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Original Articles

Two hybrid algorithms for solving split equilibrium problems

Pages 561-583 | Received 13 Sep 2016, Accepted 25 Jan 2017, Published online: 28 Feb 2017
 

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.

2010 AMS SUBJECT CLASSIFICATIONS:

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 λ1k[m,0],λ2k[1,m] for all k=1,,m. Set Qˆ1, Qˆ2 as two diagonal matrixes with eigenvalues {λ1k}k=1m and {λ2k}k=1m, respectively. Then, we make a positive semidefinite matrix N and a negative semidefinite matrix T by using full random orthogonal matrixes with Qˆ2 and Qˆ1, respectively. Finally, we set M=NT

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