Descent Derivative-Free Method Involving Symmetric Rank-One Update for Solving Convex Constrained Nonlinear Monotone Equations and Application to Image Recovery

Abstract

Many practical applications in applied sciences such as imaging, signal processing, and motion control can be reformulated into a system of nonlinear equations with or without constraints. In this paper, a new descent projection iterative algorithm for solving a nonlinear system of equations with convex constraints is proposed. The new approach is based on a modified symmetric rank-one updating formula. The search direction of the proposed algorithm mimics the behavior of a spectral conjugate gradient algorithm where the spectral parameter is determined so that the direction is sufficiently descent. Based on the assumption that the underlying function satisfies monotonicity and Lipschitz continuity, the convergence result of the proposed algorithm is discussed. Subsequently, the efficiency of the new method is revealed. As an application, the proposed algorithm is successfully implemented on image deblurring problem.