A modified PRP-type conjugate gradient projection algorithm for solving large-scale monotone nonlinear equations with convex constraint

Abstract

Conjugate gradient methods stand out as the most ideal iterative algorithms for solving nonlinear system of equations with large-dimensions. This is due to the fact that they are implemented with less memory and because of their ability to converge globally to solutions of problems considered. One of the most essential iterative method in this category is the Polak–Ribière–Polyak (PRP) scheme, which is numerically effective, but its search directions are mostly not descent directions. In this paper, based upon the adaptive PRP scheme by Yuan et al. and the projection method, a numerically efficient PRP-type scheme for system of monotone nonlinear equations is presented, where the solution is restricted to a closed convex set. Apart from the ability to satisfy the condition that is quite vital for global convergence, a distinct novelty of the new scheme is its application in compressive sensing, where it is applied to restore blurry images. The scheme’s global convergence is established with mild assumptions. Preliminary numerical results show that the method proposed is promising.