An efficient modified Polak–Ribière–Polyak conjugate gradient method with global convergence properties

The conjugate gradient is a useful tool in solving large- and small-scale unconstrained optimization problems. In addition, the conjugate gradient method can be applied in many fields, such as engineering, medical research, and computer science. In this paper, a convex combination of two different search directions is proposed. The new combination satisfies the sufficient descent condition and the convergence analysis. Moreover, a new conjugate gradient formula is proposed. The new formula satisfies the convergence properties with the descent property related to Hestenes–Stiefel conjugate gradient formula. The numerical results show that the new search direction outperforms both two search directions, making it convex between them. The numerical result includes the number of iterations, function evaluations, and central processing unit time. Finally, we present some examples about image restoration as an application of the proposed conjugate gradient method.

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“…which were presented in [15,16], respectively. In addition, Alhawarat et al [17] presented the following simple formula:…”

Section: Introductionmentioning

confidence: 99%

A Convex Combination between Two Different Search Directions of Conjugate Gradient Method and Application in Image Restoration

Alhawarat

Salleh

Masmali

2021

Mathematical Problems in Engineering

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“…Regarding the speed, memory requirements, number of iterations, function evaluations, gradient evaluations, and robustness to solve unconstrained optimization problems which have prompted the development of the CG method, the readers are advised to refer references [10][11][12][13][14][15] for more information on these new formulas.…”

Section: Introductionmentioning

confidence: 99%

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

Alhawarat

Nguyen

Sabra

et al. 2021

Journal of Mathematics

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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.

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“…Recently, Alhawarat and Salleh [14], Salleh and Alhawarat [15], and Alhawarat et al [16,17] proposed efficient CG and hybrid CG methods that fulfill the required global convergence properties. To improve the existing methods, a three-term CG technique has been introduced.…”

Section: Introductionmentioning

confidence: 99%

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

Baluch

Salleh

Alhawarat

2018

Journal of Optimization

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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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An efficient modified Polak–Ribière–Polyak conjugate gradient method with global convergence properties

Cited by 16 publications

References 17 publications

A Convex Combination between Two Different Search Directions of Conjugate Gradient Method and Application in Image Restoration

A Convex Combination between Two Different Search Directions of Conjugate Gradient Method and Application in Image Restoration

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

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

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