On The Methods For Unconstrained Optimization Problems And Theirs Realizes

Numerical methods for unconstrained optimization is an active subject in numerical anaiysis. It is very important to solve unconstrained optimization rapidly and effectively, which is not only itself of great importance but also it forms subproblems in many constrained optimization problems. Therefore, how to design fast and effective algorithms for unconstrained optimization is an important problem that optimization researchers care very much.In this paper, we proposed two new non-linear conjugate gradient methods for solving unconstrained optimization problems. We established the global convergence theory for the proposed methods and reported estensive numerical results. The major tasks in this paper include:(1) We introduced the development of the unconstrained optimization problems, and the main results obtained in this thesis. Moreover, we introduced the background and the already results of this article would need to study.(2) We presented a hybrid conjugate gradient method for unconstrained optimization based on Hestenes-Stiefel Algorithm and Dai-Yuan Algorithm, which had taken the advantages of the two alogrithms. We proved it can ensure the convergence of the new methods under the Wolfe line search and without the descent condition. Additionally, our preliminary numerical experiments were carried out, which suggested that the algorithm was validity.(3) Some modified Armijo type line search and a modified non-linear conjugate gradient method which is called MPRP method were proposed. In addition, a global convergence theory for the MPRP method with the modified Armijo type line search 3 was proved. An important property of the MPRP method is that at each iteration, the method can generate a sufficient descent direction d kwhich satisfying d kT gk= ?gk2. This property is independent of line search used, which is also one of the main differences between the MPRP method and the known non-linear conjugate gradient methods. Moreover, if exact line search is used, the MPRP method reduces to the standard PRP method. Finally, our experimental results witnessed the applied value of this new method.