Modified Conjugate Gradient Methods Based On The Dai-Liao Method

Conjugate gradient method is a powerful line search method because of its simplicity and its very low memory requirement, especially for large scale optimization problems. In this paper, combining with the characteristics of spectral conjugate gradient method and three-term conjugate gradient method, based on the Dai-Liao method we propose several modified conjugate gradient methods.In chapter 2, combining the spectral conjugate gradient method, conjugacy conditions and the su?cient descent condition, we give a family of spectral conjugate gradient method, called MSCG method, possessing su?cient descent property independent of liner search, and prove the global convergence of the MSCG method for uniformly convex functions under strong Wolfe line search. Similarly to the DL+ method, we propose a modified spectral conjugate gradient method, called MSCG+ method, and give the convergency results of the MSCG+ method for general functions under the Strong Wolfe line search. Numerical results are given in the following part.In chapter 3, combining the three-term conjugate gradient method, conjugacy conditions and the su?cient descent condition, we give a family of three-term conjugate gradient method, called MTCG method, possessing su?cient descent property independent of liner search, and prove the global convergence of the MTCG method for uniformly convex functions under strong Wolfe line search. Similarly to the DL+ method,we propose a modified three-term conjugate gradient method, called MTCG+ method,and give the convergency results of the MTCG+ method for general functions under the Strong Wolfe line search. Numerical results are given in the following part.In chapter 4, combining the formula of the scalar βkin the chapter 2 and chapter3, we propose two kinds of modified Dai-Liao conjugate gradient method, and prove the su?cient descent property and the global convergence under the strong Wolfe conditions correspondingly. Numerical results are given in the following part.The WYL method possesses the su?cient descent property and the global convergence under the strong Wolfe conditions and has nice numerical performance.In chapter 5, in order to take advantage of satisfactory theoretical and numerical characteristics of the PRP and WYL methods, we propose a hybrid PRP and WYL conjugate gradient method, called the PRP-WYL method, and show that the PRP-WYL method possesses the su?cient descent property and the global convergence under the strong Wolfe conditions. Numerical results indicate that the PRP-WYL method does have nice numerical performance. Further more, we generate the PRP-WYL method to the HS-MHS method and the LS-MLS method, the corresponding numerical results are given in the following part.