Generalized vector equilibrium problems and algorithms for variational inequality in hadamard manifolds
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Data
2016-10-20
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Universidade Federal de Goiás
Resumo
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.
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Enlargement of vector fields , Inexact proximal , Constrained optimization , Extragradient algorithm , Lower-semicontinuity , Vector equilibrium problem , Vector optimization , Hadamard manifold , Alargamento de campos vetoriais , Proximal inexato , Otimização com restrições , Algoritmo extragradiente , Semicontinuidade inferior , Problema de equilíbrio vetorial , Otimização vetorial , Variedade de Hadamard
Citação
BATISTA, E. E. A. Generalized vector equilibrium problems and algorithms for variational inequality in hadamard manifolds. 2016. 49 f. Tese (Doutorado em Matemática) - Universidade Federal de Goiás, Goiânia, 2016.