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<p>The conjugate gradient methods (CGs) have successfully solved large-scale unconstrained minimization problems. However, research activities on the methods (CGs) for some other applications are somewhat fewer. In this paper, we propose new parameters for designing two new methods as extensions and developments of the classical methods (CGs). We supply these proposed methods with a proper step size and line search technique that depends on backtracking line search to make both proposed methods capable of dealing with pretty complicated nonlinear systems. If the size of the system is very large, the task is more challenging for dealing with such a system. The two proposed methods (CGs), therefore, in this paper are abbreviated by FCGNE and HMHZS. The FCGNE method contains two new parameters of the CG method with a convexity coefficient that combines the two parameters to get a research direction. The HMHZS method is a new hybrid CG method that hybridizes three old methods (CGs) to obtain a new research direction. One task of this study is to establish the global convergence proof of the FCGNE approach by presenting some limited conditions. The effectiveness of the suggested algorithms is tested by solving a group of systems of nonlinear equations involving <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(3000 - 60000\)</EquationSource> </InlineEquation> variables. The outcomes of the proposed algorithms are compared with those of modern conjugate gradient approaches that solve the same set of test problems. Also, we adapt these methods to make them capable of dealing with image restoration issue. Fair, well-established, and appropriate evaluation criteria are utilized to compare the performances of the three methods. The numerical results demonstrate that the proposed algorithms are competitive for all test cases and outperform existing methods with respect to effectiveness, reliability, and efficiency for approximating solutions to nonlinear systems. The outstanding clarity of the repaired image demonstrates that the adaptation of a new group of modified hybrid methods CGs owns trustworthy efficiency and effectiveness to solve an image restoration problem.</p>

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Derivative-free methods (Conjugate Gradient Methods) for dealing with nonlinear equations and image restoration

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年1月24日

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  • Salem Mahdi,
  • Ibrahim Elbatal,
  • Mahmoud M. Abdelwahab,
  • Mohamed Tawhid,
  • Seyed Jalaleddin Mousavirad,
  • Ali Wagdy

摘要

The conjugate gradient methods (CGs) have successfully solved large-scale unconstrained minimization problems. However, research activities on the methods (CGs) for some other applications are somewhat fewer. In this paper, we propose new parameters for designing two new methods as extensions and developments of the classical methods (CGs). We supply these proposed methods with a proper step size and line search technique that depends on backtracking line search to make both proposed methods capable of dealing with pretty complicated nonlinear systems. If the size of the system is very large, the task is more challenging for dealing with such a system. The two proposed methods (CGs), therefore, in this paper are abbreviated by FCGNE and HMHZS. The FCGNE method contains two new parameters of the CG method with a convexity coefficient that combines the two parameters to get a research direction. The HMHZS method is a new hybrid CG method that hybridizes three old methods (CGs) to obtain a new research direction. One task of this study is to establish the global convergence proof of the FCGNE approach by presenting some limited conditions. The effectiveness of the suggested algorithms is tested by solving a group of systems of nonlinear equations involving \(3000 - 60000\) variables. The outcomes of the proposed algorithms are compared with those of modern conjugate gradient approaches that solve the same set of test problems. Also, we adapt these methods to make them capable of dealing with image restoration issue. Fair, well-established, and appropriate evaluation criteria are utilized to compare the performances of the three methods. The numerical results demonstrate that the proposed algorithms are competitive for all test cases and outperform existing methods with respect to effectiveness, reliability, and efficiency for approximating solutions to nonlinear systems. The outstanding clarity of the repaired image demonstrates that the adaptation of a new group of modified hybrid methods CGs owns trustworthy efficiency and effectiveness to solve an image restoration problem.