A Spectral Conjugate Gradient-like Method for Convex Constrained Nonlinear Monotone Equations and Signal Recovery

Abstract

Many real-world problems can be formulated as systems of nonlinear equations. Thus, finding their solutions is of paramount importance. Traditional approaches such as Newton and quasi-Newton methods for solving these systems require computing Jacobian matrix or an approximation to it at every iteration, which is very expensive especially when the dimension of the systems is large. In this work, we propose a derivative-free algorithm for solving these systems. The proposed algorithm is a combination of the popular conjugate gradient method for unconstrained optimization problems and the projection method. We prove the global convergence of the proposed algorithm under Lipschitz continuity and monotonicity assumptions on the underlying mapping. We perform numerical experiments on some test problems, and the proposed algorithm proves to be more efficient in comparison with some existing works. Finally, we give an application of the proposed algorithm in signal recovery.