Scaled Three-Term Conjugate Gradient Methods for Solving Monotone Equations with Application

Abstract

In this paper, we derived a modified conjugate gradient (CG) parameter by adopting the Birgin and Marti´nez strategy using the descent three-term CG direction and the Newton direction. The proposed CG parameter is applied and suggests a robust algorithm for solving constrained monotone equations with an application to image restoration problems. The global convergence of this algorithm is established using some proper assumptions. Lastly, the numerical comparison with some existing algorithms shows that the proposed algorithm is a robust approach for solving large-scale systems of monotone equations. Additionally, the proposed CG parameter can be used to solve the symmetric system of nonlinear equations as well as other relevant classes of nonlinear equations.

A New Hybrid Conjugate Gradient Projection Method for Solving Nonlinear Monotone Equations

“…Tese are derivative-free methods, where JKL and EMDY methods are the state-of-the-art methods. Among them, the JKL method, PDY method, and HG method use the line search of formula (17), and the EMDY method uses the line search of formula (18). All codes used were written in Matlab R2015a and run on PC with 4 GB of RAM and Windows 10 operating system.…”

“…Yin et al [16] proposed a hybrid three-term conjugate gradient projection method, whose search direction is close to the search direction generated by the memoryless BFGS method and possesses a descending characteristic independent of the line search technique as well as a trust region characteristic. Using the adaptive line search technique, the global convergence of the method is established under certain mild conditions, and numerical experiments demonstrate that the method inherits the benefcial properties of the three-term conjugate gradient method and the hybrid conjugate gradient method [17,18]. Zhou et al [19] proposed a novel hybrid PRPFR conjugate gradient method with sufcient descent and trust region properties.…”

A Scaled Dai–Yuan Projection-Based Conjugate Gradient Method for Solving Monotone Equations with Applications

“…Among the well-known efficients is the one proposed by Dai and Young [16]. Moreover, the scaling strategy proved to be an efficient strategy for enhancing the performance of CG-based algorithms, see [21]. In this work, to enhance the performance of the DY CG parameter, we present the following scaled DY CG parameter…”

A Modified Three-Term Conjugate Descent Derivative-Free Method for Constrained Nonlinear Monotone Equations and Signal Reconstruction Problems

“…However, they require computing and storing of the inverse Jacobian matrix at every iteration, which is not always easy to compute or not available due to singularity. As such, many researchers have suggested derivative-free approaches that do not require computation of the Jacobian matrix; among others are [16][17][18][19][20][21][22][23][24][25]. A modified three-term conjugate descent (CD) derivative-free method is of interest in this research.…”