Abstract
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.
Highlights
In this paper, we consider the problem of finding numerical solutions of the following large-scale nonlinear equations F ( x ) = 0, (1)where the function F : Rn −→ Rn is monotone and continuous
Global convergence of the proposed method is established if the system of nonlinear equations are Lipschitz continuous and monotone
Nonlinear monotone equations can be applied in different fields, for example, they are used as subproblems in the generalized proximal algorithms with Bregman distances [1]
Summary
Nonlinear monotone equations can be applied in different fields, for example, they are used as subproblems in the generalized proximal algorithms with Bregman distances [1]. Some monotone variational inequality problems can be converted into nonlinear monotone equations [2]. Monotone systems of equations can be applied in L1 -norm regularization sparse optimization problems (see [3,4]) and discrete mathematics such as graph theory (see [5,6]). Being aware of the important applications of nonlinear monotone equations, in recent years, many scholars have paid attention to propose efficient algorithms for solving problem (1). These algorithms are mainly divided into the following categories
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
