A note on the spectral gradient projection method for nonlinear monotone equations with applications

Abstract

In this work, we provide a note on the spectral gradient projection method for solving nonlinear equations. Motivated by recent extensions of the spectral gradient method for solving nonlinear monotone equations with convex constraints, in this paper, we note that choosing the search direction as a convex combination of two different positive spectral coefficients multiplied with the residual vector is more efficient and robust compared with the standard choice of spectral gradient coefficients combined with the projection strategy of Solodov and Svaiter (A globally convergent inexact newton method for systems of monotone equations. In: Reformulation: Nonsmooth. Piecewise Smooth, Semismooth and Smoothing Methods, pp 355–369. Springer, 1998). Under suitable assumptions, the convergence of the proposed method is established. Preliminary numerical experiments show that the method is promising. In this paper, the proposed method was used to recover sparse signal and restore blurred image arising from compressive sensing.