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.
Highlights
The fields of sciences, engineering, industry and other important areas of human endeavor employ vital applications with models in the form of systems of nonlinear equations
Careful study of results displayed in tables 1 − 6 reveals that the N I HZPM and New Enhanced Hager-Zhang Projection Method (NEHZPM) methods solve all problems successfully, and more problems with minimum number of iterations and processing time than the MHZM1, MHZM2, EPGM and CGD methods, which failed to solve some of the problems considered
It is observed from the summary table that the N I HZPM and NEHZPM methods solved 10.42% (30 out of 288) and 22.57% (65 out of 288) of all the problems in the conducted experiments with least number of iterations compared to the MHZM1, MHZM2, EPGM and CGD methods, which record 6.60% (19 out of 288), 4.51% (13 out of 288), 0% (0 out of 288), and 6.25% (18 out of 288) respectively
Summary
The fields of sciences, engineering, industry and other important areas of human endeavor employ vital applications with models in the form of systems of nonlinear equations. This paper focuses on the constrained version of (1.1), where the vector x lies in a nonempty closed convex set, say Ω ⊂ Rn. A number of applications involve the monotone equations in (1.1) and its constrained version. Several iterative schemes for finding solutions of monotone equations exists, and the most popular ones are the Newton and quasi-Newton schemes [6,7,8,9], which possess rapid convergence properties. These methods, require huge matrix storage at each iterations, which makes them unpopular when engaging problems with large dimensions
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