Convergence Of Nonlinear Conjugate Gradient Methods

Nonlinear conjugate gradient methods are fundamental iterative schemes for some large-scale nonlinear optimization problems, due to simple computation and less memory storage requirement. With rapid development of computer and occurrence of many large-scale nonlinear problems in practice, these methods become more and more important. This thesis mainly studies global convergence of the nonlinear conjugate gradient methods, and is divided into four chapters.The first chapter briefly introduces nonlinear conjugate gradient methods and their values, and their state-of-the-art developments, and key points of the researches of this thesis.The second chapter proposes a Grippo-Lucidi-type step length rule of the PRP conjugate gradient method for general function. It only uses current information of gradient and conjugate direction to determine step length, and can ensure global convergence to zero of the resulting sequence of gradient norm. Preliminary numerical tests show that it's effective and superior in practice.The third chapter proposes a class of new hybrid conjugate formulae, and the resulting conjugate gradient methods can guarantee that each search direction is a descent direction without any line search techniques. Furthermore, with Wolfe-Powell line search, they are proved to be globally convergent. Preliminary numerical results show that these hybrid conjugate gradient methods are very efficient.The fourth chapter proposes a new nonmonotone Armijo-type line search rule, and studies a class of conjugate gradient methods with property (*) under the line search rule, which include the well-known PRP method as a special case. The corresponding global convergence is proved.