scholarly journals A New Conjugate Gradient Method with Sufficient Descent Property

2021 ◽  
pp. 163-174
Author(s):  
O.B. Akinduko
Keyword(s):  
Conjugate Gradient ◽  
Large Scale ◽  
Optimization Problems ◽  
Gradient Methods ◽  
Superior Performance ◽  
Sufficient Descent Property ◽  
Unconstrained Optimization Problems ◽  
Descent Property ◽  
Sufficient Descent ◽  
Large Scale Unconstrained Optimization

In this paper, by linearly combining the numerator and denominator terms of the Dai-Liao (DL) and Bamigbola-Ali-Nwaeze (BAN) conjugate gradient methods (CGMs), a general form of DL-BAN method has been proposed. From this general form, a new hybrid CGM, which was found to possess a sufficient descent property is generated. Numerical experiment was carried out on the new CGM in comparison with four existing CGMs, using some set of large scale unconstrained optimization problems. The result showed a superior performance of new method over majority of the existing methods.

scholarly journals A decent three term conjugate gradient method with global convergence properties for large scale unconstrained optimization problems

2021 ◽  
Vol 6 (10) ◽  
pp. 10742-10764
Author(s):  
Ibtisam A. Masmali ◽  
◽  
Zabidin Salleh ◽  
Ahmad Alhawarat ◽  
◽  
...  
Keyword(s):  
Unconstrained Optimization ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Optimization Problems ◽  
Medical Science ◽  
Convergence Properties ◽  
Unconstrained Optimization Problems ◽  
Descent Property ◽  
Sufficient Descent ◽  
Large Scale Unconstrained Optimization

<abstract> <p>The conjugate gradient (CG) method is a method to solve unconstrained optimization problems. Moreover CG method can be applied in medical science, industry, neural network, and many others. In this paper a new three term CG method is proposed. The new CG formula is constructed based on DL and WYL CG formulas to be non-negative and inherits the properties of HS formula. The new modification satisfies the convergence properties and the sufficient descent property. The numerical results show that the new modification is more efficient than DL, WYL, and CG-Descent formulas. We use more than 200 functions from CUTEst library to compare the results between these methods in term of number of iterations, function evaluations, gradient evaluations, and CPU time.</p> </abstract>

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.

scholarly journals A Conjugate Gradient Type Method for the Nonnegative Constraints Optimization Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Can Li
Keyword(s):  
Conjugate Gradient ◽  
Large Scale ◽  
Optimization Problems ◽  
Gradient Methods ◽  
Type Method ◽  
Unconstrained Optimization Problems ◽  
Feasible Direction Method ◽  
Gradient Type ◽  
Large Scale Unconstrained Optimization ◽  
Nonnegative Constraints

We are concerned with the nonnegative constraints optimization problems. It is well known that the conjugate gradient methods are efficient methods for solving large-scale unconstrained optimization problems due to their simplicity and low storage. Combining the modified Polak-Ribière-Polyak method proposed by Zhang, Zhou, and Li with the Zoutendijk feasible direction method, we proposed a conjugate gradient type method for solving the nonnegative constraints optimization problems. If the current iteration is a feasible point, the direction generated by the proposed method is always a feasible descent direction at the current iteration. Under appropriate conditions, we show that the proposed method is globally convergent. We also present some numerical results to show the efficiency of the proposed method.

scholarly journals An Efficient Modified AZPRP Conjugate Gradient Method for Large-Scale Unconstrained Optimization Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Ahmad Alhawarat ◽  
Thoi Trung Nguyen ◽  
Ramadan Sabra ◽  
Zabidin Salleh
Keyword(s):  
Unconstrained Optimization ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Line Search ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Unconstrained Optimization Problems ◽  
Unconstrained Optimization Problem ◽  
Descent Property ◽  
Large Scale Unconstrained Optimization

To find a solution of unconstrained optimization problems, we normally use a conjugate gradient (CG) method since it does not cost memory or storage of second derivative like Newton’s method or Broyden–Fletcher–Goldfarb–Shanno (BFGS) method. Recently, a new modification of Polak and Ribiere method was proposed with new restart condition to give a so-call AZPRP method. In this paper, we propose a new modification of AZPRP CG method to solve large-scale unconstrained optimization problems based on a modification of restart condition. The new parameter satisfies the descent property and the global convergence analysis with the strong Wolfe-Powell line search. The numerical results prove that the new CG method is strongly aggressive compared with CG_Descent method. The comparisons are made under a set of more than 140 standard functions from the CUTEst library. The comparison includes number of iterations and CPU time.

scholarly journals The Hybrid BFGS-CG Method in Solving Unconstrained Optimization Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Mohd Asrul Hery Ibrahim ◽  
Mustafa Mamat ◽  
Wah June Leong
Keyword(s):  
Unconstrained Optimization ◽  
Hybrid Method ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Optimization Problems ◽  
Gradient Methods ◽  
Conjugate Gradient Methods ◽  
Unconstrained Optimization Problems ◽  
Large Scale Unconstrained Optimization ◽  
Quasi Newton

In solving large scale problems, the quasi-Newton method is known as the most efficient method in solving unconstrained optimization problems. Hence, a new hybrid method, known as the BFGS-CG method, has been created based on these properties, combining the search direction between conjugate gradient methods and quasi-Newton methods. In comparison to standard BFGS methods and conjugate gradient methods, the BFGS-CG method shows significant improvement in the total number of iterations and CPU time required to solve large scale unconstrained optimization problems. We also prove that the hybrid method is globally convergent.

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.

scholarly journals Two-versions of descent conjugate gradient methods for large-scale unconstrained optimization

2021 ◽  
Vol 22 (3) ◽  
pp. 1643
Author(s):  
Hawraz N. Jabbar ◽  
Basim A. Hassan
Keyword(s):  
Unconstrained Optimization ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Optimization Problems ◽  
Gradient Methods ◽  
Taylor Series Expansion ◽  
Conjugate Gradient Methods ◽  
Unconstrained Optimization Problems ◽  
Gradient Type ◽  
Large Scale Unconstrained Optimization

<p>The conjugate gradient methods are noted to be exceedingly valuable for solving large-scale unconstrained optimization problems since it needn't the storage of matrices. Mostly the parameter conjugate is the focus for conjugate gradient methods. The current paper proposes new methods of parameter of conjugate gradient type to solve problems of large-scale unconstrained optimization. A Hessian approximation in a diagonal matrix form on the basis of second and third-order Taylor series expansion was employed in this study. The sufficient descent property for the proposed algorithm are proved. The new method was converged globally. This new algorithm is found to be competitive to the algorithm of fletcher-reeves (FR) in a number of numerical experiments.</p>

scholarly journals A New Modified Three-Term Hestenes–Stiefel Conjugate Gradient Method with Sufficient Descent Property and Its Global Convergence

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Bakhtawar Baluch ◽  
Zabidin Salleh ◽  
Ahmad Alhawarat
Keyword(s):  
Global Convergence ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Line Search ◽  
Gradient Methods ◽  
Conjugate Gradient Methods ◽  
Sufficient Descent Property ◽  
Descent Property ◽  
Sufficient Descent

This paper describes a modified three-term Hestenes–Stiefel (HS) method. The original HS method is the earliest conjugate gradient method. Although the HS method achieves global convergence using an exact line search, this is not guaranteed in the case of an inexact line search. In addition, the HS method does not usually satisfy the descent property. Our modified three-term conjugate gradient method possesses a sufficient descent property regardless of the type of line search and guarantees global convergence using the inexact Wolfe–Powell line search. The numerical efficiency of the modified three-term HS method is checked using 75 standard test functions. It is known that three-term conjugate gradient methods are numerically more efficient than two-term conjugate gradient methods. Importantly, this paper quantifies how much better the three-term performance is compared with two-term methods. Thus, in the numerical results, we compare our new modification with an efficient two-term conjugate gradient method. We also compare our modification with a state-of-the-art three-term HS method. Finally, we conclude that our proposed modification is globally convergent and numerically efficient.

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