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

Abstract

Abstract 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 ) $(0, \frac{1}{ \| A \|^{2} } )$ . 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 ∞ $\{ \mu _{n} \}_{n=1}^{\infty }$ which does not require computation of ∥ A ∥ $\| A \|$ and apply the adaptive step-size rule on { μ n } n = 1 ∞ $\{ \mu _{n} \}_{n=1}^{\infty }$ 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.

Keywords

• Split equilibrium problems
• Extragradient algorithm
• Self-adaptive step-sizes
• Pseudomonotone equilibrium problems