Abstract
In this paper, two algorithms are proposed for a class of pseudomonotone and strongly pseudomonotone equilibrium problems. These algorithms can be viewed as a extension of the paper title, the extragradient algorithm with inertial effects for solving the variational inequality proposed by Dong et al. (Optimization 65:2217–2226, 2016. https://doi.org/10.1080/02331934.2016.1239266). The weak convergence of the first algorithm is well established based on the standard assumption imposed on the cost bifunction. We provide a strong convergence for the second algorithm without knowing the strongly pseudomonoton and the Lipschitz-type constants of cost bifunction. The practical interpretation of a second algorithm is that the algorithm uses a sequence of step sizes that is converging to zero and non-summable. Numerical examples are used to assist the well-established convergence result, and we see that the suggested algorithm has a competitive advantage over time of execution and the number of iterations.
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Notes
We randomly choose two diagonal matrices \(A_{1}\) and \(A_{2}\) with entries from [0, m] and \([-m, 0]\), respectively. Two random orthogonal matrices \(B_{1}\) and \(B_{2}\) are able to generate a positive semi definite matrix \(M_{1}=B_{1}A_{1}B_{1}^{T}\) and negative semi definite matrix \(M_{2}=B_{2}A_{2}B_{2}^{T}\). Finally, set \(Q=M_{1}+M_{1}^{T}\), \(S=M_{2}+M_{2}^{T}\) and \(P=Q-S\).
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Acknowledgements
The authors like to say thanks to Professor Gabor Kassay and Dr Nguyen The Vinh to providing instruction regarding the Matlab program. Finally, this project was supported by Theoretical and Computational Science (TaCS) Center, Faculty of Science, KMUTT. Habib ur Rehman was supported by the Petchra Pra Jom Klao Doctoral Scholarship, Academic for Ph.D. Program at KMUTT (Grant No. 39/2560).
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Communicated by Orizon Pereira Ferreira.
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Rehman, H.u., Kumam, P., Abubakar, A.B. et al. The extragradient algorithm with inertial effects extended to equilibrium problems. Comp. Appl. Math. 39, 100 (2020). https://doi.org/10.1007/s40314-020-1093-0
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DOI: https://doi.org/10.1007/s40314-020-1093-0
Keywords
- Extragradient method
- Inertial methods
- Equilibrium problem
- Strongly pseudomonotone bifunction
- Lipschitz-type condition