A novel class of forward-backward explicit iterative algorithms using inertial techniques to solve variational inequality problems with quasi-monotone operators

Bancha Panyanak; Chainarong Khunpanuk; Nattawut Pholasa; Nuttapol Pakkaranang; Bancha Panyanak; Chainarong Khunpanuk; Nattawut Pholasa; Nuttapol Pakkaranang

doi:10.3934/math.2023489

AIMS Mathematics

2023, Volume 8, Issue 4: 9692-9715. doi: 10.3934/math.2023489

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Research article

A novel class of forward-backward explicit iterative algorithms using inertial techniques to solve variational inequality problems with quasi-monotone operators

1.
Research Group in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
2.
Data Science Research Center, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
3.
Mathematics and Computing Science Program, Faculty of Science and Technology, Phetchabun Rajabhat University, Phetchabun 67000, Thailand
4.
School of Science, University of Phayao, Phayao 56000, Thailand

Received: 13 November 2022 Revised: 02 February 2023 Accepted: 05 February 2023 Published: 21 February 2023
MSC : 47H05, 47H10, 47J20, 47J25, 65J15, 91B50

The theory of variational inequalities is an important tool in physics, engineering, finance, and optimization theory. The projection algorithm and its variants are useful tools for determining the approximate solution to the variational inequality problem. This paper introduces three distinct extragradient algorithms for dealing with variational inequality problems involving quasi-monotone and semistrictly quasi-monotone operators in infinite-dimensional real Hilbert spaces. This problem is a general mathematical model that incorporates a set of applied mathematical models as an example, such as equilibrium models, optimization problems, fixed point problems, saddle point problems, and Nash equilibrium point problems. The proposed algorithms employ both fixed and variable stepsize rules that are iteratively transformed based on previous iterations. These algorithms are based on the fact that no prior knowledge of the Lipschitz constant or any line-search framework is required. To demonstrate the convergence of the proposed algorithms, some simple conditions are used. Numerous experiments have been conducted to highlight the numerical capabilities of algorithms.
- variational inequalities,
- Tseng's extragradient method,
- inertial-type iterative scheme,
- quasi-monotone operator,
- Lipschitz continuous operators
Citation: Bancha Panyanak, Chainarong Khunpanuk, Nattawut Pholasa, Nuttapol Pakkaranang. A novel class of forward-backward explicit iterative algorithms using inertial techniques to solve variational inequality problems with quasi-monotone operators[J]. AIMS Mathematics, 2023, 8(4): 9692-9715. doi: 10.3934/math.2023489

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Abstract

The theory of variational inequalities is an important tool in physics, engineering, finance, and optimization theory. The projection algorithm and its variants are useful tools for determining the approximate solution to the variational inequality problem. This paper introduces three distinct extragradient algorithms for dealing with variational inequality problems involving quasi-monotone and semistrictly quasi-monotone operators in infinite-dimensional real Hilbert spaces. This problem is a general mathematical model that incorporates a set of applied mathematical models as an example, such as equilibrium models, optimization problems, fixed point problems, saddle point problems, and Nash equilibrium point problems. The proposed algorithms employ both fixed and variable stepsize rules that are iteratively transformed based on previous iterations. These algorithms are based on the fact that no prior knowledge of the Lipschitz constant or any line-search framework is required. To demonstrate the convergence of the proposed algorithms, some simple conditions are used. Numerous experiments have been conducted to highlight the numerical capabilities of algorithms.

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