scholarly journals A Projection Hestenes–Stiefel Method with Spectral Parameter for Nonlinear Monotone Equations and Signal Processing

2020 ◽  
Vol 25 (2) ◽  
pp. 27
Author(s):  
Aliyu Muhammed Awwal ◽  
Lin Wang ◽  
Poom Kumam ◽  
Hassan Mohammad ◽  
Wiboonsak Watthayu
Keyword(s):  
Nonlinear Equations ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Gradient Methods ◽  
System Of Nonlinear Equations ◽  
Numerical Comparison ◽  
Sufficient Descent Condition ◽  
Monotone Equations ◽  
Matrix Free ◽  
Lipschitzian Continuous

A number of practical problems in science and engineering can be converted into a system of nonlinear equations and therefore, it is imperative to develop efficient methods for solving such equations. Due to their nice convergence properties and low storage requirements, conjugate gradient methods are considered among the most efficient for solving large-scale nonlinear equations. In this paper, a modified conjugate gradient method is proposed based on a projection technique and a suitable line search strategy. The proposed method is matrix-free and its sequence of search directions satisfies sufficient descent condition. Under the assumption that the underlying function is monotone and Lipschitzian continuous, the global convergence of the proposed method is established. The method is applied to solve some benchmark monotone nonlinear equations and also extended to solve ℓ 1 -norm regularized problems to reconstruct a sparse signal in compressive sensing. Numerical comparison with some existing methods shows that the proposed method is competitive, efficient and promising.

scholarly journals Sufficient Descent Conjugate Gradient Methods for Solving Convex Constrained Nonlinear Monotone Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
San-Yang Liu ◽  
Yuan-Yuan Huang ◽  
Hong-Wei Jiao
Keyword(s):  
Global Convergence ◽  
Projection Method ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Gradient Methods ◽  
Conjugate Gradient Methods ◽  
Monotone Equations ◽  
Mild Conditions ◽  
Sufficient Descent ◽  
Nonlinear Monotone Equations

Two unified frameworks of some sufficient descent conjugate gradient methods are considered. Combined with the hyperplane projection method of Solodov and Svaiter, they are extended to solve convex constrained nonlinear monotone equations. Their global convergence is proven under some mild conditions. Numerical results illustrate that these methods are efficient and can be applied to solve large-scale nonsmooth equations.

scholarly journals The Sufficient Descent Condition of Nonlinear Conjugate Gradient Method

2018 ◽  
Vol 7 (4.30) ◽  
pp. 458
Author(s):  
Srimazzura Basri ◽  
Mustafa Mamat ◽  
Puspa Liza Ghazali
Keyword(s):  
Conjugate Gradient ◽  
Large Scale ◽  
Gradient Methods ◽  
Sufficient Descent Condition ◽  
Nonlinear Conjugate Gradient Method ◽  
Non Linear ◽  
Nonlinear Conjugate Gradient ◽  
Sufficient Descent ◽  
Descent Condition ◽  
Scale Optimization

Non-linear conjugate gradient methods has been widely used instrumental in solving large scale optimization. These methods has been proved that only required very low memory other than its numerical efficiency. Thus, many studies have been conducted to improve these methods to find the most efficient method. In this paper, we proposed a new non-linear conjugate gradient coefficient that guarantees sufficient descent condition. Numerical tests indicate that the proposed coefficient is better than the three classical conjugate gradient coefficients.

scholarly journals Nonlinear Conjugate Gradient Methods with Sufficient Descent Condition for Large-Scale Unconstrained Optimization

2009 ◽  
Vol 2009 ◽  
pp. 1-16 ◽  
Author(s):  
Jianguo Zhang ◽  
Yunhai Xiao ◽  
Zengxin Wei
Keyword(s):  
Unconstrained Optimization ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Optimization Problems ◽  
Gradient Methods ◽  
Sufficient Descent Condition ◽  
Convergence Results ◽  
Nonlinear Conjugate Gradient ◽  
Large Scale Unconstrained Optimization ◽  
Nonlinear Conjugate Gradient Methods

Two nonlinear conjugate gradient-type methods for solving unconstrained optimization problems are proposed. An attractive property of the methods, is that, without any line search, the generated directions always descend. Under some mild conditions, global convergence results for both methods are established. Preliminary numerical results show that these proposed methods are promising, and competitive with the well-known PRP method.

scholarly journals En enhanced matrix-free method via double step length approach for solving systems of nonlinear equations

2017 ◽  
Vol 6 (4) ◽  
pp. 147 ◽  
Author(s):  
Abubakar Sani Halilu ◽  
H. Abdullahi ◽  
Mohammed Yusuf Waziri
Keyword(s):  
Nonlinear Equations ◽  
Large Scale ◽  
Step Length ◽  
Systems Of Nonlinear Equations ◽  
Test Problems ◽  
System Of Nonlinear Equations ◽  
Numerical Comparison ◽  
Inexact Line Search ◽  
Derivative Free ◽  
Variant Method

A variant method for solving system of nonlinear equations is presented. This method use the special form of iteration with two step length parameters, we suggest a derivative-free method without computing the Jacobian via acceleration parameter as well as inexact line search procedure. The proposed method is proven to be globally convergent under mild condition. The preliminary numerical comparison reported in this paper using a large scale benchmark test problems show that the proposed method is practically quite effective.

scholarly journals Efficient matrix-free direction method with line search for solving large-scale system of nonlinear equations

2020 ◽  
Vol 30 (4) ◽  
pp. 399-412
Author(s):  
Abubakar Halilu ◽  
Mohammed Waziri ◽  
Ibrahim Yusuf
Keyword(s):  
Nonlinear Equations ◽  
Large Scale ◽  
Line Search ◽  
Test Problems ◽  
System Of Nonlinear Equations ◽  
Inexact Line Search ◽  
Large Scale Problems ◽  
Scale Test ◽  
Line Search Technique ◽  
Matrix Free

We proposed a matrix-free direction with an inexact line search technique to solve system of nonlinear equations by using double direction approach. In this article, we approximated the Jacobian matrix by appropriately constructed matrix-free method via acceleration parameter. The global convergence of our method is established under mild conditions. Numerical comparisons reported in this paper are based on a set of large-scale test problems and show that the proposed method is efficient for large-scale problems.

scholarly journals A NEW THREE-TERM CONJUGATE GRADIENT-BASED PROJECTION METHOD FOR SOLVING LARGE-SCALE NONLINEAR MONOTONE EQUATIONS

2019 ◽  
Vol 24 (4) ◽  
pp. 550-563
Author(s):  
Mompati Koorapetse ◽  
Professor Kaelo
Keyword(s):  
Projection Method ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Sufficient Descent Condition ◽  
Monotone Equations ◽  
Derivative Free ◽  
Gradient Based ◽  
Sufficient Descent ◽  
Descent Condition ◽  
Nonlinear Monotone Equations

A new three-term conjugate gradient-based projection method is presented in this paper for solving large-scale nonlinear monotone equations. This method is derivative-free and it is suitable for solving large-scale nonlinear monotone equations due to its lower storage requirements. The method satisfies the sufficient descent condition FTkdk ≤ −τ‖Fk‖2, where τ > 0 is a constant, and its global convergence is also established. Numerical results show that the method is efficient and promising.

scholarly journals An efficient three-term conjugate gradient-type algorithm for monotone nonlinear equations

2020 ◽  
Author(s):  
Jamilu Sabi'u ◽  
Abdullah Shah
Keyword(s):  
Global Convergence ◽  
Nonlinear Equations ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Numerical Results ◽  
Search Direction ◽  
Projection Technique ◽  
Type Algorithm ◽  
Gradient Type

In this article, we proposed two Conjugate Gradient (CG) parameters using the modified Dai-{L}iao condition and the descent three-term CG search direction. Both parameters are incorporated with the projection technique for solving large-scale monotone nonlinear equations. Using the Lipschitz and monotone assumptions, the global convergence of methods has been proved. Finally, numerical results are provided to illustrate the robustness of the proposed methods.

scholarly journals A modified Perry-type derivative-free projection method for solving large-scale nonlinear monotone equations

2021 ◽  
Author(s):  
Mompati Koorapetse ◽  
P Kaelo ◽  
S Kooepile-Reikeletseng
Keyword(s):  
Global Convergence ◽  
Projection Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Large Scale ◽  
Numerical Results ◽  
Projection Technique ◽  
Monotone Equations ◽  
Derivative Free ◽  
Nonlinear Monotone Equations

In this paper, a new modified Perry-type derivative-free projection method for solving large-scale nonlinear monotone equations is presented. The method is developed by combining a modified Perry's conjugate gradient method with the hyperplane projection technique. Global convergence and numerical results of the proposed method are established. Preliminary numerical results show that the proposed method is promising and efficient compared to some existing methods in the literature.

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