A Couple Of Conjugate Gradient Methods For Nonlinear Equations

The main purpose of the thesis is to propose the conjugate gradient methods to solve large-scale nonlinear symmetric equations and nonlinear monotone equations. We prove that both proposed algorithm converge globally. Moreover, we show that both proposed methods are well-defined and efficient by a series of numerical experiments.In Chapter one, we introduce the primary results of unconstrained optimization, which mainly includes the definitions of optimization solution, line search, and the opti-mal condition. We also give the iterative formulation of some classic conjugate gradient method, and list the recent advances of descent conjugate gradient methods. Moreover, we give the Newton-type methods and conjugate gradient methods for nonlinear equa-tions. Finally, we list the main contribution of the thesis and some symbols which used at the context.In Chapter two, based on the sufficient descent conjugate gradient method, we pro-pose the norm descent conjugate gradient method for solving nonlinear symmetric equa-tions. The iterative form of the proposed method is very simple and requires little memory. At each iteration, the proposed method does not need to compute the Jacobian matric. With some mild conditions, we establish the convergence result of the proposed method. Finally, we do numerical experiments using the unconstrained optimization problems from CUTEr library. The numerical results show that the proposed method is well-defined and very efficient.In Chapter three, using the idea of projected algorithm by Solodov and Svaiter, we proposed a conjugate gradient method for solving nonlinear monotone equations. At each iteration, the search direction is determined by the use of the conjugate gradient method, and then the next iteration is compute using the line search technique. With some mild conditions, we show that the generated iterations converge to the solution of the equations.In Chapter four, we conclude the thesis and list some research topics.