielsen coordenates as local coordinate system of the Teichmüller space Tg.

The focus of this dissertation is solution strategies for stochastic mixed-integer programs with special structures. Motivation for the methods comes from the relatively sparse number of algorithms for solving stochastic mixed-integer programs. Two stage models with finite support are assumed throughout. The first contribution introduces the nodal decision framework under private information restrictions. Each node in the framework has control of an optimization model which may include stochastic parameters, and the nodes must coordinate toward a single objective in which a single optimal or close-to-optimal solution is desired. However, because of competitive issues, confidentiality requirements, incompatible database issues, or other complicating factors, no global view of the system is possible. An iterative methodology called the nodal decomposition-coordination algorithm (NDC) is formally developed in which each entity in the cooperation forms its own nodal deterministic or stochastic program. Lagrangian relaxation and subgradient optimization techniques are used to facilitate negotiation between the nodal decisions in the system without any one entity gaining access to the private information from other nodes. A computational study on NDC using supply chain inventory coordination problem instances demonstrates that the new methodology can obtain good solution values without violating private information restrictions. The results also show that the stochastic solutions outperform the corresponding expected value solutions. The next contribution presents a new algorithm called scenario Fenchel decomposition (SFD) for solving two-stage stochastic mixed 0-1 integer programs with special structure based on scenario decomposition of the problem and Fenchel cutting planes. The algorithm combines progressive hedging to restore nonanticipativity of the first-stage solution, and generates Fenchel cutting planes for the LP relaxations of the subproblems to recover integer solutions. A computational study SFD using instances with multiple knapsack constraint structure is given. Multiple knapsack constrained problems are chosen due to the advantages they provide when generating Fenchel cutting planes. The computational results are promising, and show that SFD is able to find optimal solutions for some problem instances in a short amount of time, and that overall, SFD outperforms the brute force method of solving the DEP.

Nesta dissertação, definimos a geometria hiperbólica usando o disco de Poincaré (D2) e o semiplano superior (H2) com as respectivas propriedades. Além disso, apresentamos algumas funções e relações importantes da geometria hiperbólica; conceituamos as superfícies de Riemann, analisando suas propriedades e representações; estudamos o espaço de Teichmüller com a devida decomposição em calças. Esses temas são ferramentas necessárias para atingir o objetivo da dissertação: definir as coordenadas de Fenchel Nielsen como um sistema de coordenadas locais do espaço de Teichmüller Tg. / In this dissertation, we defined the hyperbolic geometry using the Poincares disk (D2) and upper half-plane (H2) with its properties. Besides, we presented some functions and important relations of the hyperbolic geometry; we conceptualize the Riemann surfaces, analyzing its properties and representations; we studied the Teichmüller Space with proper decomposition pants. These themes are essential tools to reach the goal of the work: The definition of the Fenchel Nielsen coordenates as local coordinate system of the Teichmüller space Tg.

Nesta disertação estudamos algumas ferramentas básicas relacionadas aos espaços de Teichmüller. Introduzimos o espaço de Teichmüller de gênero g ≥ 1, denotado por Tg. O objetivo principal é construir as coordenadas de Fenchel-Nielsen ωG : Tg → R3g-3+ × R3g-3 para cada grafo trivalente marcado G. / In this dissertation we study some basic tools related to Teichmüller space. We introduce the Teichmüller space of genus g ≥ 1, denoted by Tg. The main goal is to construct the Fenchel-Nielsen coordinates ωG : Tg → R3g-3+ × R3g-3 to each marked cubic graph G.

With this thesis we bring some new results and improve some existing ones in conjugate duality and some of the areas it is applied in. First we recall the way Lagrange, Fenchel and Fenchel - Lagrange dual problems to a given primal optimization problem can be obtained via perturbations and we present some connections between them. For the Fenchel - Lagrange dual problem we prove strong duality under more general conditions than known so far, while for the Fenchel duality we show that the convexity assumptions on the functions involved can be weakened without altering the conclusion. In order to prove the latter we prove also that some formulae concerning conjugate functions given so far only for convex functions hold also for almost convex, respectively nearly convex functions. After proving that the generalized geometric dual problem can be obtained via perturbations, we show that the geometric duality is a special case of the Fenchel - Lagrange duality and the strong duality can be obtained under weaker conditions than stated in the existing literature. For various problems treated in the literature via geometric duality we show that Fenchel - Lagrange duality is easier to apply, bringing moreover strong duality and optimality conditions under weaker assumptions. The results presented so far are applied also in convex composite optimization and entropy optimization. For the composed convex cone - constrained optimization problem we give strong duality and the related optimality conditions, then we apply these when showing that the formula of the conjugate of the precomposition with a proper convex K - increasing function of a K - convex function on some n - dimensional non - empty convex set X, where K is a k - dimensional non - empty closed convex cone, holds under weaker conditions than known so far. Another field were we apply these results is vector optimization, where we provide a general duality framework based on a more general scalarization that includes as special cases and improves some previous results in the literature. Concerning entropy optimization, we treat first via duality a problem having an entropy - like objective function, from which arise as special cases some problems found in the literature on entropy optimization. Finally, an application of entropy optimization into text classification is presented.

Nesta disertação estudamos algumas ferramentas básicas relacionadas aos espaços de Teichmüller. Introduzimos o espaço de Teichmüller de gênero g ≥ 1, denotado por Tg. O objetivo principal é construir as coordenadas de Fenchel-Nielsen ωG : Tg → R3g-3+ × R3g-3 para cada grafo trivalente marcado G. / In this dissertation we study some basic tools related to Teichmüller space. We introduce the Teichmüller space of genus g ≥ 1, denoted by Tg. The main goal is to construct the Fenchel-Nielsen coordinates ωG : Tg → R3g-3+ × R3g-3 to each marked cubic graph G.

Depuis son introduction, la programmation mathématique à deux niveaux suscite un intérêt toujours croissant. En effet, vu ses applications dans une multitude de problèmes concrets (problèmes de gestion, planification économique, chimie, sciences environnementales,...), beaucoup de recherches ont été effectuées afin de contribuer à la résolution de cette classe de problèmes. Cette thèse est consacrée à l'étude de quelques classes de problèmes d'optimisation à deux niveaux, à savoir, les problèmes à deux niveaux forts, les problèmes à deux niveaux forts-faibles et les problèmes à deux niveaux semi-vectoriels. Le premier chapitre est consacré aux rappels de quelques définitions et résultats de topologie et d'analyse convexe que nous avons utilisé dans la suite. Dans le deuxième chapitre, nous avons rappelé quelques résultats théoriques et algorithmiques établis dans la littérature pour la résolution de quelques classes de problèmes d'optimisation à deux niveaux. Le troisième chapitre est consacré à l'étude d'un problème à deux niveaux fort-faible (SWBL). Vu la difficulté que présente cette classe de problèmes dans l'étude de l'existence de solutions, et afin de donner de nouvelles perspectives à leur résolution, nous avons procédé à une régularisation du problème. Sous des conditions suffisantes et via cette régularisation, nous avons montré que le problème (SWBL) admet au moins une solution. Dans le quatrième chapitre, nous avons donné une approche de dualité à un problème d'optimisation à deux niveaux fort (S). Cette approche est basée sur l'utilisation d'une régularisation et la dualité de Fenchel-Lagrange. En utilisant cette approche, nous avons donné des conditions nécessaires d'optimalité pour le problème (S). Enfin, des conditions suffisantes d'optimalité sont obtenues pour (S) sans utiliser l'approche. Une application concrète est donnée sur l'allocation de ressources. Dans le cinquième chapitre, nous avons étudié un problème à deux niveaux semi-vectoriel (SVBL). Pour ce problème, nous avons donné une approche de dualité en utilisant une régularisation, une scalarisation et la dualité de Fenchel-Lagrange. Puis, via cette approche et sous des hypothèses appropriées, nous avons donné des conditions nécessaires d'optimalité pour une classe de solutions du problème (SVBL). Finalement, des conditions suffisantes d'optimalité sont établies sont établies sans utiliser l'approche de dualité. / Since its introduction, the class of tao-level programming problems has attracted increasing interest. Indeed, because of its applications in a multitude of concrete problems (management problems, economic planning, chemistry, environmental sciences,...), several researchers have been interested in the study of such class of problems. This thesis deals with the study of some classes of two-level optimization problems, namely, strong two-level problems, strong-weak two-level problems and semi-vectorial two-level problems. In the first chapter, we have recalled some definitions and results related to topology and convex analysis that we have used in our study. In the second chapter, we have discussed some theoretical and algorithmic results established in the literature for solving some classes of two-level optimization problems. The third chapter deals with strong-weak Stackelberg problems. As it is well-known, such a class of problems presents difficulties in its study concerning the existence of solutions. So that, for a strong-weak two-level optimization problem, we have first given a regularization. Then, via this regularization and under appropriate assumptions we have shown the existence of solutions to such a problem. This result generalizes the one given in the literature for weak Stackelberg problems. In the fourth chapter, we have given a duality approach for a strong two-level programming problem (S). The duality approach is based on the use of a regularization and the Fenchel-Lagrange duality. Then, via this approach, we have given necessary optimality conditions for (S). Finally, sufficient optimality conditions are given for the initial problem (S). An application to a two-level resource allocation problem is given. In the fifth chapter, we have considered a semivectorial two-level programming problem (SVBL) where the upper and lower levels are vectorial and scalar respectively. For such a problem, we have given a duality approach based on the use of a regularization, a scalarization and the Fenchel-Lagrange duality. Then, via this approach we have established necessary optimality conditions for (SVBL). Finally, we have given sufficient optimality conditions without using the duality approach.

The standard approach to pricing financial derivatives is to determine the discounted, risk-neutral expected payoff under a model. This model-based approach leaves us prone to model risk, as no model can fully capture the complex behaviour of asset prices in the real world. Alternatively, we could use the prices of some liquidly traded options to deduce no-arbitrage conditions on the contingent claim in question. Since the reference prices are taken from the market, we are not required to postulate a model and thus the conditions found have to hold under any model. In this thesis we are interested in the pricing of American put options using the latter approach. To this end, we will assume that European options on the same underlying and with the same maturity are liquidly traded in the market. We can then use the market information incorporated into these prices to derive a set of no-arbitrage conditions that are valid under any model. Furthermore, we will show that in a market trading only finitely many American and co-terminal European options it is always possible to decide whether the prices are consistent with a model or there has to exist arbitrage in the market.