An Optimal Extension of the Polak-Ribiere-Polyak Conjugate Gradient Method

Based on a singular value analysis on an extension of the Polak-Ribiere-Polyak method, a nonlinear conjugate gradient method with the following two optimal features is proposed: the condition number of its search direction matrix is minimum and also, the distance of its search direction from the search direction of a descent nonlinear conjugate gradient method proposed by Zhang et al. is minimum. Under proper conditions, global convergence of the method can be achieved. To enhance eciency of the proposed method, Powell's truncation of the conjugate gradient parameters is used. The method is computationally compared with the nonlinear conjugate gradient method proposed by Zhang et al. and a modified Polak-Ribiere-Polyak method proposed by Yuan. Results of numerical comparisons show eciency of the proposed method in the sense of the Dolan-More performance profile.