Multiple search direction conjugate gradient method II: theory and numerical experiments

2004 ◽  
Vol 81 (10) ◽  
pp. 1289-1307 ◽  
Author(s):  
Tongxiang Gu ◽  
Xingping Liu ◽  
Zeyao Mo ◽  
Xuebin Chi
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Numerical Experiments ◽  
Search Direction

scholarly journals New Investigation for the Liu-Story Scaled Conjugate Gradient Method for Nonlinear Optimization

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Eman T. Hamed ◽  
Rana Z. Al-Kawaz ◽  
Abbas Y. Al-Bayati
Keyword(s):  
Nonlinear Optimization ◽  
Unconstrained Optimization ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Numerical Experiments ◽  
Scalar Parameter ◽  
Scaled Conjugate Gradient ◽  
Wolfe Conditions ◽  
Descent Condition

This article considers modified formulas for the standard conjugate gradient (CG) technique that is planned by Li and Fukushima. A new scalar parameter θkNew for this CG technique of unconstrained optimization is planned. The descent condition and global convergent property are established below using strong Wolfe conditions. Our numerical experiments show that the new proposed algorithms are more stable and economic as compared to some well-known standard CG methods.

A Two-Level Preconditioned Conjugate-Gradient Method in Distorted and Structured Grids

2012 ◽  
Vol 4 (2) ◽  
pp. 238-249
Author(s):  
Qiaolin He
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Numerical Experiments ◽  
Preconditioned Conjugate Gradient ◽  
Structured Grids ◽  
Structured Grid ◽  
Cpu Time ◽  
Linear Correction ◽  
Quadratic Element

AbstractIn this paper, we propose a new two-level preconditioned C-G method which uses the quadratic smoothing and the linear correction in distorted but topo-logically structured grid. The CPU time of this method is less than that of the multigrid preconditioned C-G method (MGCG) using the quadratic element, but their accuracy is almost the same. Numerical experiments and eigenvalue analysis are given and the results show that the proposed two-level preconditioned method is efficient.

A MODIFIED FR CONJUGATE GRADIENT METHOD FOR COMPUTING -EIGENPAIRS OF SYMMETRIC TENSORS

2016 ◽  
Vol 94 (3) ◽  
pp. 411-420
Author(s):  
MEILAN ZENG ◽  
GUANGHUI ZHOU
Keyword(s):  
Global Convergence ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Numerical Experiments ◽  
Gradient Methods ◽  
Conjugate Gradient Methods ◽  
Power Method ◽  
Symmetric Tensors

This paper proposes improvements to the modified Fletcher–Reeves conjugate gradient method (FR-CGM) for computing $Z$-eigenpairs of symmetric tensors. The FR-CGM does not need to compute the exact gradient and Jacobian. The global convergence of this method is established. We also test other conjugate gradient methods such as the modified Polak–Ribière–Polyak conjugate gradient method (PRP-CGM) and shifted power method (SS-HOPM). Numerical experiments of FR-CGM, PRP-CGM and SS-HOPM show the efficiency of the proposed method for finding $Z$-eigenpairs of symmetric tensors.

scholarly journals A Modified Nonlinear Conjugate Gradient Method for Engineering Computation

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Tiefeng Zhu ◽  
Zaizai Yan ◽  
Xiuyun Peng
Keyword(s):  
Global Convergence ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Numerical Experiments ◽  
General Criterion ◽  
Mild Conditions ◽  
Nonlinear Conjugate Gradient Method ◽  
Engineering Computation ◽  
Nonlinear Conjugate Gradient

A general criterion for the global convergence of the nonlinear conjugate gradient method is established, based on which the global convergence of a new modified three-parameter nonlinear conjugate gradient method is proved under some mild conditions. A large amount of numerical experiments is executed and reported, which show that the proposed method is competitive and alternative. Finally, one engineering example has been analyzed for illustrative purposes.

A Hybrid Nonlinear Conjugate Gradient Method with Sufficient Descent Property

2011 ◽  
Vol 58-60 ◽  
pp. 943-949
Author(s):  
Wan You Cheng ◽  
Xue Jie Liu
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Numerical Experiments ◽  
Test Problems ◽  
Sufficient Descent Property ◽  
Nonlinear Conjugate Gradient Method ◽  
Descent Property ◽  
Nonlinear Conjugate Gradient ◽  
Sufficient Descent

In this paper, on the basis of the recently developed HZ (Hager-Zhang) method [SIAM J. Optim., 16 (2005), pp. 170-192], we propose a hybrid descent conjugate gradient method which reserves the sufficient descent property of the HZ method. Under suitable conditions, we prove the global convergence of the proposed method. Extensive numerical experiments show that the method is promising for the test problems from the CUTE library.

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.

MATRIX ANALYSES ON THE DAI–LIAO CONJUGATE GRADIENT METHOD

2019 ◽  
Vol 61 (02) ◽  
pp. 195-203
Author(s):  
Z. AMINIFARD ◽  
S. BABAIE-KAFAKI
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Search Direction ◽  
Condition Numbers ◽  
Test Problems ◽  
Spectral Condition ◽  
Testing Environment ◽  
Measure Function ◽  
Descent Property

Some optimal choices for a parameter of the Dai–Liao conjugate gradient method are proposed by conducting matrix analyses of the method. More precisely, first the $\ell _{1}$ and $\ell _{\infty }$ norm condition numbers of the search direction matrix are minimized, yielding two adaptive choices for the Dai–Liao parameter. Then we show that a recent formula for computing this parameter which guarantees the descent property can be considered as a minimizer of the spectral condition number as well as the well-known measure function for a symmetrized version of the search direction matrix. Brief convergence analyses are also carried out. Finally, some numerical experiments on a set of test problems related to constrained and unconstrained testing environment, are conducted using a well-known performance profile.

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.

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