Methods for constrained nonlinear systems: inexact Newton-like conditional gradient and Levenberg-Marquardt with inexact projections

• Ano de defesa: 2019
• Autor(a) principal: Oliveira, Fabrícia Rodrigues de
• Orientador(a): Gonçalves, Max Leandro Nobre
• Banca de defesa: Melo, Jefferson Divino Gonçalves de, Ferreira, Orizon Pereira, Gonçalves, Douglas Soares, Haeser, Gabriel, Andreani, Roberto
• 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: Sistemas não lineares restritos; Método do tipo Newton gradiente condicional; Estratégia de busca não monótona e livre de derivadas; Projeções inexatas; Convergência local e global
• Palavras-chave em Inglês: Constrained nonlinear systems; Inexact Newton-like conditional gradient method; Nonmonotone and derivative-free line search; Inexact projections; Local and global convergence
• Área do conhecimento CNPq: CIENCIAS EXATAS E DA TERRA::MATEMATICA
• Link de acesso: http://repositorio.bc.ufg.br/tede/handle/tede/9486

In this work, we propose and analyze some methods to solve constrained nonlinear systems of equations. First, we present a method that combines the inexact Newton-like method with a specialized version of the conditional gradient method (also known as Frank-Wolfe method). Using a majorant condition, which allows us to prove in a unified way convergence results for some classes functions, the local convergence of the proposed method as well as results on its rates are established. Second, we present a global version of the previous method by means of a derivative-free and nonmonotone line search strategy. Under appropriate conditions the global convergence of the proposed method is proved. Third, based on the well-known Levenberg-Marquardt method, we also propose a Levenberg-Marquardt method with inexact projections which combines the unconstrained Levenberg-Marquardt method with a notion of inexact projetions. In this case, the local convergence of the proposed method is proved using the error bound condition that is weaker than the standard condition full-rank of the Jacobian. Moreover, we also present a global version of the latter method by means of a nonmonotone line search technique. Finally, numerical experiments are also reported to illustrate the performances of the proposed schemes.