Least-Square-Based Three-Term Conjugate Gradient Projection Method for ℓ1-Norm Problems with Application to Compressed Sensing

Ibrahim, Abdulkarim Hassan; Abubakar, Auwal Bala; Abubakar, Jamilu; Muhammad, Abubakar Bakoji

doi:10.3390/math8040602

Cited by 28 publications

(4 citation statements)

References 34 publications

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“…Let x 0 be a starting point that satisfies Hypothesis 1. Regard any algorithm to be of Form (18), such that the d k satisfies (19) and the α k satisfies conditions (31) and (32). Hence, the following inequality is met.…”

Section: Convergence Analysismentioning

confidence: 99%

“…Additionally, CG parameters have shown remarkable superiority in solving problems involving systems of nonlinear equations (see, for example, [26][27][28][29][30][31][32][33][34][35]). According to previous successful uses of CG techniques to solve different applications problems, many authors have adapted CG methods such that they are capable of dealing with image restoration problems (see, for example, [25,[35][36][37][38][39][40][41][42][43]).…”

Section: Introductionmentioning

confidence: 99%

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A Family of Developed Hybrid Four-Term Conjugate Gradient Algorithms for Unconstrained Optimization with Applications in Image Restoration

Ali

Mahdi

2023

Symmetry

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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.

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Section: Convergence Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Family of Developed Hybrid Four-Term Conjugate Gradient Algorithms for Unconstrained Optimization with Applications in Image Restoration

Ali

Mahdi

2023

Symmetry

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“…Let assumptions 1 -3 be fulfilled. If { } and { } are sequences defined by (19) and (20) in Algorithm 1, then { } and { } are bounded. Furthermore,…”

Section: Lemma 33mentioning

confidence: 99%

“…For more on the conjugate gradient algorithms, the interested reader is referred to [1,2,3,4,5,6,7,8,9,18,19,20,21,22,26,29,35].…”

Section: Introductionmentioning

confidence: 99%

Derivative-free HS-DY-type method for solving nonlinear equations and image restoration

Abubakar

Ibrahim

Rilwan

2020

Heliyon

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A derivative-free conjugate gradient algorithm for solving nonlinear equations and image restoration is proposed. The conjugate gradient (CG) parameter of the proposed algorithm is a convex combination of Hestenes-Stiefel (HS) and Dai-Yuan (DY) type CG parameters. The search direction is descent and bounded. Under suitable assumptions, the convergence of the proposed hybrid algorithm is obtained. Using some benchmark test problems, the proposed algorithm is shown to be efficient compared with existing algorithms. In addition, the proposed algorithm is effectively applied to solve image restoration problems.

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Another hybrid approach for solving monotone operator equations and application to signal processing

Abubakar

Ibrahim

Kura³

et al. 2022

Math Methods in App Sciences

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This paper presents a hybrid conjugate gradient (CG) approach for solving nonlinear equations and signal reconstruction. The CG parameter of the approach is a convex combination of the Dai‐Yuan (DY)‐like and Hestenes‐Stiefel (HS)‐like parameters. Independent of any line search, the search direction is descent and bounded. Under some reasonable assumptions, the global convergence of the hybrid approach is proved. Numerical experiments on some benchmark test problems show that the proposed approach is efficient compared with some existing algorithms. Finally, the proposed approach is applied in signal reconstruction.

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Least-Square-Based Three-Term Conjugate Gradient Projection Method for ℓ1-Norm Problems with Application to Compressed Sensing

Cited by 28 publications

References 34 publications

A Family of Developed Hybrid Four-Term Conjugate Gradient Algorithms for Unconstrained Optimization with Applications in Image Restoration

A Family of Developed Hybrid Four-Term Conjugate Gradient Algorithms for Unconstrained Optimization with Applications in Image Restoration

Derivative-free HS-DY-type method for solving nonlinear equations and image restoration

Another hybrid approach for solving monotone operator equations and application to signal processing

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