Methods for vector optimization: trust region and proximal on riemannian manifolds and Newton with variable order

• Ano de defesa: 2017
• Autor(a) principal: Pereira, Yuri Rafael Leite
• Orientador(a): Bento, Glaydston de Carvalho
• Banca de defesa: Bento, Glaydston de Carvalho, Ferreira, Orizon Pereira, Pérez, Luís Román Lucambio, Cruz Neto, João Xavier da, Santos, Paulo Sérgio Marques dos
• 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: Optimização vetorial; Optimização multiobjetivo; Região de confiança; Método do ponto proximal; Variedades riemannianas; Ordem variável
• Palavras-chave em Inglês: Vector optimization; Multiobjective optimization; Trust region; Proximal point method; Newton method; Riemannian manifolds; Variable order
• Área do conhecimento CNPq: CIENCIAS EXATAS E DA TERRA::MATEMATICA
• Link de acesso: http://repositorio.bc.ufg.br/tede/handle/tede/7791

In 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.