A spectral conjugate gradient projection algorithm to solve the large-scale system of monotone nonlinear equations with application to compressed sensing

Abstract

In this paper, a derivative-free spectral projection technique to solve a system of large-scale nonlinear monotone equations is presented. The primary motivation is to use the appropriate structure of spectral conjugate gradient directions in the projection algorithms. The new direction is derivative-free and requires a little storage and computation. So, it is an appropriate direction to use in large-scale projection algorithms. We prove the global convergence and R-linear convergence rate of the proposed algorithm under some suitable conditions. Numerical experiments show a promising behaviour of the proposed algorithm to deal with large-scale monotone equations. Additionally, as a practical application, we use the new method to solve the -norm regularization problems to reconstruct a sparse signal in compressed sensing.