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Item Um algoritmo proximal com quase-distância(Universidade Federal de Goiás, 2015-02-25) Assunção Filho, Pedro Bonfim de; Bento, Glaydston de Carvalho; http://lattes.cnpq.br/1089906772427394; Bento, Glaydston de Carvalho Bento; Cruz Neto, João Xavier da; Ferreira, Orizon PereiraIn this work, based in [1, 18], we study the convergence of method of proximal point (MPP) regularized by a quasi-distance, applied to an optimization problem. The objective function considered not is necessarily convex and satisfies the property of Kurdyka- Lojasiewicz around by their generalized critical points. More specifically, we will show that any limited sequence, generated from MPP, converge the a generalized critical point.Item Proximal point methods for multiobjective optimization in riemannian manifolds(Universidade Federal de Goiás, 2019-02-26) Meireles, Lucas Vidal de; Bento, Glaydston de Carvalho; http://lattes.cnpq.br/1089906772427394; Bento, Glaydston de Carvalho; Oliveira, Paulo Roberto; Santos, Paulo Sérgio Marques dos; Cruz Neto, João Xavier da; Ferreira, Orizon PereiraIn this work, two different proximal-type methods are investigated in the Riemannian context, namely, an exact and an inexact version. Two strategies were used to analyze these methods. For the exact version, we used a direct approach by investigating the regularized problem, not considering any convexity assumption over the constraint sets, that determine the vectorial improvement steps, which replaces the classical approach via scalarization. To study the inexact version, a definition of the approximate Pareto efficient solution is introduced. For the convex case on Hadamard manifolds, full convergence of both methods to a weak Pareto optimal point is obtained.Item Methods for vector optimization: trust region and proximal on riemannian manifolds and Newton with variable order(Universidade Federal de Goiás, 2017-08-28) Pereira, Yuri Rafael Leite; Pereira, Orizon Ferreira; http://lattes.cnpq.br/0201145506453251; Bento, Glaydston de Carvalho; http://lattes.cnpq.br/1089906772427394; Bento, Glaydston de Carvalho; Ferreira, Orizon Pereira; Pérez, Luís Román Lucambio; Cruz Neto, João Xavier da; Santos, Paulo Sérgio Marques dosIn this work, we will analyze three types of method to solve vector optimization problems in different types of context. First, we will present the trust region method for multiobjective optimization in the Riemannian context, which retrieves the classical trust region method for minimizing scalar functions. Under mild assumptions, we will show that each accumulation point of the generated sequences by the method, if any, is Pareto critical. Next, the proximal point method for vector optimization and its inexact version will be extended from Euclidean space to the Riemannian context. Under suitable assumptions on the objective function, the well-definedness of the methods will be established. Besides, the convergence of any generated sequence, to a weak efficient point, will be obtained. The last method to be investigated is the Newton method to solve vector optimization problem with respect to variable ordering structure. Variable ordering structures are set-valued map with cone values that to each element associates an ordering. In this analyze we will prove the convergence of the sequence generated by the algorithm of Newton method and, moreover, we also will obtain the rate of convergence under variable ordering structures satisfying mild hypothesis.Item Sobre a convergência de métodos de descida em otimização não-suave: aplicações à ciência comportamental(Universidade Federal de Goiás, 2017-02-03) Sousa Júnior, Valdinês Leite de; Ferreira, Orizon Pereira; http://lattes.cnpq.br/0201145506453251; Bento, Glaydston de Carvalho; http://lattes.cnpq.br/1089906772427394; Bento, Glaydston de Carvalho; Ferreira, Orizon Pereira; Melo, Jefferson Divino Gonçalves de; Cruz Neto, João Xavier da; Santos, Sandra AugustaIn this work, we investigate four different types of descent methods: a dual descent method in the scalar context and a multiobjective proximal point methods (one exact and two inexact versions). The first one is restricted to functions that satisfy the Kurdyka-Lojasiewicz property, where it is used a quasi-distance as a regularization function. In the next three methods, the objective is to study the convergence of a multiobjective proximal methods (exact an inexact) for a particular class of multiobjective functions that are not necessarily differentiable. For the inexact methods, we choose a proximal distance as the regularization term. Such a well-known distance allows us to analyze the convergence of the method under various settings. Applications in behavioral sciences are analyzed in the sense of the variational rationality approach.