A New Alternative Regularization Method for Solving Generalized Equilibrium Problems
Abstract
:1. Introduction
Algorithm 1: Tseng’s extragradient method (TEGM) |
Initialization: Set and let be arbitrary. Step 1. Given , compute
where is chosen to be the largest satisfying the following: If , then stop and is the solution of the VIP (3). Otherwise, go to Step 2. Step 2. Compute
Set and return to Step 1. |
Algorithm 2: Tseng’s extragradient method with viscosity technique (TEGMV) |
Initialization: Set and let be arbitrary. Step 1. Given , compute
where is chosen to be the largest satisfying the following: If , then stop and is the solution of VIP. Otherwise, go to Step 2. Step 2. Compute
Set and return to Step 1. |
2. Preliminaries
- (1)
- monotone on C, if
- (2)
- L- Lipschitz continuous on C, if there exists such that
- (A1)
- , ;
- (A2)
- is monotone;
- (A3)
- for all is convex and lower semicontinuous;
- (A4)
- for all
- (A4′)
- for every and satisfy
- (A4″)
- is jointly weakly upper semicontinuous on in the sense that, if and , converges weakly to x and y, respectively, then as (see, e.g., [20]).
- (i)
- is single-valued;
- (ii)
- is a firmly nonexpansive mapping, i.e., for all
- (iii)
- (iv)
- is nonempty closed and convex.
- (i)
- is single-valued;
- (ii)
- is a firmly nonexpansive mapping;
- (iii)
- (iv)
- is nonempty, closed and convex.
3. Main Results
- (i)
- for each the has a unique solution ;
- (ii)
- , and , ;
- (iii)
- , .
- (C1)
- and ;
- (C2)
- .
Algorithm 3: The Tseng’s extragradient method with regularization (TEGMR) |
Initialization: Set and let be arbitrary. Step 1. Given , compute
where is chosen to be the largest satisfying the following: Step 2. Compute
Set and return to Step 1. |
4. Application to Split Minimization Problems
Algorithm 4: The Tseng’s extragradient method with regularization for minimization problems |
Initialization: Set and let be arbitrary. Step 1. Given , compute
where is chosen to be the largest satisfying the following: Step 2. Compute
Set and return to Step 1. |
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Song, Y.; Bazighifan, O. A New Alternative Regularization Method for Solving Generalized Equilibrium Problems. Mathematics 2022, 10, 1350. https://doi.org/10.3390/math10081350
Song Y, Bazighifan O. A New Alternative Regularization Method for Solving Generalized Equilibrium Problems. Mathematics. 2022; 10(8):1350. https://doi.org/10.3390/math10081350
Chicago/Turabian StyleSong, Yanlai, and Omar Bazighifan. 2022. "A New Alternative Regularization Method for Solving Generalized Equilibrium Problems" Mathematics 10, no. 8: 1350. https://doi.org/10.3390/math10081350
APA StyleSong, Y., & Bazighifan, O. (2022). A New Alternative Regularization Method for Solving Generalized Equilibrium Problems. Mathematics, 10(8), 1350. https://doi.org/10.3390/math10081350