On the Parallel Subgradient Extragradient Rule for Solving Systems of Variational Inequalities in Hadamard Manifolds

In a Hadamard manifold, let the VIP and SVI represent a variational inequality problem and a system of variational inequalities, respectively, where the SVI consists of two variational inequalities which are of symmetric structure mutually. This article designs two parallel algorithms to solve the S...

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Published inSymmetry (Basel) Vol. 13; no. 8; p. 1496
Main Authors Wang, Chun-Yan, Ceng, Lu-Chuan, He, Long, Hu, Hui-Ying, Zhao, Tu-Yan, Wang, Dan-Qiong, Fan, Hong-Ling
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.08.2021
Subjects
Algorithms
convex set
Fields (mathematics)
Hadamard manifold
Inequalities
Manifolds
Mapping
Mathematical functions
Mathematical programming
monotone vector fields
Optimization
parallel subgradient extragradient rule
system of variational inequalities
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Summary:In a Hadamard manifold, let the VIP and SVI represent a variational inequality problem and a system of variational inequalities, respectively, where the SVI consists of two variational inequalities which are of symmetric structure mutually. This article designs two parallel algorithms to solve the SVI via the subgradient extragradient approach, where each algorithm consists of two parts which are of symmetric structure mutually. It is proven that, if the underlying vector fields are of monotonicity, then the sequences constructed by these algorithms converge to a solution of the SVI. We also discuss applications of these algorithms for approximating solutions to the VIP. Our theorems complement some recent and important ones in the literature.
ISSN:2073-8994
2073-8994
DOI:10.3390/sym13081496