Two hybrid algorithms for solving split equilibrium problems

ABSTRACT

The paper considers split equilibrium problems (EPs) in Hilbert spaces and proposes two hybrid algorithms for finding their solution approximations. Three methods including the diagonal subgradient method, the projection method and the proximal method have been used to design the algorithms. Using the diagonal subgradient method for EPs has allowed us to reduce complex computations on bifunctions and feasible sets. The first algorithm is designed with two projections on feasible set and with the prior knowledge of operator norm while the second algorithm is simpler in computations where only one projection on feasible set needs to be implemented and the information of operator norm is not necessary to construct solution approximations. The strongly convergent theorems are established under suitable assumptions imposed on equilibrium bifunctions. The computational performance of the proposed algorithms over existing methods is also illustrated by several preliminary numerical experiments.

KEYWORDS:

• Split equilibrium problem
• split inverse problem
• projection method
• diagonal subgradient method
• proximal method

2010 AMS SUBJECT CLASSIFICATIONS:

• 90C33
• 68W10
• 65K10

Acknowledgments

The author would like to thank the Associate Editor and three anonymous referees for their valuable comments and suggestions which helped us very much in improving the original version of this paper. The guidance of Profs. P. K. Anh and L. D. Muu is gratefully acknowledged.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. Choose randomly ${\lambda }_{1k}\in [-m,0],{\lambda }_{2k}\in [1,m]$ for all $k=1,\dots ,m$. Set ${\hat{Q}}_1$, ${\hat{Q}}_2$ as two diagonal matrixes with eigenvalues $\{{\lambda }_{1k}{\}}_{k=1}^m$ and $\{{\lambda }_{2k}{\}}_{k=1}^m$, respectively. Then, we make a positive semidefinite matrix N and a negative semidefinite matrix T by using full random orthogonal matrixes with ${\hat{Q}}_2$ and ${\hat{Q}}_1$, respectively. Finally, we set M=N−T