Two new Hager–Zhang iterative schemes with improved parameter choices for monotone nonlinear systems and their applications in compressed sensing

Abstract

Notwithstanding its efficiency and nice attributes, most research on the Hager–Zhang (HZ) iterative scheme are focused on unconstrained minimization problems. Inspired by this and recent extensions of the one-parameter HZ scheme to system of nonlinear monotone equations, two new HZ-type iterative methods are developed in this paper for solving system of monotone equations with convex constraint. This is achieved by developing two HZ-type search directions with new parameter choices combined with the popular projection method. The first parameter choice is obtained by minimizing the condition number of a modified HZ direction matrix, while the second choice is realized using singular value analysis and minimizing the spectral condition number of a nonsingular HZ search direction matrix. Interesting properties of the schemes include solving non-smooth problems and satisfying the inequality that is vital for global convergence. Using standard assumptions, global convergence of the schemes are proven and numerical experiments with recent methods in the literature, indicate that the methods proposed are promising. The effectiveness of the schemes are further demonstrated by their applications to sparse signal and image reconstruction problems, where they outperform some recent schemes in the literature.