New extragradient methods for solving variational inequality problems and fixed point problems
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In this paper, we introduce two new iterative algorithms for finding a common element of the set of fixed points of a quasi-nonexpansive mapping and the set of solutions of the variational inequality problem with a monotone and Lipschitz continuous mapping in real Hilbert spaces, by combining a modified Tseng’s extragradient scheme with the Mann approximation method. We prove weak and strong convergence theorems for the sequences generated by these iterative algorithms. The main advantages of our algorithms are that the construction of solution approximations and the proof of convergence of the algorithms are performed without the prior knowledge of the Lipschitz constant of cost operators. Finally, we provide numerical experiments to show the efficiency and advantage of the proposed algorithms.
KeywordsVariational inequality problem fixed point problem extragradient method subgradient extragradient method Tseng’s extragradient method Mann method Halpern method
Mathematics Subject Classification47H10 47J25 47H45 65J15
The authors would like to thank Professor Simeon Reich and the referee(s) for their comments on the manuscript which helped us very much in improving and presenting the original version of this paper. The second author was partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the project 101.01-2017.315.
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