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In Rivaie et al [M. Rivaie, M. Mustafa, L.W. June and I. Mohd, A new class of nonlinear conjugate gradient coefficient with global convergence properties, Appl. Math. Comp, 218(2012), 11323-11332], an efficient CG algorithm has been proposed for solving unconstrained optimization problems. However, due to a wrong inequality (3.3) used in Rivaie et al., the proof of theorem 2 and the global convergence theorem 3 are not correct. We present the necessary corrections, then the proposed method in Rivaie et al still converges globally. Finally, we report some numerical comparisons.

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The goal of this paper is to extend the modified Hestenes-Stiefel method to solve large-scale nonlinear monotone equations. The method is presented by combining the hyperplane projection method (Solodov, M.V.; Svaiter, B.F. A globally convergent inexact Newton method for systems of monotone equations, in: M. Fukushima, L. Qi (Eds.)Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publishers. 1998, 355-369) and the modified Hestenes-Stiefel method in Dai and Wen (Dai, Z.; Wen, F. Global convergence of a modified Hestenes-Stiefel nonlinear conjugate gradient method with Armijo line search. Numer Algor. 2012, 59, 79-93). In addition, we propose a new line search for the derivative-free method. Global convergence of the proposed method is established if the system of nonlinear equations are Lipschitz continuous and monotone. Preliminary numerical results are given to test the effectiveness of the proposed method.

In this paper, we propose a derivative-free method for solving large-scale nonlinear monotone equations. It combines the modified Perry's conjugate gradient method [19] for unconstrained optimization problems and the hyperplane projection method [8]. We prove that the proposed method converges globally if the equations are monotone and Lipschitz continuous without differentiability requirement on the equations, which makes it possible to solve some nonsmooth equations. Another good property of the proposed method is that it is suitable to solve large-scale nonlinear monotone equations due to its lower storage requirement. Preliminary numerical results show that the proposed method is promising.

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