A Family of Developed Hybrid Four-Term Conjugate Gradient Algorithms for Unconstrained Optimization with Applications in Image Restoration
The most important advantage of conjugate gradient methods (CGs) is that these methods have low memory requirements and convergence speed. This paper contains two main parts that deal with two application problems, as follows. In the first part, three new parameters of the CG methods are designed an...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-06-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/6/1203 |
| _version_ | 1797592566308274176 |
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| author | Eltiyeb Ali Salem Mahdi |
| author_facet | Eltiyeb Ali Salem Mahdi |
| author_sort | Eltiyeb Ali |
| collection | DOAJ |
| description | The most important advantage of conjugate gradient methods (CGs) is that these methods have low memory requirements and convergence speed. This paper contains two main parts that deal with two application problems, as follows. In the first part, three new parameters of the CG methods are designed and then combined by employing a convex combination. The search direction is a four-term hybrid form for modified classical CG methods with some newly proposed parameters. The result of this hybridization is the acquisition of a newly developed hybrid CGCG method containing four terms. The proposed CGCG has sufficient descent properties. The convergence analysis of the proposed method is considered under some reasonable conditions. A numerical investigation is carried out for an unconstrained optimization problem. The comparison between the newly suggested algorithm (CGCG) and five other classical CG algorithms shows that the new method is competitive with and in all statuses superior to the five methods in terms of efficiency reliability and effectiveness in solving large-scale, unconstrained optimization problems. The second main part of this paper discusses the image restoration problem. By using the adaptive median filter method, the noise in an image is detected, and then the corrupted pixels of the image are restored by using a new family of modified hybrid CG methods. This new family has four terms: the first is the negative gradient; the second one consists of either the HS-CG method or the HZ-CG method; and the third and fourth terms are taken from our proposed CGCG method. Additionally, a change in the size of the filter window plays a key role in improving the performance of this family of CG methods, according to the noise level. Four famous images (test problems) are used to examine the performance of the new family of modified hybrid CG methods. The outstanding clearness of the restored images indicates that the new family of modified hybrid CG methods has reliable efficiency and effectiveness in dealing with image restoration problems. |
| first_indexed | 2024-03-11T01:53:04Z |
| format | Article |
| id | doaj.art-663993c1aaf34aa1aa2ffdbbce570dfa |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T01:53:04Z |
| publishDate | 2023-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-663993c1aaf34aa1aa2ffdbbce570dfa2023-11-18T12:50:53ZengMDPI AGSymmetry2073-89942023-06-01156120310.3390/sym15061203A Family of Developed Hybrid Four-Term Conjugate Gradient Algorithms for Unconstrained Optimization with Applications in Image RestorationEltiyeb Ali0Salem Mahdi1Department of Mathematics, College of Science and Arts Sharourah, Najran University, P.O. Box 1988, Najran 68341, Saudi ArabiaDepartment of Mathematics & Computer Science, Faculty of Science, Alexandria University, Alexandria 5424041, EgyptThe most important advantage of conjugate gradient methods (CGs) is that these methods have low memory requirements and convergence speed. This paper contains two main parts that deal with two application problems, as follows. In the first part, three new parameters of the CG methods are designed and then combined by employing a convex combination. The search direction is a four-term hybrid form for modified classical CG methods with some newly proposed parameters. The result of this hybridization is the acquisition of a newly developed hybrid CGCG method containing four terms. The proposed CGCG has sufficient descent properties. The convergence analysis of the proposed method is considered under some reasonable conditions. A numerical investigation is carried out for an unconstrained optimization problem. The comparison between the newly suggested algorithm (CGCG) and five other classical CG algorithms shows that the new method is competitive with and in all statuses superior to the five methods in terms of efficiency reliability and effectiveness in solving large-scale, unconstrained optimization problems. The second main part of this paper discusses the image restoration problem. By using the adaptive median filter method, the noise in an image is detected, and then the corrupted pixels of the image are restored by using a new family of modified hybrid CG methods. This new family has four terms: the first is the negative gradient; the second one consists of either the HS-CG method or the HZ-CG method; and the third and fourth terms are taken from our proposed CGCG method. Additionally, a change in the size of the filter window plays a key role in improving the performance of this family of CG methods, according to the noise level. Four famous images (test problems) are used to examine the performance of the new family of modified hybrid CG methods. The outstanding clearness of the restored images indicates that the new family of modified hybrid CG methods has reliable efficiency and effectiveness in dealing with image restoration problems.https://www.mdpi.com/2073-8994/15/6/1203developed deterministic methodconjugate gradient method (CG)unconstrained optimization problemssufficient descent conditionglobal convergenceadaptive median filter method |
| spellingShingle | Eltiyeb Ali Salem Mahdi A Family of Developed Hybrid Four-Term Conjugate Gradient Algorithms for Unconstrained Optimization with Applications in Image Restoration Symmetry developed deterministic method conjugate gradient method (CG) unconstrained optimization problems sufficient descent condition global convergence adaptive median filter method |
| title | A Family of Developed Hybrid Four-Term Conjugate Gradient Algorithms for Unconstrained Optimization with Applications in Image Restoration |
| title_full | A Family of Developed Hybrid Four-Term Conjugate Gradient Algorithms for Unconstrained Optimization with Applications in Image Restoration |
| title_fullStr | A Family of Developed Hybrid Four-Term Conjugate Gradient Algorithms for Unconstrained Optimization with Applications in Image Restoration |
| title_full_unstemmed | A Family of Developed Hybrid Four-Term Conjugate Gradient Algorithms for Unconstrained Optimization with Applications in Image Restoration |
| title_short | A Family of Developed Hybrid Four-Term Conjugate Gradient Algorithms for Unconstrained Optimization with Applications in Image Restoration |
| title_sort | family of developed hybrid four term conjugate gradient algorithms for unconstrained optimization with applications in image restoration |
| topic | developed deterministic method conjugate gradient method (CG) unconstrained optimization problems sufficient descent condition global convergence adaptive median filter method |
| url | https://www.mdpi.com/2073-8994/15/6/1203 |
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