Research On Conjugate Gradient Methods For Unconstrained Optimization

Many problems in practice can be characterized by a constrained optimization problem,which is turned into a unconstrained optimization problem by using Lagrangian or penalty function method. It can be solved by the steepest descent method, Newton’s method, conjugate gradient methods, etc. In all of these methods, the steepest descent method is simple in computation and requires small storage space, but slow in convergence. The Newton’s method has fast convergence rate, but cost more storge space for large-scale problems. Conjugate gradient methods overcome the shortcomings of the above both, which only use the objective function value and gradient function value and converge to the ideal points quickly. Thus, conjugate gradient methods play an important role in solving large-scale unconstrained optimization problems. Further study of the conjugate gradient methods has become a hot issue. In this paper, two new conjugate gradient methods are presented through improving conjugate gradient method proposed recently. The main work is summarized as follows.1. A new conjugate gradient method(MRMIL) is presented through improving RMIL method proposed by Rivaie. Then, it is true that the search direction has sufficient descent property. Under Wolfe line search, the MRMIL has global convergence. Finally,through comparing three kinds of conjugate gradient methods for convex functions and non-convex functions, the results show that MRMIL is feasible and effective, and is more suitable for solving convex functions.2. Firstly, the parameter ? is expand in MRMIL. Secondly, a new spectral conjugate gradient method(NRMIL) is presented through adding a spectral coefficient. Thirdly,the descent property of search direction produced by NRMIL method is proved, and its global convergence is testified under Armijo and Wolfe line search. Finally, the numerical experiments show that NRMIL method is effective.