Improvement And Generalization Of WYL And HZ Conjugate Gradient Algorithms

In this paper,based on conjugate gradient parameters of Du et al.and conjugate gradient parameter of HZ,some improved and generalized conjugate gradient algorithms are proposed,which have better theoretical results and numerical results respectively.In the first chapter,the basic knowledge relative to nonlinear conjugate gradient methods?several classical conjugate gradient methods and the results of their global convergence are introduced.In the second chapter,based on conjugate gradient parameters proposed by Du et al.,we propose three improved nonlinear conjugate gradient methods,namely MNVPRP*,MNVHS*and MNVLS*,the theoretical results of the modified methods are better.The descent and global convergence properties of MNVPRP*,MNVHS*and MNVLS*methods are proved under the Wolfe line search conditions.Numerical results show that MNVHS*method is better than NVHS*method and MNVLS*method is better than NVLS*method.In the third chapter,based on the HZ conjugate gradient method in 2006,a modified HZ conjugate gradient method(called MHZ method)is proposed.Further,the generalized HZ method(called GHZ method)is proposed.It can be proved that the GHZ method is globally convergent on the Wolfe line search or the Goldstein line search for the uniformly convex function.Furthermore,based on the idea of HZ+ method,the truncation of GHZ method(called GHZ+ method)is proposed,we can get that GHZ+ method is globally convergent for general functions under Wolfe line search conditions.In 2011,Dai YuHong proposed the GSD conjugate gradient methods.Based on this method,the generalized GHZ method(called GGHZ method)was further proposed.Under the appropriate assumption,it is proved that the GGHZ method is globally convergent to the uniformly convex function under the Wolfe line search or the Goldstein line search conditions.Taking the truncation of the GGHZ method(called GGHZ+ method),we can get the global convergence of the GGHZ+ method to the general function under the Wolfe line search conditions.Finally,two groups of special GGHZ+ methods are taken,and the numerical results show that the methods are effective.