Doutorado em Matemática (IME)
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Item Generalized vector equilibrium problems and algorithms for variational inequality in hadamard manifolds(Universidade Federal de Goiás, 2016-10-20) Batista, Edvaldo Elias de Almeida; Bento, Glaydston de Carvalho; http://lattes.cnpq.br/1089906772427394; Ferreira, Orizon Pereira; http://lattes.cnpq.br/0201145506453251; Ferreira, Orizon Pereira; http://lattes.cnpq.br/0201145506453251; Bento, Glaydston de Carvalho; http://lattes.cnpq.br/1089906772427394; Cruz Neto, João Xavier da; Peréz, Luís Román Lucambio; Alves, Maicon MarquesIn this thesis, we study variational inequalities and generalized vector equilibrium problems. In Chapter 1, several results and basic definitions of Riemannian geometry are listed; we present the concept of the monotone vector field in Hadamard manifolds and many of their properties, besides, we introduce the concept of enlargement of a monotone vector field, and we display its properties in a Riemannian context. In Chapter 2, an inexact proximal point method for variational inequalities in Hadamard manifolds is introduced, and its convergence properties are studied; see [7]. To present our method, we generalize the concept of enlargement of monotone operators, from a linear setting to the Riemannian context. As an application, an inexact proximal point method for constrained optimization problems is obtained. In Chapter 3, we present an extragradient algorithm for variational inequality associated with the point-to-set vector field in Hadamard manifolds and study its convergence properties; see [8]. In order to present our method, the concept of enlargement of maximal monotone vector fields is used and its lower-semicontinuity is established to obtain the convergence of the method in this new context. In Chapter 4, we present a sufficient condition for the existence of a solution to the generalized vector equilibrium problem on Hadamard manifolds using a version of the KnasterKuratowski-Mazurkiewicz Lemma; see [6]. In particular, the existence of solutions to optimization, vector optimization, Nash equilibria, complementarity, and variational inequality is a special case of the existence result for the generalized vector equilibrium problem.