In this paper, we consider a multivariate spectral DY-type projection method for solving nonlinear monotone equations with convex constraints. The search direction of the proposed method combines those of the multivariate spectral gradient method and DY conjugate gradient method. With no need for the derivative information, the proposed method is very suitable to solve large-scale nonsmooth monotone equations. Under appropriate conditions, we prove the global convergence and R-linear convergence rate of the proposed method. The preliminary numerical results also indicate that the proposed method is robust and effective.

Many problems arising from machine learning, compressive sensing, linear inverse problem, and statistical inference involve finding sparse solutions to under-determined or ill-conditioned equations. In this paper, a gradient projection method is proposed to recover sparse signal in compressive sensing by solving the nonlinear convex constrained equations. The global convergence is established with the backtracking line search. Preliminary numerical experiments coping with the sparse signal reconstruction in compressive sensing are performed, which show that the proposed method is very effective and stable.