A Projection Hestenes–Stiefel Method with Spectral Parameter for Nonlinear Monotone Equations and Signal Processing

A number of practical problems in science and engineering can be converted into a system of nonlinear equations and therefore, it is imperative to develop efficient methods for solving such equations. Due to their nice convergence properties and low storage requirements, conjugate gradient methods are considered among the most efficient for solving large-scale nonlinear equations. In this paper, a modified conjugate gradient method is proposed based on a projection technique and a suitable line search strategy. The proposed method is matrix-free and its sequence of search directions satisfies sufficient descent condition. Under the assumption that the underlying function is monotone and Lipschitzian continuous, the global convergence of the proposed method is established. The method is applied to solve some benchmark monotone nonlinear equations and also extended to solve ${\ell }_1$ -norm regularized problems to reconstruct a sparse signal in compressive sensing. Numerical comparison with some existing methods shows that the proposed method is competitive, efficient and promising.