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Some three-term RMIL conjugate gradient methods with descent property for solving optimization problems with application.

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Title: Some three-term RMIL conjugate gradient methods with descent property for solving optimization problems with application.
Authors: Salihu, Nasiru1,2 (AUTHOR) nsalihu@mau.edu.ng, Kumam, Poom1 (AUTHOR), Salihu, Suraj3 (AUTHOR), Seangwattana, Thidaporn4 (AUTHOR)
Source: RAIRO: Operations Research (2804-7303). Mar/Apr2026, Vol. 60 Issue 2, p1269-1286. 18p.
Subjects: Conjugate gradient methods, Quasi-Newton methods, Mathematical optimization, Motion control devices
Abstract: To propose a conjugate gradient (CG) scheme with a promising structure, it has been observed that Newton's direction is optimal when the current iteration is near the solution, and the objective function behaves like a quadratic. However, for large-scale problems, a method that does not require second-derivative information is often necessary. Therefore, to develop a more effective scheme for handling complex problems, we apply the standard secant equation to construct a combination of three-term CG search directions using the βkRMIL β k RMIL $ \beta_{k}^{\text{RMIL}} $ method. This combination approximates the quasi-Newton direction and ensures sufficient descent. Furthermore, we establish the global convergence of the scheme under mild assumptions, demonstrating that the algorithm is robust and reliable compared to earlier CG methods. Finally, we showcase the efficiency of the proposed scheme by applying it to solve a three degrees of freedom (3DOF) motion control model. [ABSTRACT FROM AUTHOR]
Copyright of RAIRO: Operations Research (2804-7303) is the property of EDP Sciences and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Some three-term RMIL conjugate gradient methods with descent property for solving optimization problems with application.
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  Data: <searchLink fieldCode="AR" term="%22Salihu%2C+Nasiru%22">Salihu, Nasiru</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> nsalihu@mau.edu.ng</i><br /><searchLink fieldCode="AR" term="%22Kumam%2C+Poom%22">Kumam, Poom</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Salihu%2C+Suraj%22">Salihu, Suraj</searchLink><relatesTo>3</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Seangwattana%2C+Thidaporn%22">Seangwattana, Thidaporn</searchLink><relatesTo>4</relatesTo> (AUTHOR)
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  Data: <searchLink fieldCode="JN" term="%22RAIRO%3A+Operations+Research+%282804-7303%29%22">RAIRO: Operations Research (2804-7303)</searchLink>. Mar/Apr2026, Vol. 60 Issue 2, p1269-1286. 18p.
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  Data: <searchLink fieldCode="DE" term="%22Conjugate+gradient+methods%22">Conjugate gradient methods</searchLink><br /><searchLink fieldCode="DE" term="%22Quasi-Newton+methods%22">Quasi-Newton methods</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+optimization%22">Mathematical optimization</searchLink><br /><searchLink fieldCode="DE" term="%22Motion+control+devices%22">Motion control devices</searchLink>
– Name: Abstract
  Label: Abstract
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  Data: To propose a conjugate gradient (CG) scheme with a promising structure, it has been observed that Newton's direction is optimal when the current iteration is near the solution, and the objective function behaves like a quadratic. However, for large-scale problems, a method that does not require second-derivative information is often necessary. Therefore, to develop a more effective scheme for handling complex problems, we apply the standard secant equation to construct a combination of three-term CG search directions using the βkRMIL β k RMIL $ \beta_{k}^{\text{RMIL}} $ method. This combination approximates the quasi-Newton direction and ensures sufficient descent. Furthermore, we establish the global convergence of the scheme under mild assumptions, demonstrating that the algorithm is robust and reliable compared to earlier CG methods. Finally, we showcase the efficiency of the proposed scheme by applying it to solve a three degrees of freedom (3DOF) motion control model. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of RAIRO: Operations Research (2804-7303) is the property of EDP Sciences and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1051/ro/2026031
    Languages:
      – Code: eng
        Text: English
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      Pagination:
        PageCount: 18
        StartPage: 1269
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      – SubjectFull: Conjugate gradient methods
        Type: general
      – SubjectFull: Quasi-Newton methods
        Type: general
      – SubjectFull: Mathematical optimization
        Type: general
      – SubjectFull: Motion control devices
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      – TitleFull: Some three-term RMIL conjugate gradient methods with descent property for solving optimization problems with application.
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            – D: 01
              M: 03
              Text: Mar/Apr2026
              Type: published
              Y: 2026
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