Projection-Based Method for Finding Zeros of Nonlinear Equations

Following Andrei's approach of convexly integrating the Hestenes–Stiefel and Dai–Yuan conjugate gradient parameters, this article proposes a hybrid method that combines the Hestenes–Stiefel and Dai–Yuan like conjugate gradient method to solve constrained nonlinear equations involving monotone mappings. The hybridization parameter is determined by solving a least squares problem aimed at minimizing the distance between the search directions of the hybrid parameter and those of a three-term projection method that possesses a descent property. Under certain appropriate conditions, the global convergence of the method is established. Furthermore, two types of numerical experiments are presented: (i) tests of nonlinear equations involving monotone mappings and (ii) image restoration problems. The numerical experiments demonstrate the effectiveness of the proposed method in solving constrained nonlinear equations and in restoring blurred and noisy images, outperforming the compared methods.