A modified projection and contraction method for solving a variational inequality problem in Hilbert spaces

Limei Xue; Jianmin Song; Shenghua Wang; Limei Xue; Jianmin Song; Shenghua Wang

doi:10.3934/math.2025279

AIMS Mathematics

2025, Volume 10, Issue 3: 6128-6143. doi: 10.3934/math.2025279

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A modified projection and contraction method for solving a variational inequality problem in Hilbert spaces

1.
School of Mathematics and Science, Hebei GEO University, Shijiazhuang, 050031, China
2.
Department of Mathematics and Physics, North China Electric Power University, Baoding 071003, China

Received: 21 November 2024 Revised: 01 March 2025 Accepted: 10 March 2025 Published: 19 March 2025
MSC : 47H05, 47J20, 65K15

In this paper, we proposed a modified projection and contraction method for solving a variational inequality problem in Hilbert spaces. In the existing projection and contraction methods for solving the variational inequality problem, the sequence $ \{\beta_n\} $ has a similar computation manner, which is computed in a self-adaptive manner, but the sequence $ \{\beta_n\} $ in our method is a sequence of numbers in (0, 1) given in advance, which is the main difference of our method with the existing projection and contraction methods. A line search is used to deal with the unknown Lipschitz constant of the mapping. The strong convergence of the proposed method is proved under certain conditions. Finally, some numerical examples are presented to illustrate the effectiveness of our method and compare the computation results with some related methods in the literature. The numerical results show that our method has an obvious competitive advantage compared with the related methods.
- variational inequality,
- projection method,
- golden ratio,
- Hilbert space
Citation: Limei Xue, Jianmin Song, Shenghua Wang. A modified projection and contraction method for solving a variational inequality problem in Hilbert spaces[J]. AIMS Mathematics, 2025, 10(3): 6128-6143. doi: 10.3934/math.2025279

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Abstract

In this paper, we proposed a modified projection and contraction method for solving a variational inequality problem in Hilbert spaces. In the existing projection and contraction methods for solving the variational inequality problem, the sequence $ \{\beta_n\} $ has a similar computation manner, which is computed in a self-adaptive manner, but the sequence $ \{\beta_n\} $ in our method is a sequence of numbers in (0, 1) given in advance, which is the main difference of our method with the existing projection and contraction methods. A line search is used to deal with the unknown Lipschitz constant of the mapping. The strong convergence of the proposed method is proved under certain conditions. Finally, some numerical examples are presented to illustrate the effectiveness of our method and compare the computation results with some related methods in the literature. The numerical results show that our method has an obvious competitive advantage compared with the related methods.

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