Abstract
One of the fastest growing and efficient methods for solving the unconstrained minimization problem is the conjugate gradient method (CG). Recently, considerable efforts have been made to extend the CG method for solving monotone nonlinear equations. In this research article, we present a modification of the Fletcher–Reeves (FR) conjugate gradient projection method for constrained monotone nonlinear equations. The method possesses sufficient descent property and its global convergence was proved using some appropriate assumptions. Two sets of numerical experiments were carried out to show the good performance of the proposed method compared with some existing ones. The first experiment was for solving monotone constrained nonlinear equations using some benchmark test problem while the second experiment was applying the method in signal and image recovery problems arising from compressive sensing.
Highlights
In this paper, we are considering a system of nonlinear monotone equations of the formF ( x ) = 0, subject to x ∈ E, (1)where E ⊆ Rn is closed and convex, F : Rn → Rm, (m ≥ n) is continuous and monotone, which means h F ( x ) − F (y), ( x − y)i ≥ 0, Mathematics 2019, 7, 745; doi:10.3390/math7080745 ∀ x, y ∈ Rn .www.mdpi.com/journal/mathematicsA well-known fact is that under the above assumption, the solution set of (1) is convex unless is empty
Inspired by some the above proposals, we present a simple modification of the Fletcher–Reeves (FR) conjugate gradient method [19] considered in [12] to solve nonlinear monotone equations with convex constraints
To test the performance of the proposed method, we compare it with accelerated conjugate gradient descent (ACGD) and projected Dai-Yuan (PDY) methods in [27,28], respectively
Summary
Auwal Bala Abubakar 1,2 , Poom Kumam 1,3,4, * , Hassan Mohammad 2 , Aliyu Muhammed Awwal 1,5 and Kanokwan Sitthithakerngkiet 6. KMUTTFixed Point Research Laboratory, SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand.
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