Generalized vector equilibrium problems and algorithms for variational inequality in hadamard manifolds

• Ano de defesa: 2016
• Autor(a) principal: Batista, Edvaldo Elias de Almeida
• Orientador(a): Ferreira, Orizon Pereira
• Banca de defesa: Ferreira, Orizon Pereira , Bento, Glaydston de Carvalho , Cruz Neto, João Xavier da, Peréz, Luís Román Lucambio, Alves, Maicon Marques
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Universidade Federal de Goiás
• Programa de Pós-Graduação: Programa de Pós-graduação em Matemática (IME)
• Departamento: Instituto de Matemática e Estatística - IME (RG)
• País: Brasil
• Palavras-chave em Português: Alargamento de campos vetoriais; Proximal inexato; Otimização com restrições; Algoritmo extragradiente; Semicontinuidade inferior; Problema de equilíbrio vetorial; Otimização vetorial; Variedade de Hadamard
• Palavras-chave em Inglês: Enlargement of vector fields; Inexact proximal; Constrained optimization; Extragradient algorithm; Lower-semicontinuity; Vector equilibrium problem; Vector optimization; Hadamard manifold
• Área do conhecimento CNPq: CIENCIAS EXATAS E DA TERRA::MATEMATICA
• Link de acesso: http://repositorio.bc.ufg.br/tede/handle/tede/6562

In 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.