A Subgradient Extragradient Framework Incorporating a Relaxation and Dual Inertial Technique for Variational Inequalities

Habib ur Rehman, Kanokwan Sitthithakerngkiet and Thidaporn Seangwattana ()
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Habib ur Rehman: School of Mathematics, Zhejiang Normal University, Jinhua 321004, China
Kanokwan Sitthithakerngkiet: Intelligent 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, Thailand
Thidaporn Seangwattana: Faculty of Science Energy and Environment, King Mongkut’s University of Technology North Bangkok (KMUTNB), Rayong Campus, Rayong 21120, Thailand

Mathematics, 2024, vol. 13, issue 1, 1-32

Abstract: 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 [ 0 , 1 ] . We establish R -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.

Keywords: variational inequality problem; double inertial steps; subgradient extragradient method; pseudomonotone operator (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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