Extragradient method and golden ratio method for equilibrium problems on Hadamard manifolds

ABSTRACT

In this paper, we present two algorithms for solving equilibrium problems on Hadamard manifolds. The two algorithms use the extragradient model and the golden ratio model, respectively, which are two classical models for solving the equilibrium problem in linear space. In each iteration, the step sizes of the two algorithms only depend on the value of the initial parameters and the information of the current iteration. Moreover, compared with the first algorithm, the second algorithm only needs to solve one quadratic programming problem per iteration, which can greatly reduce the computational complexity of the algorithm when the structure of the feasible region is complex. Under mild conditions, we prove that the sequence generated by each algorithm converges to some equilibrium point. Also, we use an experiment to verify the effectiveness of the algorithms.

Keywords:

• Hadamard manifolds
• equilibrium problem
• extragradient method
• golden ratio method
• Lipschitz-type condition

2010 AMS Subject Classifications:

• 32Q05
• 47J20
• 90C25
• 90C33
• 90C52

Acknowledgements

The authors are grateful to the editors and anonymous reviewers for their valuable comments and encouragement.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The project was supported by the National Natural Science Foundation of China [grant numbers 61877046,11602184], and the third author was supported by the Hongliu Foundation of First-class Disciplines of Lanzhou University of Technology.