We investigate the impact of cash reserves upon the optimal behaviour of a modelled firm that has uncertain future revenues. To achieve this, we build up a corporate financing model of a firm from a Real Options foundation, with the option to close as a core business decision maintained throughout. We model the firm by employing an optimal stochastic control mathematical approach, which is based upon a partial differential equations perspective. In so doing, we are able to assess the incremental impacts upon the optimal operation of the cash constrained firm, by sequentially including: an optimal dividend distribution; optimal equity financing; and optimal debt financing (conducted in a novel equilibrium setting between firm and creditor). We present efficient numerical schemes to solve these models, which are generally built from the Projected Successive Over Relaxation (PSOR) method, and the Semi-Lagrangian approach. Using these numerical tools, and our gained economic insights, we then allow the firm the option to also expand the operation, so they may also take advantage of favourable economic conditions.

Since augmented Lagrangian methods were introduced by Powell and Hestenes, this class of methods has been investigated very intensively. While the finite dimensional case has been treated in a satisfactory manner, the infinite dimensional case is studied much less. The general approach to solve an infinite dimensional optimization problem subject to equality constraints is as follows: First one proves convergence for a basic algorithm in the Hilbert space setting. Then one discretizes the given spaces and operators in order to make numerical computations possible. Finally, one constructs a discretized version of the infinite dimensional method and tries to transfer the convergence results to the finite dimensional version of the basic algorithm. In this thesis we discuss a globally convergent augmented Lagrangian algorithm and discretize it in terms of functional analytic restriction operators. Given this setting, we prove global convergence of the discretized version of this algorithm to a stationary point of the infinite dimensional optimization problem. The proposed algorithm includes an explicit rule of how to update the discretization level and the penalty parameter from one iteration to the next one - questions that had been unanswered so far. In particular the latter update rule guarantees that the penalty parameters stay bounded away from zero which prevents the Hessian of the discretized augmented Lagrangian functional from becoming more and more ill conditioned. / Master of Science

This thesis is concerned with the numerical simulation of high Reynolds number, three-dimensional, incompressible flows in open domains. Many problems treated in Computational Fluid Dynamics (CFD) occur in free space: e.g., external aerodynamics past vehicles, bluff bodies or aircraft; shear flows such as shear layers or jets. In observing all these flows, we can remark that they are often unsteady, appear chaotic with the presence of a large range of eddies, and are mainly dominated by convection. For years, it was shown that Lagrangian Vortex Element Methods (VEM) are particularly well appropriate for simulating such flows. In VEM, two approaches are classically used for solving the Poisson equation. The first one is the Biot-Savart approach where the Poisson equation is solved using the Green's function approach. The unbounded domain is thus implicitly taken into account. In that case, Parallel Fast Multipole (PFM) solvers are usually used. The second approach is the Vortex-In-Cell (VIC) method where the Poisson equation is solved on a grid using fast grid solvers. This requires to impose boundary conditions or to assume periodicity. An important difference is that fast grid solvers are much faster than fast multipole solvers. We here combine these two approaches by taking the advantages of each one and, eventually, we obtain an efficient VIC-PFM method to solve incompressible flows in open domain. The major interest of this combination is its computational efficiency: compared to the PFM solver used alone, the VIC-PFM combination is 15 to 20 times faster. The second major advantage is the possibility to run Large Eddy Simulations (LES) at high Reynolds number. Indeed, as a part of the operations are done in an Eulerian way (i.e. on the VIC grid), all the existing subgrid scale (SGS) models used in classical Eulerian codes, including the recent "multiscale" models, can be easily implemented.

This thesis is concerned with the numerical simulation of high Reynolds number, three-dimensional, incompressible flows in open domains. Many problems treated in Computational Fluid Dynamics (CFD) occur in free space: e.g., external aerodynamics past vehicles, bluff bodies or aircraft; shear flows such as shear layers or jets. In observing all these flows, we can remark that they are often unsteady, appear chaotic with the presence of a large range of eddies, and are mainly dominated by convection. For years, it was shown that Lagrangian Vortex Element Methods (VEM) are particularly well appropriate for simulating such flows. In VEM, two approaches are classically used for solving the Poisson equation. The first one is the Biot-Savart approach where the Poisson equation is solved using the Green's function approach. The unbounded domain is thus implicitly taken into account. In that case, Parallel Fast Multipole (PFM) solvers are usually used. The second approach is the Vortex-In-Cell (VIC) method where the Poisson equation is solved on a grid using fast grid solvers. This requires to impose boundary conditions or to assume periodicity. An important difference is that fast grid solvers are much faster than fast multipole solvers. We here combine these two approaches by taking the advantages of each one and, eventually, we obtain an efficient VIC-PFM method to solve incompressible flows in open domain. The major interest of this combination is its computational efficiency: compared to the PFM solver used alone, the VIC-PFM combination is 15 to 20 times faster. The second major advantage is the possibility to run Large Eddy Simulations (LES) at high Reynolds number. Indeed, as a part of the operations are done in an Eulerian way (i.e. on the VIC grid), all the existing subgrid scale (SGS) models used in classical Eulerian codes, including the recent "multiscale" models, can be easily implemented.

Cette thèse présente l'étude et le développement de méthodes numériques pour la résolution d'équations de transport, en particulier d'une méthode de remapping bidimensionnel dont un avantage important par rapport aux algorithmes existants est la propriété de conservation de la masse. De nombreux cas-tests permettront de comparer ces approches entre elles ainsi qu'à des méthodes de référence. On s'intéressera en particulier aux équations dites de Vlasov-Poisson et du Centre-Guide, qui apparaissent très classiquement dans le cadre de la physique des plasmas. / This PhD thesis presents the study and development of numerical methods for the resolution of transport equations, in particular a bidimensional remapping method whose main advantage over existing algorithms is the property of mass conservation. Numerous test cases are presented in order to compare these approaches with regard to the others and with reference methods. Focus is made on the so-called Vlasov-Poisson and Center-Guide equations, that appear very classically in the domain of plasma physics.

Este trabalho tem o objetivo de utilizar métodos de otimização não linear a fim de desenvolver uma metodologia eficiente para alocação de bancos de capacitores visando a eliminar violações de tensão em redes de distribuição. A aplicação de capacitores em paralelo a sistemas elétricos de potência é comumente empregada com o intuito de se obter melhor controle do fluxo de potência, gerenciamento do perfil de tensão, correção do fator de potência e minimização de perdas. Tendo em vista estes benefícios, a metodologia deste trabalho se dará por meio da resolução de um problema de programação não linear associada com a aproximação linear da relação potência reativa versus tensão para determinar o número, a localização e o dimensionamento dos bancos capacitores ao longo das linhas de distribuição. Desta forma, pretende-se minimizar a injeção de reativos e reduzir as perdas ativas totais de modo que todas as restrições de operação e de carga sejam atendidas. Os resultados são avaliados pelo programa GAMS (General Algebraic Modeling System), pelo MATLAB TM (Matrix Laboratory) e por um programa elaborado em Fortran, sendo possível analisar e descrever as contribuições alcançadas pelo presente trabalho, considerando que este é um tema de grande relevância para a operação e planejamento da expansão dos sistemas elétricos de potência. / This work aims to use nonlinear optimization methods to develop an efficient methodology for capacitor banks allocation to eliminate voltage violations in distribution networks. The application of capacitors in parallel to the electric power systems are commonly employed in order to have better control of power flow, voltage profile management, power factor correction and loss minimization. To achieve these benefits, the methodology of this work will be done through the resolution of a nonlinear programming problem associated with the linear approach of Voltage Variations versus Reactive Power Variation, calculating the number, location and optimal design of capacitor banks along distribution lines. Thus, it looks forward to minimize reactive power injection and reduce losses subject to meeting the operating and the loading constraints. The results are evaluated by the program GAMS TM (General Algebraic Modeling System), by Matlab TM (Matrix Laboratory) and by a program written in FORTRAN TM, being able to analyze and describe the contributions achieved by this work, considering it is a topic of great relevance to the operation and expansion planning of electric power systems.

Esta tese é um estudo acerca do Problema de Equilíbrio de Nash Generalizado (GNEP). Na primeira parte, faremos um resumo dos principais conceitos sobre GNEPs, a relação com outros problemas já conhecidos e comentaremos brevemente os principais métodos já feitos até esta data para resolver numericamente este tipo de problema. Na segunda parte, estudamos condições de otimalidade e condições de qualificação (CQ) para GNEPs, fazendo uma analogia como em otimização. Estendemos os conceitos de cone tangente, normal, gerado pelas restrições ativas, linearizado e polar para a estrutura dos GNEPs. Cada CQ de otimização gera dois tipos de CQ para GNEPs, sendo que a denotada por CQ-GNEP é mais forte e útil para a análise de algoritmos para GNEPs. Mostramos que as condições de qualificação para GNEPs deste tipo em alguns casos não guardam a mesma relação que em otimização. Estendemos também o conceito de Aproximadamente Karush-KuhnTucker (AKKT) de otimização para GNEPs, o AKKT-GNEP. É bem conhecido que AKKT é uma genuína condição de otimalidade em otimização, mas para o caso dos GNEPs mostramos que isto não ocorre em geral. Por outro lado, AKKT-GNEP é satisfeito, por exemplo, em qualquer solução de um GNEP conjuntamente convexo, desde que seja um equilíbrio bvariacional. Com isso em mente, definimos um método do tipo Lagrangiano Aumentado para o GNEP usando penalidades quadráticas e exponenciais e estudamos as propriedades de otimalidade e viabilidade dos pontos limites de sequências geradas pelo algoritmo. Finalmente alguns critérios para resolver os subproblemas e resultados numéricos são apresentados. / This thesis is a study about the generalized Nash equilibrium problem (GNEP). In the first part we will summarize the main concepts about GNEPs, the relationship with other known problems and we will briefly comment on the main methods already done in order to solve these problems numerically. In the second part we study optimality conditions and constraint qualification (CQ) for GNEPs making an analogy with the optimization case. We extend the concepts of the tangent, normal and generated by the active cones, linear and polar cone to the structure of the GNEPs. Each optimization CQ generates two types of CQs for GNEPs, with the one called CQ-GNEP being the strongest and most useful for analyzing the algorithms for GNEPs. We show that the qualification conditions for GNEPs of this type in some cases do not have the same relation as in optimization. We also extend the Approximate Karush- Kuhn-Tucker (AKKT) concept used in optimization for GNEPs to AKKT-GNEP. It is well known that AKKT is a genuine optimality condition in optimization but for GNEPs we show that this does not occur in general. On the other hand, AKKT-GNEP is satisfied, for example, in any solution of a jointly convex GNEP, provided that it is a b-variational equilibrium. With this in mind, we define Augmented Lagrangian methods for the GNEP, using the quadratic and the exponential penalties, and we study the optimality and feasibility properties of the sequence of points generated by the algorithms. Finally some criteria to solve the subproblems and numerical results are presented.

Este trabalho insere-se no contexto de métodos de multiplicadores para a resolução de problemas de programação convexa semidefinida e a análise de suas propriedades através do método proximal aplicado sobre o problema dual. Nosso foco será uma subclasse de problemas de programação convexa semidefinida com restrições afins, para a qual estudaremos relações de dualidade e condições para a existência de soluções dos problemas primal e dual. Em seguida, analisaremos dois métodos de multiplicadores para resolver essa classe de problemas e que são extensões de métodos conhecidos para programação não-linear. O primeiro, proposto por Doljansky e Teboulle, aborda um método de ponto proximal interior entrópico e sua conexão com um método de multiplicadores exponenciais. O segundo, apresentado por Mosheyev e Zibulevsky, estende para a classe de problemas de nosso interesse um método de lagrangianos aumentados suaves proposto por Ben-Tal e Zibulevsky. Por fim, apresentamos os resultados de testes numéricos feitos com o algoritmo proposto por Mosheyev e Zibulevsky, analisando diferentes escolhas de parâmetros, o aproveitamento do padrão de esparsidade das matrizes do problema e critérios para a resolução aproximada dos subproblemas irrestritos que devem ser resolvidos a cada iteração desse algoritmo de lagrangianos aumentados. / This work deals with multiplier methods to solve semidefinite convex programming problems and the analysis of their proprieties based on the proximal point method applied on the dual problem. We focus on a subclass of semidefinite programming problems with affine constraints, for which we study duality relations an conditions for the existence of solutions of the primal and dual problems. Afterwards, we analyze two multiplier methods to solve this class of problems which are extensions of known methods in nonlinear programming. The first one, introduced by Doljansky e Teboulle, approaches an entropic interior proximal algorithm and their relationship with an exponential multiplier method. The second one, presented by Mosheyev e Zibulevsky, extends a smooth augmented Lagrangian method proposed by Ben-Tal and Zibulevsky for the problems of our interest. Finally, we present the results of numerical experiments for the algorithm proposed by Mosheyev e Zibulevsky, analyzing some choices of parameters, the sparsity patterns of matrices of the problem and criteria to accept approximate solutions of the unconstrained subproblems that must be solved at each iteration of the augmented Lagrangian method.

Este trabalho tem o objetivo de utilizar métodos de otimização não linear a fim de desenvolver uma metodologia eficiente para alocação de bancos de capacitores visando a eliminar violações de tensão em redes de distribuição. A aplicação de capacitores em paralelo a sistemas elétricos de potência é comumente empregada com o intuito de se obter melhor controle do fluxo de potência, gerenciamento do perfil de tensão, correção do fator de potência e minimização de perdas. Tendo em vista estes benefícios, a metodologia deste trabalho se dará por meio da resolução de um problema de programação não linear associada com a aproximação linear da relação potência reativa versus tensão para determinar o número, a localização e o dimensionamento dos bancos capacitores ao longo das linhas de distribuição. Desta forma, pretende-se minimizar a injeção de reativos e reduzir as perdas ativas totais de modo que todas as restrições de operação e de carga sejam atendidas. Os resultados são avaliados pelo programa GAMS (General Algebraic Modeling System), pelo MATLAB TM (Matrix Laboratory) e por um programa elaborado em Fortran, sendo possível analisar e descrever as contribuições alcançadas pelo presente trabalho, considerando que este é um tema de grande relevância para a operação e planejamento da expansão dos sistemas elétricos de potência. / This work aims to use nonlinear optimization methods to develop an efficient methodology for capacitor banks allocation to eliminate voltage violations in distribution networks. The application of capacitors in parallel to the electric power systems are commonly employed in order to have better control of power flow, voltage profile management, power factor correction and loss minimization. To achieve these benefits, the methodology of this work will be done through the resolution of a nonlinear programming problem associated with the linear approach of Voltage Variations versus Reactive Power Variation, calculating the number, location and optimal design of capacitor banks along distribution lines. Thus, it looks forward to minimize reactive power injection and reduce losses subject to meeting the operating and the loading constraints. The results are evaluated by the program GAMS TM (General Algebraic Modeling System), by Matlab TM (Matrix Laboratory) and by a program written in FORTRAN TM, being able to analyze and describe the contributions achieved by this work, considering it is a topic of great relevance to the operation and expansion planning of electric power systems.

Esta tese é um estudo acerca do Problema de Equilíbrio de Nash Generalizado (GNEP). Na primeira parte, faremos um resumo dos principais conceitos sobre GNEPs, a relação com outros problemas já conhecidos e comentaremos brevemente os principais métodos já feitos até esta data para resolver numericamente este tipo de problema. Na segunda parte, estudamos condições de otimalidade e condições de qualificação (CQ) para GNEPs, fazendo uma analogia como em otimização. Estendemos os conceitos de cone tangente, normal, gerado pelas restrições ativas, linearizado e polar para a estrutura dos GNEPs. Cada CQ de otimização gera dois tipos de CQ para GNEPs, sendo que a denotada por CQ-GNEP é mais forte e útil para a análise de algoritmos para GNEPs. Mostramos que as condições de qualificação para GNEPs deste tipo em alguns casos não guardam a mesma relação que em otimização. Estendemos também o conceito de Aproximadamente Karush-KuhnTucker (AKKT) de otimização para GNEPs, o AKKT-GNEP. É bem conhecido que AKKT é uma genuína condição de otimalidade em otimização, mas para o caso dos GNEPs mostramos que isto não ocorre em geral. Por outro lado, AKKT-GNEP é satisfeito, por exemplo, em qualquer solução de um GNEP conjuntamente convexo, desde que seja um equilíbrio bvariacional. Com isso em mente, definimos um método do tipo Lagrangiano Aumentado para o GNEP usando penalidades quadráticas e exponenciais e estudamos as propriedades de otimalidade e viabilidade dos pontos limites de sequências geradas pelo algoritmo. Finalmente alguns critérios para resolver os subproblemas e resultados numéricos são apresentados. / This thesis is a study about the generalized Nash equilibrium problem (GNEP). In the first part we will summarize the main concepts about GNEPs, the relationship with other known problems and we will briefly comment on the main methods already done in order to solve these problems numerically. In the second part we study optimality conditions and constraint qualification (CQ) for GNEPs making an analogy with the optimization case. We extend the concepts of the tangent, normal and generated by the active cones, linear and polar cone to the structure of the GNEPs. Each optimization CQ generates two types of CQs for GNEPs, with the one called CQ-GNEP being the strongest and most useful for analyzing the algorithms for GNEPs. We show that the qualification conditions for GNEPs of this type in some cases do not have the same relation as in optimization. We also extend the Approximate Karush- Kuhn-Tucker (AKKT) concept used in optimization for GNEPs to AKKT-GNEP. It is well known that AKKT is a genuine optimality condition in optimization but for GNEPs we show that this does not occur in general. On the other hand, AKKT-GNEP is satisfied, for example, in any solution of a jointly convex GNEP, provided that it is a b-variational equilibrium. With this in mind, we define Augmented Lagrangian methods for the GNEP, using the quadratic and the exponential penalties, and we study the optimality and feasibility properties of the sequence of points generated by the algorithms. Finally some criteria to solve the subproblems and numerical results are presented.