A new extragradient algorithm with adaptive step-size for solving split equilibrium problems

• Published in: Journal of inequalities and applications Vol. 2021; no. 1; pp. 1 - 14
• Main Authors: Suleiman, Yusuf I., Kumam, Poom, Rehman, Habib ur, Kumam, Wiyada
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 02.08.2021; Springer Nature B.V; SpringerOpen
• Subjects: Adaptive algorithms; Algorithms; Analysis; Applications of Mathematics; Computation; Convergence; Equilibrium; Extragradient algorithm; Iterative methods; Mathematics; Mathematics and Statistics; Numerical analysis; Pseudomonotone equilibrium problems; Robustness (mathematics); Self-adaptive step-sizes; Split equilibrium problems; Extragradient algorithm; 47H09; Split equilibrium problems; 65K10; Self-adaptive step-sizes; 49M37; 47H05; Pseudomonotone equilibrium problems
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-021-02668-x

He (J. Inequal. Appl. 2012:Article ID 162 2012 ) introduced the proximal point CQ algorithm (PPCQ) for solving the split equilibrium problem (SEP). However, the PPCQ converges weakly to a solution of the SEP and is restricted to monotone bifunctions. In addition, the step-size used in the PPCQ is a fixed constant μ in the interval ( 0 , 1 ∥ A ∥ 2 ) . This often leads to excessive numerical computation in each iteration, which may affect the applicability of the PPCQ. In order to overcome these intrinsic drawbacks, we propose a robust step-size { μ n } n = 1 ∞ which does not require computation of ∥ A ∥ and apply the adaptive step-size rule on { μ n } n = 1 ∞ in such a way that it adjusts itself in accordance with the movement of associated components of the algorithm in each iteration. Then, we introduce a self-adaptive extragradient-CQ algorithm (SECQ) for solving the SEP and prove that our proposed SECQ converges strongly to a solution of the SEP with more general pseudomonotone equilibrium bifunctions. Finally, we present a preliminary numerical test to demonstrate that our SECQ outperforms the PPCQ.