Métodos Do Tipo Newton Aplicados A Métodos De Restauração Inexata

• Autor(a) principal: Herrera, Francis Lorena Larreal [UNIFESP]
• Data de Publicação: 2017
• Tipo de documento: Dissertação
• Idioma: por
• Título da fonte: Repositório Institucional da UNIFESP
• Texto Completo: https://sucupira.capes.gov.br/sucupira/public/consultas/coleta/trabalhoConclusao/viewTrabalhoConclusao.jsf?popup=true&id_trabalho=5004460; https://repositorio.unifesp.br/handle/11600/50651
• topic: Newton's Method; Brent's Method; Non-Linear Optimization; Kkt Conditions; Inexact Restoration Methods; Método De Newton; Método De Brent; Otimização Não Linear; Condições Kkt; Restauração Inexata

In this dissertation, we study Brent’s method to solve systems of equations and their relation with Inexata Restoration methods. Brent’s method solves a non-linear system by dividing it into blocks and considering linearizations of these blocks in each iteration. We reconstruct a proof of a theorem in which are established the conditions so that the point sequence generated by Brent’s method has local quadratic convergence to the system solution. Inexact Restoration methods are developed to solved constrained optimization problems and the have the characteristic of dividing each iteration into two phases. In the first one, they seek to improve viability and, in the second, optimality. So, it is natural to think that Inexact Restoration methods look to solve the KKT system by dividing it into two blocks. For this reason, it seems evident the existence of a relation between Brent’s and Inexact Restoration methods. Considering this, we present a quadratic local convergence result for the point sequences generated by the Inexact Restoration methods, derived from adaptations in the convergence demonstration of Brent’s method. After that, we propose two iterative computational methods for optimization, introducing small modifications in the Inexact Restoration method. We show that these two methods also have quadratic convergence and we discuss possible advantages and disadvantages of each one of them. Finally we briefly comment some ideas about how these methods could be inserted into a scheme with global convergence.