Converg??ncia completa do m??todo do gradiente com busca linear exata e inexata

• Author(s): Sousa, Jeanne Moreira de
• Date: 2015
• Origin: Oasisbr
• Subject(s): Converg??ncia completa; M??todo do gradiente; Busca linear exata e inexata; CI??NCIAS EXATAS E DA TERRA: MATEM??TICA
• Document Type: Master thesis
• Language: Portuguese

Description

Made available in DSpace on 2015-04-22T22:16:10Z (GMT). No. of bitstreams: 1 Dissertacao Jeanne Moreira.pdf: 447774 bytes, checksum: 635ca33ffaf3746929571ab0aabcfd32 (MD5) Previous issue date: 2008-12-29

FAPEAM - Funda????o de Amparo ?? Pesquisa do Estado do Amazonas

In this work we use the gradient method to minimize, without restrictions, convex and pseudoconvex continuously differentiable functions. An important theme considered is the path length determination. We have that, when minimizing pseudoconvex functions, the linear search is exact. In this case, we present the first algorithm to obtain the path length, where will be included a quadratic regularization term, in the proximal point method sense. When dealing with the minimization of convex functions case, we have that the linear search is not exact. To obtain the path length, two algorithms are presented: the former needs that the gradient of the objective function satisfies a Lipschitz condition with a known constant L > 0. The latter is based on the work of Dennis-Schnabel (see [4]). The three process are based on the quasi-Fej??r convergence principle. Although these descent methods need that the objective functions to be minimized have bounded level sets, in order to establish that the limit points are stationary, this approach guarantees the complete convergence of every sequence to a minimizer of the function without the hypothesis of bounded level sets.

Neste trabalho utilizamos o m??todo do gradiente para minimizar, sem restri????es, fun????es continuamente diferenci??veis pseudo-convexas e convexas. Um termo considerado importante ?? o c??lculo do comprimento do passo. Na minimiza????o de fun????es pseudo-convexas a busca linear ?? exata. Neste caso, apresentamos o primeiro algoritmo para o c??lculo do comprimento do passo, onde ?? acrescentado um termo de regulariza????o quadr??tico no sentido do m??todo do ponto proximal. Posteriormente, na minimiza????o de fun????es convexas, a busca linear ?? inexata. Para o c??lculo do comprimento do passo apresentamos dois algoritmos: um necessita que o gradiente da fun????o objetivo satisfa??a uma condi????o de Lipschitz com constante L > 0 conhecida, e o outro ?? baseado no trabalho desenvolvido por Dennis-Schnabel (ver [4]). Os tr??s processos baseiam-se na no????o da quase-Fej??r converg??ncia. Embora os m??todos de descida necessitem que a fun????o objetivo a ser minimizada possua conjuntos de n??veis limitados a fim de estabelecer que os pontos de acumula????o sejam estacion??rios, nesta abordagem ?? garantida a converg??ncia completa de toda sequ??ncia para um minimizador da fun????o sem a hip??tese de limita????o do conjunto de n??vel.