An Accelerated Conjugate Gradient Algorithm for Solving Nonlinear Monotone Equations and Image Restoration Problems

Haishan Feng,Tingting Li

doi:10.1155/2020/7945467

Abstract

Combining the three-term conjugate gradient method of Yuan and Zhang and the acceleration step length of Andrei with the hyperplane projection method of Solodov and Svaiter, we propose an accelerated conjugate gradient algorithm for solving nonlinear monotone equations in this paper. The presented algorithm has the following properties: (i) All search directions generated by the algorithm satisfy the sufficient descent and trust region properties independent of the line search technique. (ii) A derivative-free search technique is proposed along the direction to obtain the step length αk. (iii) If ϕk=−αkhk−hwkTdk>0, then an acceleration scheme is used to modify the step length in a multiplicative manner and create a point. (iv) If the point satisfies the given condition, then it is the next point; otherwise, the hyperplane projection technique is used to obtain the next point. (v) The global convergence of the proposed algorithm is established under some suitable conditions. Numerical comparisons with other conjugate gradient algorithms show that the accelerated computing scheme is more competitive. In addition, the presented algorithm can also be applied to image restoration.

Highlights

In this paper, the following nonlinear equation is considered: h(x) 0, subject to x ∈ Rn, (1)where h: Rn ⟶ Rn is continuous and monotone, and h(x) satisfies (h(x) − h(y))T(x − y) ≥ 0, ∀x, y ∈ Rn. (2)It is not difficult to show that the solution set of monotone equation (1), unless empty, is convex. is problem has many significant applications in applied mathematics, economics, and engineering
Some derivative-free line search techniques [2,3,4,5] were proposed to search for step length αk
Inspired by the above discussions, we proposed an accelerated conjugate gradient algorithm that combines the three-term PRP (TTPRP) method, the acceleration step length, and the hyperplane projection method. e main contributions of the algorithm are as follows: An accelerated conjugate gradient algorithm is introduced for solving nonlinear monotone equations

Summary

Introduction

The following nonlinear equation is considered: h(x) 0, subject to x ∈ Rn,. E specific process of the hyperplane projection method is as follows: let xk be the current iteration point, and obtain a point wk xk + αkdk along a certain line search direction dk such that h(wk)T(xk − wk) > 0. All search directions of the algorithm satisfy the sufficient descent condition All search directions of the algorithm belong to a trust region e global convergence of the presented algorithm is proved e numerical results show that the proposed algorithm is more effective for nonlinear monotone equations e algorithm can be applied to restore an original image from an image damaged by impulse noise is paper is organized as follows: we discuss the ATTPRP algorithm and global convergence analysis.

Accelerated Algorithm and Convergence Analysis

Numerical Experiments

Findings

Conclusions