Hybrid Inertial Accelerated Algorithms for Solving Split Equilibrium and Fixed Point Problems
Abstract
:1. Introduction
Algorithm 1: The general alternative regularization algorithm. |
Initialization: Set and let be arbitrary. Step 1. Compute: , where is chosen to be the largest satisfying the following: If , then set and go to Step 3. Otherwise, do Step 2. Step 2. Compute: Step 3. Compute , where Set and return to Step 1. |
Algorithm 2: The new inertial Tseng’s extragradient iterative algorithm. |
Initialization: Set , and let be arbitrary. Step 1. Given and , , compute
Step 2. Compute , where is chosen to be the largest satisfying the following: If or , then stop. Otherwise, go to Step 3. Step 3. Compute
Step 4. Compute . Set and return to Step 1. |
2. Preliminaries
- (1)
- nonexpansive, if
- (2)
- γ-contractive, if there exists such that
- (3)
- quasi-nonexpansive, if and
- (4)
- α-strongly pseudo-contractive, if there exists a constant , such that
- (5)
- pseudo-monotone, if
- (6)
- k-demicontractive, if and there exists , such that
- (7)
- k-demimetric, if and there exists , such that
- (A1)
- , ;
- (A2)
- F is monotone, i.e., , ;
- (A3)
- For all
- (A4)
- For all is convex and lower semicontinuous.
- (1)
- if
- (2)
- T is a quasi-nonexpansive mapping for ;
- (3)
- is a closed convex subset of .
- (i)
- ,
- (ii)
- ,
- (iii)
- .
- (i)
- is single-valued;
- (ii)
- is a firmly nonexpansive mapping, i.e., for all
- (iii)
- (iv)
- is closed and convex.
3. Main Results
- (C1)
- , and ;
- (C2)
- , ;
- (C3)
- and , where is generated by Algorithm 3;
- (C4)
- and .
Algorithm 3: The hybrid inertial accelerated algorithm. |
Initialization: Set and let be arbitrary. Step 1. Given and , compute
Step 2. Compute , where is chosen to be the largest satisfying the following: If , then stop and is a solution of the SEP. Otherwise, Step 3. Compute
Step 4. Compute , where Set and return to Step 1. |
- (1)
- ;
- (2)
- .
- (1)
- If is a solution of the , then ;
- (2)
- Assume that . Then, is a solution of the .
- The proofs of our main results are simple and different from those in early and recent literature manly due to Lemma 8. More precisely, Lemma 8 together with Lemma 13 presents an interesting and simple method to prove under the conditions and .
- Theorem 2 extends, improves, and develops the corresponding results in [12,13,14,17] from finding a solution for the VIP, a solution for the the EP, or a common solution for the SEP and the fixed point problem for demicontractive mappings to finding a common solution for the SEP and the fixed point problem for demimetric mappings. Moreover, our proof is also different from the one used in those paper.
Algorithm 4: The hybrid inertial accelerated algorithm for the SVIP and the fixed point problem. |
Initialization: Set and let be arbitrary. Step 1. Given and , compute
Step 2. Compute , where is chosen to be the largest satisfying the following: If , then stop and is a solution of the SVIP. Otherwise, Step 3. Compute
Step 4. Compute , where Set and return to Step 1. |
4. Application to Split Minimization Problems
Algorithm 5: The general alternative regularization algorithm. |
Initialization: Set and let be arbitrary. Step 1.
Given and , compute
Step 2.
Compute , where is chosen to be the largest satisfying the following:
If , then stop and is a solution of the . Otherwise, Step 3.
Compute
Step 4.
Compute , where Set and return to
Step 1. |
5. Numerical Examples
6. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Err | Algorithm 1 | Algorithm 2 | Algorithm 3 | |||
---|---|---|---|---|---|---|
1.0947111779190 | 25 | 9.8932802287582 | 11 | 2.5084018650195 | 9 | |
1.0947111779190 | 25 | 5.0103649467704 | 14 | 2.5084018650195 | 9 | |
1.0732462528617 | 26 | 6.8899484598943 | 16 | 3.3005287697625 | 10 |
Err | Algorithm 1 | Algorithm 2 | Algorithm 3 | |||
---|---|---|---|---|---|---|
3.8495238046047 | 36 | 5.5670721630783 | 14 | 3.8440832718020 | 12 | |
2.6366601401402 | 37 | 7.6554982887714 | 16 | 3.8440832718020 | 12 | |
2.6366601401402 | 37 | 3.9225807673322 | 19 | 3.8440832718020 | 13 |
Err | Algorithm 1 | Algorithm 2 | Algorithm 3 | |||
---|---|---|---|---|---|---|
6.7539211424883 | 31 | 5.2608831941090 | 13 | 7.1926227474796 | 10 | |
5.3602548749907 | 32 | 7.2227962115800 | 15 | 8.5626461279519 | 11 | |
5.3602548749907 | 32 | 9.9463481554871 | 17 | 8.5626461279519 | 11 |
Err | Algorithm 1 | Algorithm 2 | Algorithm 3 | |||
---|---|---|---|---|---|---|
3.0475500166712 | 30 | 3.7577737100778 | 13 | 9.5530385596659 | 10 | |
3.0475500166712 | 30 | 5.1591401511286 | 15 | 9.5530385596659 | 10 | |
2.4979918169436 | 31 | 7.1045343967765 | 17 | 1.1372664951983 | 11 |
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Song, Y. Hybrid Inertial Accelerated Algorithms for Solving Split Equilibrium and Fixed Point Problems. Mathematics 2021, 9, 2680. https://doi.org/10.3390/math9212680
Song Y. Hybrid Inertial Accelerated Algorithms for Solving Split Equilibrium and Fixed Point Problems. Mathematics. 2021; 9(21):2680. https://doi.org/10.3390/math9212680
Chicago/Turabian StyleSong, Yanlai. 2021. "Hybrid Inertial Accelerated Algorithms for Solving Split Equilibrium and Fixed Point Problems" Mathematics 9, no. 21: 2680. https://doi.org/10.3390/math9212680
APA StyleSong, Y. (2021). Hybrid Inertial Accelerated Algorithms for Solving Split Equilibrium and Fixed Point Problems. Mathematics, 9(21), 2680. https://doi.org/10.3390/math9212680