Hybrid Conjugate Gradient Methods Based On The DAI-LIAO-Type And WEI-YAO-LIU-Type Methods

Nonlinear conjugate gradient methods are a class of important methods for solving large-scale unconstrained optimization problems. The DAI-LIAO-Type and WEI-YAO-LIU-Type methods are two subclasses of very effective nonlinear conjugate gradient methods. Focusing on the sufficient descent property, global convergence and numerical performance of conjugate gradient methods, we propose several hybrid conjugate gradient methods on the basis of the DAI-LIAO-Type and Wei-Yao-Liu-type methods. These methods posses the sufficient descent property and are globally convergent for general functions, and have nice numerical performance. This thesis is organised as follows.In Chapter 1, the related concepts of conjugate gradient methods are introduced, and the current research status of them are stated.In Chapter 2, two new hybrid conjugate gradient methods, named the DLWYL1 and the DLWYL2 respectively, are proposed on the basis of two existing DAI-LIAO-Type and Wei-Yao-Liu-type hybrid methods. It is proved that the DLWYL1 and DLWYL2 methods with the strong Wolfe line search have the sufficient descent property, and are globally convergent for general nonlinear functions. The numerical results reported in this chapter show that the performance of the DLWYL1 and DLWYL2 methods are comparable with those of some favorable conjugate gradient methods.In Chapter 3, two hybrid conjugate gradient methods with disturbance factors, named the DLWYL-D and the DLMHS-D respectively, are proposed on the basis of some modified conjugate gradient methods with disturbance factors and two existing DAI-LIAO-Type and Wei-Yao-Liu-type hybrid methods. It is proved that the DLWYL-D and DLMHS-D methods with the strong Wolfe line search have the sufficient descent property, and are globally convergent for general nonlinear functions. The numerical results show that the performance of the DLWYL-D and DLMHS-D methods proposed in this chapter are slightly better than those of some good conjugate gradient methods.In Chapter 4, basing on a existing DAI-LIAO-Type and Wei-Yao-Liu-type hybrid method and a modified version of it, and using the secant condition and a modified secant condition, we propose two hybrid conjugate gradient methods whose directions approximate the quasi-Newton direction and a modified quasi-Newton direction respectively. These two methods are called the DLWYL-QN and DLWYL-MQN methods respectively. It is proved that the DLWYL-QN method with the strong Wolfe line search have the sufficient descent property and is globally convergent for uniquely convex functions, and the DLWYL-MQN method with the strong Wolfe line search have the sufficient descent property and is globally convergent for general nonlinear functions. The numerical results reported in this chapter show that the performance of the DLWYL-QN and DLWYL-MQN methods are slightly better than those of some very favorable conjugate gradient methods.