On the Parallel Subgradient Extragradient Rule for Solving Systems of Variational Inequalities in Hadamard Manifolds
Abstract
:1. Introduction
2. Preliminaries
- (i)
- ;
- (ii)
- ;
- (iii)
- .
- (i)
- and for all .
- (ii)
- If and , then .
- (iii)
- Given and , if and , then .
- (iv)
- For each , the map , defined by , is continuous on .
- (i)
- solves the VIP (4);
- (ii)
- for some ;
- (iii)
- for all ;
- (iv)
- , with .
- (i)
- Assume that (resp., ) are three angles of (resp., ) at three vertices (resp., ). Then, the inequalities hold: .
- (ii)
- Assume that the point z lies in the geodesic joining u to v and is its comparison point in the interval satisfying and . Then, the inequality holds: .
3. Algorithms and Convergence Criteria
3.1. The First Parallel Algorithm
Algorithm 1: The first parallel algorithm for the SVI. |
Iteration Steps: Compute below: |
Step 1. Compute |
Step 2. Construct |
and calculate |
Step 3. Calculate |
Iteration Steps: Compute below: |
Step 1. Compute |
Step 2. Construct |
and calculate |
Step 3. Calculate |
3.2. The Second Parallel Algorithm
, and compute |
Iteration Steps: Compute and below: |
Step 1. Construct |
and calculate |
Step 2. Calculate |
where |
and for , and compute |
Iteration Steps: Compute and below: |
Step 1. Construct |
and calculate |
Step 2. Calculate |
where |
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Wang, C.-Y.; Ceng, L.-C.; He, L.; Hu, H.-Y.; Zhao, T.-Y.; Wang, D.-Q.; Fan, H.-L. On the Parallel Subgradient Extragradient Rule for Solving Systems of Variational Inequalities in Hadamard Manifolds. Symmetry 2021, 13, 1496. https://doi.org/10.3390/sym13081496
Wang C-Y, Ceng L-C, He L, Hu H-Y, Zhao T-Y, Wang D-Q, Fan H-L. On the Parallel Subgradient Extragradient Rule for Solving Systems of Variational Inequalities in Hadamard Manifolds. Symmetry. 2021; 13(8):1496. https://doi.org/10.3390/sym13081496
Chicago/Turabian StyleWang, Chun-Yan, Lu-Chuan Ceng, Long He, Hui-Ying Hu, Tu-Yan Zhao, Dan-Qiong Wang, and Hong-Ling Fan. 2021. "On the Parallel Subgradient Extragradient Rule for Solving Systems of Variational Inequalities in Hadamard Manifolds" Symmetry 13, no. 8: 1496. https://doi.org/10.3390/sym13081496
APA StyleWang, C.-Y., Ceng, L.-C., He, L., Hu, H.-Y., Zhao, T.-Y., Wang, D.-Q., & Fan, H.-L. (2021). On the Parallel Subgradient Extragradient Rule for Solving Systems of Variational Inequalities in Hadamard Manifolds. Symmetry, 13(8), 1496. https://doi.org/10.3390/sym13081496