Some New Nonlinear Conjugate Gradient Methods

This paper consists of four chapters. In the first chapter, we simply review origin, development and characteristics of nonlnear conjugate gradient method for solving unconstrained optimization, introduce some important formulas of the method. Nonlinear conjugate gradient method is a very important method because it has advantages that its algorithm is very simple and it does not require the storage of any matrices. It is very suited to solving some large scale problems.In chapter 2, we propose a new formula of main parameter βk of conjugate gradient method that is related to decrease of objective function, test its numerical effects and analyze its convergent properties. By restricting the new formula βk to nonnegative domains, we construct two hybrid formulas and discuss their convergent properties under some weak conditions. Their numerical results show that they are efficient and robust. Their effects are in some extent better than PRPs one.In chapter 3, by minimizing a quadratic model restricting next search direction to a subset of Span{-g_k,d_k-1}, we propose other formula of parameter β^*k Convergences of the new method and its hybrid formula are proved under monotone line search. Their numerical results show that they are efficient. Based on the papers [10] and [19], we propose a new nonmonotone conjugate gradient algorithm, prove its convergence and test the new nonmonotone algorihm.In the last chapter, we make further discussion on conjugate gradient method. Firstly, we introduce two generalized conjugate gradient methods. Some of the existing problems on conjugate gradient are presented in the last section.