A Modified Nonlinear Conjugate Gradient Method with the Armijo Line Search and Its Application

Abstract

In this article, a modified Polak-Ribière-Polyak (PRP) conjugate gradient method is proposed for image restoration. The presented method can generate sufficient descent directions without any line search conditions. Under some mild conditions, this method is globally convergent with the Armijo line search. Moreover, the linear convergence rate of the modified PRP method is established. The experimental results of unconstrained optimization, image restoration, and compressive sensing show that the proposed method is promising and competitive with other conjugate gradient methods.

A Modified Conjugate Gradient Method for Solving Large-Scale Nonlinear Equations

“…In this section, we develop a modified nonlinear PRP method for solving the nonlinear equations with convex constraints. Based on the modified nonlinear PRP method [20], we now introduce our method for solving (1). Inspired by (8), we define d k as…”

“…Based on the idea of [18,19], Zhang et al [20] proposed a new modified nonlinear PRP method in which β k is defined by…”

“…Other related works can be found [3, 5-8, 10, 22-30]. In this paper, we combined the projection method [3], the modified nonlinear PRP conjugate gradient method for unconstrained optimization [20] and the iterative method [10] and proposed a modified nonlinear conjugate gradient method for solving large-scale nonlinear monotone equations with convex constrains.…”

Gaussian noise parameter estimation based on multiple singular value decomposition and non‐linear fitting

“…In addition, if the increment is too large, the accuracy of the local approximation will be reduced. To overcome the problem of excessive increment, backtracking Armijo line search [36,37] is introduced, which can adaptively adjust the step size to ensure the global optimality of the solution. Backtracking Armijo line search is an important method for solving unconstrained optimization problems, which first moves a large step to estimate function along the search direction and then continuously reduces the step size in an iterative form until the Armijo rule is satisfied.…”

A q-Polak–Ribière–Polyak conjugate gradient algorithm for unconstrained optimization problems

“…The engineering Muskingum model and image restoration problems were used to determine the interesting aspects of the given algorithm [30]. The generalized conjugate gradient algorithms were studied for solving large-scale unconstrained optimization problems within the real world applications, and two open problems were formulated [31][32][33].…”