An Accelerated Extragradient Method for Solving Pseudomonotone Equilibrium Problems with Applications
Abstract
:1. Introduction
2. Preliminaries
- (1)
- strongly monotone if:
- (2)
- monotone if:
- (3)
- strongly pseudomonotone if:
- (4)
- pseudomonotone if:
- (i)
- For every exists;
- (ii)
- each sequentially weak cluster limit point of belongs to set
- (1)
- for all , and f is pseudomonotone on
- (2)
- f satisfies the Lipschitz-type condition on through and
- (3)
- for each and satisfy ;
- (4)
- is subdifferentiable and convex on for each
3. An Accelerated Method for Pseudomonotone Equilibrium Problems and Its Convergence Analysis
Algorithm 1 (Accelerated method for pseudomonotone equilibrium problems) |
|
4. Applications of the Main Results
- (i)
- L-Lipschitz continuous on if:
- (ii)
- pseudomonotone on if:
- (G1)
- G is pseudomonotone on with solution set ;
- (G2)
- G is L-Lipschitz continuous on with ;
- (G3)
- and satisfy
- (i)
- Choose , and a sequence such that:
- (ii)
- Let satisfy such that:
- (iii)
- Set , and compute:
- (iv)
- Next, stepsize is obtained as follows:
- (i)
- (ii)
- sequentially weakly continuous on if:
- (i)
- Choose , and a sequence such that:
- (ii)
- Let satisfy such that:
- (iii)
- Set , and compute:
- (iv)
- Next, stepsize is obtained as follows:
5. Numerical Experiments
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Number of Iterations | CPU Time in Seconds | |||||
---|---|---|---|---|---|---|
Alg1 | Alg2 | Alg1 | Alg2 | |||
0.40 | 0.22 | 0.45 | 14 | 2 | 0.9134 | 0.5424 |
0.40 | 0.17 | 0.35 | 18 | 2 | 0.8615 | 0.5223 |
0.40 | 0.12 | 0.32 | 20 | 2 | 1.0815 | 0.5112 |
0.40 | 0.07 | 0.25 | 22 | 2 | 1.4219 | 0.5367 |
0.40 | 0.02 | 0.05 | 26 | 2 | 1.7329 | 0.5181 |
Number of Iterations | CPU Time in Seconds | |||||
---|---|---|---|---|---|---|
Alg1 | Alg2 | Alg1 | Alg2 | |||
0.80 | 0.22 | 0.16 | 21 | 2 | 1.0482 | 0.0811 |
0.60 | 0.22 | 0.16 | 15 | 2 | 0.8676 | 0.0626 |
0.40 | 0.22 | 0.16 | 12 | 2 | 1.0545 | 0.0791 |
0.20 | 0.22 | 0.16 | 11 | 2 | 0.09923 | 0.0892 |
0.05 | 0.22 | 0.16 | 19 | 2 | 1.09151 | 0.0788 |
Number of Iterations | CPU Time in Seconds | |||
---|---|---|---|---|
Alg1 | Alg2 | Alg1 | Alg2 | |
0.10 | 42 | 8 | 1.5851 | 0.2822 |
0.15 | 29 | 8 | 1.3148 | 0.2433 |
0.40 | 28 | 7 | 1.1278 | 0.2662 |
0.55 | 37 | 8 | 1.2211 | 0.2745 |
0.70 | 47 | 8 | 1.7188 | 0.2279 |
0.85 | 49 | 8 | 1.6188 | 0.2179 |
Number of Iterations | CPU Time in Seconds | |||
---|---|---|---|---|
Alg1 | Alg2 | Alg1 | Alg2 | |
22 | 7 | 1.0321 | 0.1634 | |
24 | 7 | 1.0945 | 0.1858 | |
25 | 7 | 1.0328 | 0.2012 | |
30 | 7 | 1.0517 | 0.2020 | |
35 | 8 | 1.0919 | 0.1428 |
Number of Iterations | CPU Time in Seconds | |||
---|---|---|---|---|
Alg1 | Alg2 | Alg1 | Alg2 | |
37 | 4 | 4.9566 | 0.5460 | |
41 | 2 | 5.2378 | 0.4331 | |
48 | 3 | 6.4556 | 0.3945 | |
51 | 6 | 6.6756 | 0.4945 | |
56 | 6 | 6.8713 | 0.5108 |
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Wairojjana, N.; Rehman, H.u.; Argyros, I.K.; Pakkaranang, N. An Accelerated Extragradient Method for Solving Pseudomonotone Equilibrium Problems with Applications. Axioms 2020, 9, 99. https://doi.org/10.3390/axioms9030099
Wairojjana N, Rehman Hu, Argyros IK, Pakkaranang N. An Accelerated Extragradient Method for Solving Pseudomonotone Equilibrium Problems with Applications. Axioms. 2020; 9(3):99. https://doi.org/10.3390/axioms9030099
Chicago/Turabian StyleWairojjana, Nopparat, Habib ur Rehman, Ioannis K. Argyros, and Nuttapol Pakkaranang. 2020. "An Accelerated Extragradient Method for Solving Pseudomonotone Equilibrium Problems with Applications" Axioms 9, no. 3: 99. https://doi.org/10.3390/axioms9030099
APA StyleWairojjana, N., Rehman, H. u., Argyros, I. K., & Pakkaranang, N. (2020). An Accelerated Extragradient Method for Solving Pseudomonotone Equilibrium Problems with Applications. Axioms, 9(3), 99. https://doi.org/10.3390/axioms9030099