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

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Hongbo Guan ◽  
Sheng Wang
Keyword(s):  
Nonlinear Equations ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Numerical Experiments ◽  
Large Scale ◽  
Globally Convergent ◽  
Weaker Conditions ◽  
Modified Conjugate Gradient Method

In this paper, we propose a modified Polak–Ribière–Polyak (PRP) conjugate gradient method for solving large-scale nonlinear equations. Under weaker conditions, we show that the proposed method is globally convergent. We also carry out some numerical experiments to test the proposed method. The results show that the proposed method is efficient and stable.

scholarly journals A Scaled Conjugate Gradient Method for Solving Monotone Nonlinear Equations with Convex Constraints

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Sheng Wang ◽  
Hongbo Guan
Keyword(s):  
Nonlinear Equations ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Large Scale ◽  
Optimization Problems ◽  
Convex Constraints ◽  
Type Method ◽  
Scaled Conjugate Gradient ◽  
Unconstrained Optimization Problems

Based on the Scaled conjugate gradient (SCALCG) method presented by Andrei (2007) and the projection method presented by Solodov and Svaiter, we propose a SCALCG method for solving monotone nonlinear equations with convex constraints. SCALCG method can be regarded as a combination of conjugate gradient method and Newton-type method for solving unconstrained optimization problems. So, it has the advantages of the both methods. It is suitable for solving large-scale problems. So, it can be applied to solving large-scale monotone nonlinear equations with convex constraints. Under reasonable conditions, we prove its global convergence. We also do some numerical experiments show that the proposed method is efficient and promising.

scholarly journals A Parallel Preconditioned Modified Conjugate Gradient Method for Large Sylvester Matrix Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Junxia Hou ◽  
Quanyi Lv ◽  
Manyu Xiao
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
High Performance ◽  
Large Scale ◽  
Computational Effort ◽  
Control Problems ◽  
Sylvester Matrix Equation ◽  
Modified Conjugate Gradient Method

Computational effort of solving large-scale Sylvester equationsAX+XB+F=Ois frequently hindered in dealing with many complex control problems. In this work, a parallel preconditioned algorithm for solving it is proposed based on combination of a parameter iterative preconditioned method and modified form of conjugate gradient (MCG) method. Furthermore, Schur’s inequality and modified conjugate gradient method are employed to overcome the involved difficulties such as determination of parameter and calculation of inverse matrix. Several numerical results finally show that high performance of proposed parallel algorithm is obtained both in convergent rate and in parallel efficiency.

scholarly journals A Modified Fletcher–Reeves Conjugate Gradient Method for Monotone Nonlinear Equations with Some Applications

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 745 ◽  
Author(s):  
Auwal Bala Abubakar ◽  
Poom Kumam ◽  
Hassan Mohammad ◽  
Aliyu Muhammed Awwal ◽  
Kanokwan Sitthithakerngkiet
Keyword(s):  
Nonlinear Equations ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Numerical Experiments ◽  
Unconstrained Minimization ◽  
Research Article ◽  
Descent Property ◽  
Constrained Nonlinear Equations ◽  
Sufficient Descent

One of the fastest growing and efficient methods for solving the unconstrained minimization problem is the conjugate gradient method (CG). Recently, considerable efforts have been made to extend the CG method for solving monotone nonlinear equations. In this research article, we present a modification of the Fletcher–Reeves (FR) conjugate gradient projection method for constrained monotone nonlinear equations. The method possesses sufficient descent property and its global convergence was proved using some appropriate assumptions. Two sets of numerical experiments were carried out to show the good performance of the proposed method compared with some existing ones. The first experiment was for solving monotone constrained nonlinear equations using some benchmark test problem while the second experiment was applying the method in signal and image recovery problems arising from compressive sensing.

scholarly journals A Descent Four-Term Conjugate Gradient Method with Global Convergence Properties for Large-Scale Unconstrained Optimisation Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Ahmad Alhawarat ◽  
Ghaliah Alhamzi ◽  
Ibitsam Masmali ◽  
Zabidin Salleh
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Large Scale ◽  
Medical Science ◽  
Gradient Methods ◽  
Numerical Results ◽  
Unconstrained Optimisation ◽  
Descent Condition ◽  
Modified Conjugate Gradient Method

The conjugate gradient method is a useful method to solve large-scale unconstrained optimisation problems and to be used in some applications in several fields such as engineering, medical science, image restorations, neural network, and many others. The main benefit of the conjugate gradient method is not using the second derivative or its approximation, such as Newton’s method or its approximation. Moreover, the algorithm of the conjugate gradient method is simple and easy to apply. This study proposes a new modified conjugate gradient method that contains four terms depending on popular two- and three-term conjugate gradient methods. The new algorithm satisfies the descent condition. In addition, the new CG algorithm possesses the convergence property. In the numerical results part, we compare the new algorithm with famous methods such as CG-Descent. We conclude from numerical results that the new algorithm is more efficient than other popular CG methods such as CG-Descent 6.8 in terms of number of function evaluations, number of gradient evaluations, number of iterations, and CPU time.

scholarly journals On the Strong Convergence of a Sufficient Descent Polak-Ribière-Polyak Conjugate Gradient Method

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Min Sun ◽  
Jing Liu
Keyword(s):  
Strong Convergence ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Large Scale ◽  
Optimization Problems ◽  
Line Searches ◽  
Unconstrained Optimization Problems ◽  
Sufficient Descent ◽  
Large Scale Unconstrained Optimization

Recently, Zhang et al. proposed a sufficient descent Polak-Ribière-Polyak (SDPRP) conjugate gradient method for large-scale unconstrained optimization problems and proved its global convergence in the sense thatlim infk→∞∥∇f(xk)∥=0when an Armijo-type line search is used. In this paper, motivated by the line searches proposed by Shi et al. and Zhang et al., we propose two new Armijo-type line searches and show that the SDPRP method has strong convergence in the sense thatlimk→∞∥∇f(xk)∥=0under the two new line searches. Numerical results are reported to show the efficiency of the SDPRP with the new Armijo-type line searches in practical computation.

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