Optimization Based Methods for Solving the Equilibrium Problems with Applications in Variational Inequality Problems and Solution of Nash Equilibrium Models
Abstract
:1. Introduction
2. Preliminaries
- (1)
- strongly monotone if:
- (2)
- monotone if:
- (3)
- strongly pseudomonotone if:
- (4)
- pseudomonotone if:
- (5)
- satisfying the Lipschitz-type condition on K if there exist two numbers such that:
- (i)
- with
- (ii)
- (i)
- For exists;
- (ii)
- Every weak cluster point of the sequence belongs to K.
3. An Algorithm for the Pseudomonotone Equilibrium Problem and Its Convergence Analysis
Algorithm 1 Two-step proximal iterative method for pseudomonotone EP. |
|
- (A1)
- f is pseudomonotone on K with for all ;
- (A2)
- f satisfies the Lipschitz-type condition on with and ;
- (A3)
- for each and with ;
- (A4)
- is subdifferentiable and convex on for each
4. An Algorithm for the Strongly Pseudomonotone Equilibrium Problem and Its Convergence Analysis
- (B1)
- f is strongly pseudomonotone on K with ;
- (B2)
- f satisfies the Lipschitz-type condition on with and ;
- (B3)
- is sub-differentiable and convex on for each fixed
Algorithm 2 Two-step proximal iterative method for strongly pseudomonotone EP. |
|
5. Application to Variational Inequality Problems
- (1)
- strongly pseudomonotone on K if
- (2)
- pseudomonotone on K if
- (3)
- satisfying being L-Lipschitz continuous on K if
- (G1)
- G is strongly pseudomonotone on K, and is a nonempty solution set;
- (G2)
- G is pseudomonotone on K, and is a nonempty solution set;
- (G3)
- G is L-Lipschitz continuous upon K for positive constant
- (G4)
- for every and satisfying
- (i)
- Choose , and Set:
- (ii)
- Assume , and are known for Construct a half space:
- (i)
- Choose , and Set:
- (ii)
- Assume that and are known for Construct a half space:
6. Numerical Experiments
6.1. Nash–Cournot Equilibrium Model of Electricity Markets
Algorithm 1 Comparison with Other Existing Methods
6.2. Nash–Cournot Oligopolistic Equilibrium Model
Algorithm 2 Comparison with Other Existing Methods
7. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Unit j | ||||||
---|---|---|---|---|---|---|
1 | 0.0400 | 2.00 | 0.00 | 2.0000 | 1.0000 | 25.0000 |
2 | 0.0350 | 1.75 | 0.00 | 1.7500 | 1.0000 | 28.5714 |
3 | 0.1250 | 1.00 | 0.00 | 1.0000 | 1.0000 | 8.0000 |
4 | 0.0116 | 3.25 | 0.00 | 3.2500 | 1.0000 | 86.2069 |
5 | 0.0500 | 3.00 | 0.00 | 3.0000 | 1.0000 | 20.0000 |
6 | 0.0500 | 3.00 | 0.00 | 3.0000 | 1.0000 | 20.0000 |
j | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | |
80 | 80 | 50 | 55 | 30 | 40 |
u | n | ||||||
---|---|---|---|---|---|---|---|
Algo1 | 0.16 | 0.012 | 50 | 614 | 15.511285 | ||
Algo1 | 0.11 | 0.012 | 50 | 659 | 17.624634 | ||
Algo1 | 0.06 | 0.012 | 50 | 704 | 17.143282 | ||
Algo1 | 0.01 | 0.012 | 50 | 748 | 18.846679 | ||
Algo1 | 0.001 | 0.012 | 50 | 756 | 19.411861 |
u | n | ||||||
---|---|---|---|---|---|---|---|
Algo1 | 0.16 | 0.012 | 0.1 | 756 | 23.649384 | ||
Algo1 | 0.16 | 0.012 | 1 | 698 | 19.827807 | ||
Algo1 | 0.16 | 0.012 | 5 | 647 | 18.198626 | ||
Algo1 | 0.16 | 0.012 | 10 | 630 | 17.335370 | ||
Algo1 | 0.16 | 0.012 | 50 | 614 | 16.943758 |
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Rehman, H.u.; Kumam, P.; Argyros, I.K.; Shutaywi, M.; Shah, Z. Optimization Based Methods for Solving the Equilibrium Problems with Applications in Variational Inequality Problems and Solution of Nash Equilibrium Models. Mathematics 2020, 8, 822. https://doi.org/10.3390/math8050822
Rehman Hu, Kumam P, Argyros IK, Shutaywi M, Shah Z. Optimization Based Methods for Solving the Equilibrium Problems with Applications in Variational Inequality Problems and Solution of Nash Equilibrium Models. Mathematics. 2020; 8(5):822. https://doi.org/10.3390/math8050822
Chicago/Turabian StyleRehman, Habib ur, Poom Kumam, Ioannis K. Argyros, Meshal Shutaywi, and Zahir Shah. 2020. "Optimization Based Methods for Solving the Equilibrium Problems with Applications in Variational Inequality Problems and Solution of Nash Equilibrium Models" Mathematics 8, no. 5: 822. https://doi.org/10.3390/math8050822
APA StyleRehman, H. u., Kumam, P., Argyros, I. K., Shutaywi, M., & Shah, Z. (2020). Optimization Based Methods for Solving the Equilibrium Problems with Applications in Variational Inequality Problems and Solution of Nash Equilibrium Models. Mathematics, 8(5), 822. https://doi.org/10.3390/math8050822