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.

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.

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

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