The Convergence Of A Modified PRP Conjugate Gradient Method

Conjugate gradient methods are important iterative methods for solving optimization problems. They are particularly welcome in the solution of large-scale optimization problems. A good property of the conjugate gradient method is its lower storage and good convergence property. However, most existing conjugate gradient methods do not guarantee to get descent directions for the objective function. In this paper, we study a conjugate gradient method for solving unconstrained optimization problem. We call the method modified PRP method—MPRP method. An attractive property of the MPRP method is that at each iteration, it generates a sufficient descent direction for the objective function. This property does not depend upon the line search used. Moreover, if exact line search is used, the method reduces to the ordinary PRP method.Firstly, we investigate the global convergence property of the MPRP method with the strong Wolfe line search. Under weak conditions, we establish the global convergence of the MPRP method. We also present some numerical experiences to test the performance of the MPRP method and compare it with the ordinary PRP method. The results show that the MPRP method does work well.Secondly, we consider the use of the nonmonotone line search technique in the MPRP method. We also obtain the global convergence of the MPRP method with the nonmonotone line search under appropriate conditions. We also test the method with different parameters via numerical experiments. Taking into account that monotone line search has some advantage for some nonlinear problems, we introduce a hybrid line search technique in the MPRP method. Specifically, at each iteration, we first use a monotone line search for some steps and then turn to the nonmonotone search. Our numerical experiments show that the performance of this hybrid line search is very good.