A Subgradient Extragradient Framework Incorporating a Relaxation and Dual Inertial Technique for Variational Inequalities
This paper presents an enhanced algorithm designed to solve variational inequality problems that involve a pseudomonotone and Lipschitz continuous operator in real Hilbert spaces. The method integrates a dual inertial extrapolation step, a relaxation step, and the subgradient extragradient technique...
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2024-12-01
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| author | Habib ur Rehman Kanokwan Sitthithakerngkiet Thidaporn Seangwattana |
| author_facet | Habib ur Rehman Kanokwan Sitthithakerngkiet Thidaporn Seangwattana |
| author_sort | Habib ur Rehman |
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| description | This paper presents an enhanced algorithm designed to solve variational inequality problems that involve a pseudomonotone and Lipschitz continuous operator in real Hilbert spaces. The method integrates a dual inertial extrapolation step, a relaxation step, and the subgradient extragradient technique, resulting in faster convergence than existing inertia-based subgradient extragradient methods. A key feature of the algorithm is its ability to achieve weak convergence without needing a prior guess of the operator’s Lipschitz constant in the problem. Our method encompasses a range of subgradient extragradient techniques with inertial extrapolation steps as particular cases. Moreover, the inertia in our algorithm is more flexible, chosen from the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>. We establish <i>R</i>-linear convergence under the added hypothesis of strong pseudomonotonicity and Lipschitz continuity. Numerical findings are presented to showcase the algorithm’s effectiveness, highlighting its computational efficiency and practical relevance. A notable conclusion is that using double inertial extrapolation steps, as opposed to the single step commonly seen in the literature, provides substantial advantages for variational inequalities. |
| format | Article |
| id | doaj-art-dee28a03fae14d6daede73f101d8c1ab |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-dee28a03fae14d6daede73f101d8c1ab2025-01-10T13:18:21ZengMDPI AGMathematics2227-73902024-12-0113113310.3390/math13010133A Subgradient Extragradient Framework Incorporating a Relaxation and Dual Inertial Technique for Variational InequalitiesHabib ur Rehman0Kanokwan Sitthithakerngkiet1Thidaporn Seangwattana2School of Mathematics, Zhejiang Normal University, Jinhua 321004, ChinaIntelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok (KMUTNB), Wongsawang, Bangsue, Bangkok 10800, ThailandFaculty of Science Energy and Environment, King Mongkut’s University of Technology North Bangkok (KMUTNB), Rayong Campus, Rayong 21120, ThailandThis paper presents an enhanced algorithm designed to solve variational inequality problems that involve a pseudomonotone and Lipschitz continuous operator in real Hilbert spaces. The method integrates a dual inertial extrapolation step, a relaxation step, and the subgradient extragradient technique, resulting in faster convergence than existing inertia-based subgradient extragradient methods. A key feature of the algorithm is its ability to achieve weak convergence without needing a prior guess of the operator’s Lipschitz constant in the problem. Our method encompasses a range of subgradient extragradient techniques with inertial extrapolation steps as particular cases. Moreover, the inertia in our algorithm is more flexible, chosen from the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>. We establish <i>R</i>-linear convergence under the added hypothesis of strong pseudomonotonicity and Lipschitz continuity. Numerical findings are presented to showcase the algorithm’s effectiveness, highlighting its computational efficiency and practical relevance. A notable conclusion is that using double inertial extrapolation steps, as opposed to the single step commonly seen in the literature, provides substantial advantages for variational inequalities.https://www.mdpi.com/2227-7390/13/1/133variational inequality problemdouble inertial stepssubgradient extragradient methodpseudomonotone operator |
| spellingShingle | Habib ur Rehman Kanokwan Sitthithakerngkiet Thidaporn Seangwattana A Subgradient Extragradient Framework Incorporating a Relaxation and Dual Inertial Technique for Variational Inequalities Mathematics variational inequality problem double inertial steps subgradient extragradient method pseudomonotone operator |
| title | A Subgradient Extragradient Framework Incorporating a Relaxation and Dual Inertial Technique for Variational Inequalities |
| title_full | A Subgradient Extragradient Framework Incorporating a Relaxation and Dual Inertial Technique for Variational Inequalities |
| title_fullStr | A Subgradient Extragradient Framework Incorporating a Relaxation and Dual Inertial Technique for Variational Inequalities |
| title_full_unstemmed | A Subgradient Extragradient Framework Incorporating a Relaxation and Dual Inertial Technique for Variational Inequalities |
| title_short | A Subgradient Extragradient Framework Incorporating a Relaxation and Dual Inertial Technique for Variational Inequalities |
| title_sort | subgradient extragradient framework incorporating a relaxation and dual inertial technique for variational inequalities |
| topic | variational inequality problem double inertial steps subgradient extragradient method pseudomonotone operator |
| url | https://www.mdpi.com/2227-7390/13/1/133 |
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