Research On The Conjugate Gradient Method

Conjugate gradient method is an important method for solving large-scale nonlinear opti-mization. It only uses the objective function values and gradient values, but it overcomes the slow convergence disadvantage of the steepest descent method, also it has advantages in no matrix storage and quadratic termination property. Based on the above advantages, the conju-gate gradient method is considered indispensable in solving large-scale optimization problems of many areas, such as large industrial applications, aerospace, chemical engineering and so on. Thus it becomes a hot research direction. So, in this article, we do some research on the conjugate gradient method as follow:(1) In the first chapter, we introduce briefly the basic knowledge of the conjugate gradient method and a few classical methods, and also introduce some new research directions of the conjugate gradient method and the main results obtained in this article.(2) In the second chapter, a new spectral conjugate gradient method is proposed base on the DY conjugate gradient method. First, the method is proved to be sufficiently decent without any line search rules, and we propose a new algorithm. Then, we prove the global convergence of the new algorithm with Wolfe line search, for solving the optimization problem of uniformly convex function.(3) In the third chapter, a new nonmonotone spectral conjugate gradient method is pro-posed base on the technique of nonmonotone line search proposed by Grippo. First, the method is proved to be sufficiently decent without any line search rules, and we propose a new algorithm. Then, we prove the global convergence of the new algorithm with nonmonotone Aromijo-type line search, for solving the optimization problem.