scholarly journals A Modified PRP-CG Type Derivative-Free Algorithm with Optimal Choices for Solving Large-Scale Nonlinear Symmetric Equations

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 234
Author(s):  
Jamilu Sabi’u ◽  
Kanikar Muangchoo ◽  
Abdullah Shah ◽  
Auwal Bala Abubakar ◽  
Lateef Olakunle Jolaoso
Keyword(s):  
Global Convergence ◽  
Nonlinear Equations ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Search Direction ◽  
Large Scale Systems ◽  
Measure Function ◽  
Derivative Free ◽  
Newton Direction

Inspired by the large number of applications for symmetric nonlinear equations, this article will suggest two optimal choices for the modified Polak–Ribiére–Polyak (PRP) conjugate gradient (CG) method by minimizing the measure function of the search direction matrix and combining the proposed direction with the default Newton direction. In addition, the corresponding PRP parameters are incorporated with the Li and Fukushima approximate gradient to propose two robust CG-type algorithms for finding solutions for large-scale systems of symmetric nonlinear equations. We have also demonstrated the global convergence of the suggested algorithms using some classical assumptions. Finally, we demonstrated the numerical advantages of the proposed algorithms compared to some of the existing methods for nonlinear symmetric equations.

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 Simple Three-term Conjugate Gradient Algorithm for Solving Symmetric Systems of Nonlinear Equations

2016 ◽  
Vol 5 (3) ◽  
pp. 118 ◽  
Author(s):  
M. Y. Waziri ◽  
L. Muhammad ◽  
J. Sabi’u
Keyword(s):  
Nonlinear Equations ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Conjugate Gradient Algorithm ◽  
Gradient Algorithm ◽  
Systems Of Nonlinear Equations ◽  
Large Scale Systems ◽  
Large Scale Problems ◽  
Derivative Free ◽  
Symmetric Systems

<p>This paper presents a simple three-terms Conjugate Gradient algorithm for solving Large-Scale systems of nonlinear equations without computing Jacobian and gradient via the special structure of the underlying function. This three term CG of the proposed method has an advantage of solving relatively large-scale problems, with lower storage requirement compared to some existing methods. By incoporating the Powel restart approach in to the algorithm, we prove the global convergence of the proposed method with a derivative free line search under suitable assumtions. The numerical results are presented which show that the proposed method is promising.</p>

scholarly journals A Derivative-Free Conjugate Gradient Method and Its Global Convergence for Solving Symmetric Nonlinear Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Mohammed Yusuf Waziri ◽  
Jamilu Sabi’u
Keyword(s):  
Global Convergence ◽  
Nonlinear Equations ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Systems Of Nonlinear Equations ◽  
Test Problems ◽  
Storage Requirement ◽  
Large Scale Problems ◽  
Derivative Free ◽  
Symmetric Systems

We suggest a conjugate gradient (CG) method for solving symmetric systems of nonlinear equations without computing Jacobian and gradient via the special structure of the underlying function. This derivative-free feature of the proposed method gives it advantage to solve relatively large-scale problems (500,000 variables) with lower storage requirement compared to some existing methods. Under appropriate conditions, the global convergence of our method is reported. Numerical results on some benchmark test problems show that the proposed method is practically effective.

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.

scholarly journals A Derivative-Free Decent Method Via Acceleration Parameter for Solving Systems of Nonlinear Equations

2019 ◽  
Vol 2 (3) ◽  
pp. 1-4
Author(s):  
Abubakar Sani Halilu ◽  
M K Dauda ◽  
M Y Waziri ◽  
M Mamat
Keyword(s):  
Nonlinear Equations ◽  
Large Scale ◽  
Line Search ◽  
Main Idea ◽  
Step Length ◽  
Descent Method ◽  
Systems Of Nonlinear Equations ◽  
Inexact Line Search ◽  
Large Scale Systems ◽  
Derivative Free

An algorithm for solving large-scale systems of nonlinear equations based on the transformation of the Newton method with the line search into a derivative-free descent method is introduced. Main idea used in the algorithm construction is to approximate the Jacobian by an appropriate diagonal matrix. Furthermore, the step length is calculated using inexact line search procedure. Under appropriate conditions, the proposed method is proved to be globally convergent under mild conditions. The numerical results presented show the efficiency of the proposed method.

scholarly journals A method with inertial extrapolation step for convex constrained monotone equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Abdulkarim Hassan Ibrahim ◽  
Poom Kumam ◽  
Auwal Bala Abubakar ◽  
Jamilu Abubakar
Keyword(s):  
Global Convergence ◽  
Theoretical Analysis ◽  
Nonlinear Equations ◽  
Large Scale ◽  
Descent Direction ◽  
Monotone Equations ◽  
Derivative Free ◽  
Speed Up ◽  
Sufficient Descent ◽  
Inertial Extrapolation

AbstractIn recent times, various algorithms have been incorporated with the inertial extrapolation step to speed up the convergence of the sequence generated by these algorithms. As far as we know, very few results exist regarding algorithms of the inertial derivative-free projection method for solving convex constrained monotone nonlinear equations. In this article, the convergence analysis of a derivative-free iterative algorithm (Liu and Feng in Numer. Algorithms 82(1):245–262, 2019) with an inertial extrapolation step for solving large scale convex constrained monotone nonlinear equations is studied. The proposed method generates a sufficient descent direction at each iteration. Under some mild assumptions, the global convergence of the sequence generated by the proposed method is established. Furthermore, some experimental results are presented to support the theoretical analysis of the proposed method.

scholarly journals Enhanced Derivative-Free Conjugate Gradient Method for Solving Symmetric Nonlinear Equations

2016 ◽  
Vol 5 (1) ◽  
pp. 50
Author(s):  
Jamilu Sabi'u
Keyword(s):  
Global Convergence ◽  
Nonlinear Equations ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Line Search ◽  
Jacobian Matrix ◽  
Derivative Free ◽  
Matrix Free ◽  
Gradient Approach

In this article, an enhanced conjugate gradient approach for solving symmetric nonlinear equations is propose without computing the Jacobian matrix. This approach is completely derivative and matrix free. Using classical assumptions the proposed method has global convergence with nonmonotone line search. Some reported numerical results shows the approach is promising.<p style="margin: 0px; text-indent: 0px; -qt-block-indent: 0; -qt-paragraph-type: empty;"> </p>

scholarly journals An Inexact Optimal Hybrid Conjugate Gradient Method for Solving Symmetric Nonlinear Equations

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1829
Author(s):  
Jamilu Sabi’u ◽  
Kanikar Muangchoo ◽  
Abdullah Shah ◽  
Auwal Bala Abubakar ◽  
Kazeem Olalekan Aremu
Keyword(s):  
Nonlinear Equations ◽  
Conjugate Gradient ◽  
Convex Combination ◽  
Iteration Process ◽  
Nonlinear Problems ◽  
Conjugacy Condition ◽  
Derivative Free ◽  
Numerical Performance ◽  
Hybrid Conjugate Gradient Method ◽  
Newton Direction

This article presents an inexact optimal hybrid conjugate gradient (CG) method for solving symmetric nonlinear systems. The method is a convex combination of the optimal Dai–Liao (DL) and the extended three-term Polak–Ribiére–Polyak (PRP) CG methods. However, two different formulas for selecting the convex parameter are derived by using the conjugacy condition and also by combining the proposed direction with the default Newton direction. The proposed method is again derivative-free, therefore the Jacobian information is not required throughout the iteration process. Furthermore, the global convergence of the proposed method is shown using some appropriate assumptions. Finally, the numerical performance of the method is demonstrated by solving some examples of symmetric nonlinear problems and comparing them with some existing symmetric nonlinear equations CG solvers.

scholarly journals Global Convergence of a New Coefficient Nonlinear Conjugate Gradient Method

2018 ◽  
Vol 11 (3) ◽  
pp. 1188
Author(s):  
Nur Syarafina Mohamed ◽  
Mustafa Mamat ◽  
Mohd Rivaie ◽  
Shazlyn Milleana Shaharuddin
Keyword(s):  
Global Convergence ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Line Search ◽  
Optimization Problems ◽  
Search Direction ◽  
Nonlinear Conjugate Gradient Method ◽  
Unconstrained Optimization Problems ◽  
Nonlinear Conjugate Gradient ◽  
Large Scale Unconstrained Optimization

Nonlinear conjugate gradient (CG) methods are widely used in optimization field due to its efficiency for solving a large scale unconstrained optimization problems. Many studies and modifications have been developed in order to improve the method. The method is known to possess sufficient descend condition and its global convergence properties under strong Wolfe-Powell search direction. In this paper, the new coefficient of CG method is presented. The global convergence and sufficient descend properties of the new coefficient are established by using strong Wolfe-Powell line search direction. Results show that the new coefficient is able to globally converge under certain assumptions and theories.

scholarly journals A Cauchy Point Direction Trust Region Algorithm for Nonlinear Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Zhou Sheng ◽  
Dan Luo
Keyword(s):  
Global Convergence ◽  
Nonlinear Equations ◽  
Large Scale ◽  
Convex Combination ◽  
Trust Region ◽  
Numerical Results ◽  
The Other ◽  
Search Direction ◽  
Trust Region Algorithm ◽  
Descent Property

In this paper, a Cauchy point direction trust region algorithm is presented to solve nonlinear equations. The search direction is an optimal convex combination of the trust region direction and the Cauchy point direction with the sufficiently descent property and the automatic trust region property. The global convergence of the proposed algorithm is proven under some conditions. The preliminary numerical results demonstrate that the proposed algorithm is promising and has better convergence behaviors than the other two existing algorithms for solving large-scale nonlinear equations.

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