scholarly journals A MODIFIED PROJECTED CONJUGATE GRADIENT ALGORITHM FOR UNCONSTRAINED OPTIMIZATION PROBLEMS

2013 ◽  
Vol 54 (3) ◽  
pp. 143-152 ◽  
Author(s):  
SHUAI HUANG ◽  
ZHONG WAN ◽  
SONGHAI DENG
Keyword(s):  
Conjugate Gradient ◽  
Line Search ◽  
Optimization Problems ◽  
Gradient Algorithm ◽  
Test Problems ◽  
Optimal Solutions ◽  
Conjugacy Condition ◽  
Unconstrained Optimization Problems ◽  
Sufficient Descent ◽  
Descent Directions

AbstractWe propose a modified projected Polak–Ribière–Polyak (PRP) conjugate gradient method, where a modified conjugacy condition and a method which generates sufficient descent directions are incorporated into the construction of a suitable conjugacy parameter. It is shown that the proposed method is a modification of the PRP method and generates sufficient descent directions at each iteration. With an Armijo-type line search, the theory of global convergence is established under two weak assumptions. Numerical experiments are employed to test the efficiency of the algorithm in solving some benchmark test problems available in the literature. The numerical results obtained indicate that the algorithm outperforms an existing similar algorithm in requiring fewer function evaluations and fewer iterations to find optimal solutions with the same tolerance.

New hyrid conjugate gradient method as a convex combination of HZ and CD methods

2021 ◽  
pp. 2150187
Author(s):  
Amira Hamdi ◽  
Badreddine Sellami ◽  
Mohammed Belloufi
Keyword(s):  
Global Convergence ◽  
Conjugate Gradient ◽  
Line Search ◽  
Optimization Problems ◽  
Convex Combination ◽  
Gradient Algorithm ◽  
Unconstrained Optimization Problems ◽  
Gradient Parameter ◽  
Sufficient Descent ◽  
Wolfe Line Search

In this paper, a new hybrid conjugate gradient algorithm is proposed for solving unconstrained optimization problems, the conjugate gradient parameter [Formula: see text] is computed as a convex combination of [Formula: see text] and [Formula: see text]. Under the wolfe line search, we prove the sufficient descent and the global convergence. Numerical results are reported to show the effectiveness of our procedure.

scholarly journals An Extension of Polak-Ribière-Polyak Method using Exact Line Search

2019 ◽  
Vol 8 (2S3) ◽  
pp. 368-373
Keyword(s):  
Conjugate Gradient ◽  
Large Scale ◽  
Line Search ◽  
Optimization Problems ◽  
Test Problems ◽  
Exact Line Search ◽  
Unconstrained Optimization Problems ◽  
Nonlinear Conjugate Gradient ◽  
Sufficient Descent ◽  
Large Scale Unconstrained Optimization

Lately, many large-scale unconstrained optimization problems rely upon nonlinear conjugate gradient (CG) methods. Many areas such as engineering, and computer science have benefited because of its simplicity, fast and low memory requirements. Many modified coefficients have appeared recently, all of which to improve these methods. This paper considers an extension conjugate gradient method of PolakRibière-Polyak using exact line search to show that it holds for some properties such as sufficient descent and global convergence. A set of 113 test problems is used to evaluate the performance of the proposed method and get compared to other existing methods using the same line search.

Global convergence of a modified Fletcher–Reeves conjugate gradient method with Wolfe line search

2019 ◽  
Vol 13 (04) ◽  
pp. 2050081
Author(s):  
Badreddine Sellami ◽  
Mohamed Chiheb Eddine Sellami
Keyword(s):  
Conjugate Gradient ◽  
Line Search ◽  
Optimization Problems ◽  
Gradient Methods ◽  
Conjugate Gradients ◽  
Gradient Algorithm ◽  
Conjugacy Condition ◽  
Sufficient Descent Condition ◽  
Strong Wolfe Line Search ◽  
Wolfe Line Search

In this paper, we are concerned with the conjugate gradient methods for solving unconstrained optimization problems. we propose a modified Fletcher–Reeves (abbreviated FR) [Function minimization by conjugate gradients, Comput. J. 7 (1964) 149–154] conjugate gradient algorithm satisfying a parametrized sufficient descent condition with a parameter [Formula: see text] is proposed. The parameter [Formula: see text] is computed by means of the conjugacy condition, thus an algorithm which is a positive multiplicative modification of the Hestenes and Stiefel (abbreviated HS) [Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standards Sec. B 48 (1952) 409–436] algorithm is obtained, which produces a descent search direction at every iteration that the line search satisfies the Wolfe conditions. Under appropriate conditions, we show that the modified FR method with the strong Wolfe line search is globally convergent of uniformly convex functions. We also present extensive preliminary numerical experiments to show the efficiency of the proposed method.

scholarly journals Global convergence of conjugate gradient method in unconstrained optimization problems

2019 ◽  
Vol 38 (7) ◽  
pp. 227-231
Author(s):  
Huda Younus Najm ◽  
Eman T. Hamed ◽  
Huda I. Ahmed
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Line Search ◽  
Optimization Problems ◽  
Inexact Line Search ◽  
Sufficient Descent Condition ◽  
Unconstrained Optimization Problems ◽  
Sufficient Descent ◽  
Descent Condition

In this study, we propose a new parameter in the conjugate gradient method. It is shown that the new method fulfils the sufficient descent condition with the strong Wolfe condition when inexact line search has been used. The numerical results of this suggested method also shown that this method outperforms to other standard conjugate gradient method.

scholarly journals A q-Polak–Ribière–Polyak conjugate gradient algorithm for unconstrained optimization problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shashi Kant Mishra ◽  
Suvra Kanti Chakraborty ◽  
Mohammad Esmael Samei ◽  
Bhagwat Ram
Keyword(s):  
Unconstrained Optimization ◽  
Conjugate Gradient ◽  
Optimization Problems ◽  
Gradient Algorithm ◽  
Test Problems ◽  
Conjugate Gradient Algorithms ◽  
Gradient Algorithms ◽  
Unconstrained Optimization Problems ◽  
Wolfe Conditions ◽  
Nonlinear Unconstrained Optimization

AbstractA Polak–Ribière–Polyak (PRP) algorithm is one of the oldest and popular conjugate gradient algorithms for solving nonlinear unconstrained optimization problems. In this paper, we present a q-variant of the PRP (q-PRP) method for which both the sufficient and conjugacy conditions are satisfied at every iteration. The proposed method is convergent globally with standard Wolfe conditions and strong Wolfe conditions. The numerical results show that the proposed method is promising for a set of given test problems with different starting points. Moreover, the method reduces to the classical PRP method as the parameter q approaches 1.

scholarly journals A COMPARATIVE STUDY OF SOME MODIFICATIONS OF CG METHODS UNDER EXACT LINE SEARCH

2020 ◽  
Vol 1 (1) ◽  
pp. 12-17
Author(s):  
Yasir Salih ◽  
Mustafa Mamat ◽  
Sukono Sukono
Keyword(s):  
Conjugate Gradient ◽  
Numerical Experiments ◽  
Line Search ◽  
Optimization Problems ◽  
Gradient Methods ◽  
Test Problems ◽  
Convergence Properties ◽  
Exact Line Search ◽  
Unconstrained Optimization Problems ◽  
Nonlinear Unconstrained Optimization

Conjugate Gradient (CG) method is a technique used in solving nonlinear unconstrained optimization problems. In this paper, we analysed the performance of two modifications and compared the results with the classical conjugate gradient methods of. These proposed methods possesse global convergence properties for general functions using exact line search. Numerical experiments show that the two modifications are more efficient for the test problems compared to classical CG coefficients.

scholarly journals Modified Three-Term Conjugate Gradient Method and Its Applications

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Jiankun Liu ◽  
Shouqiang Du
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Line Search ◽  
Optimization Problems ◽  
Unconstrained Optimization Problems ◽  
Armijo Line Search ◽  
Global Convergence Property ◽  
Sufficient Descent ◽  
The Given

We propose a modified three-term conjugate gradient method with the Armijo line search for solving unconstrained optimization problems. The proposed method possesses the sufficient descent property. Under mild assumptions, the global convergence property of the proposed method with the Armijo line search is proved. Due to simplicity, low storage, and nice convergence properties, the proposed method is used to solve M-tensor systems and a kind of nonsmooth optimization problems with l1-norm. Finally, the given numerical experiments show the efficiency of the proposed method.

scholarly journals A new hybrid conjugate gradient algorithm for unconstrained optimization with inexact line search

2020 ◽  
Vol 20 (2) ◽  
pp. 939
Author(s):  
Fanar N. Jardow ◽  
Ghada M. Al-Naemi
Keyword(s):  
Unconstrained Optimization ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Line Search ◽  
Optimization Problems ◽  
Convex Combination ◽  
Gradient Algorithm ◽  
Inexact Line Search ◽  
Unconstrained Optimization Problems ◽  
Large Scale Unconstrained Optimization

Many researchers are interested for developed and improved the conjugate gradient method for solving large scale unconstrained optimization problems. In this work a new parameter  will be presented as a convex combination between RMIL and MMWU. The suggestion method always produces a descent search direction at each iteration. Under Strong Wolfe Powell (SWP) line search conditions, the global convergence of the proposed method is established. The preliminary numerical comparisons with some others CG methods have shown that this new method is efficient and robust in solving all given problems.

scholarly journals A Modified Sufficient Descent Polak–Ribiére–Polyak Type Conjugate Gradient Method for Unconstrained Optimization Problems

Algorithms ◽  
2018 ◽  
Vol 11 (9) ◽  
pp. 133 ◽  
Author(s):  
Xiuyun Zheng ◽  
Jiarong Shi
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Line Search ◽  
Optimization Problems ◽  
Nonlinear Conjugate Gradient Method ◽  
Modified Method ◽  
Unconstrained Optimization Problems ◽  
Nonlinear Conjugate Gradient ◽  
Sufficient Descent

In this paper, a modification to the Polak–Ribiére–Polyak (PRP) nonlinear conjugate gradient method is presented. The proposed method always generates a sufficient descent direction independent of the accuracy of the line search and the convexity of the objective function. Under appropriate conditions, the modified method is proved to possess global convergence under the Wolfe or Armijo-type line search. Moreover, the proposed methodology is adopted in the Hestenes–Stiefel (HS) and Liu–Storey (LS) methods. Extensive preliminary numerical experiments are used to illustrate the efficiency of the proposed method.

scholarly journals A Conjugate Gradient Method for Unconstrained Optimization Problems

2009 ◽  
Vol 2009 ◽  
pp. 1-14 ◽  
Author(s):  
Gonglin Yuan
Keyword(s):  
Unconstrained Optimization ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Line Search ◽  
Optimization Problems ◽  
Unconstrained Optimization Problems ◽  
Sufficient Descent ◽  
Search Rule ◽  
Better Than

A hybrid method combining the FR conjugate gradient method and the WYL conjugate gradient method is proposed for unconstrained optimization problems. The presented method possesses the sufficient descent property under the strong Wolfe-Powell (SWP) line search rule relaxing the parameterσ<1. Under the suitable conditions, the global convergence with the SWP line search rule and the weak Wolfe-Powell (WWP) line search rule is established for nonconvex function. Numerical results show that this method is better than the FR method and the WYL method.

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