Songnian He, Lili Liu, Xiaolong Qin, A self-adaptive hybrid steepest descent algorithm for solving a class of variational inequalities, Vol. 2018 (2018), Article ID 49, pp. 1-9

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DOI: 10.23952/jnfa.2018.49

Received October 4, 2018; Accepted December 21, 2019; Published December 31, 2018

 

Abstract. Let \mathcal{H} be a real Hilbert space. In this paper, we propose a new self-adaptive hybrid steepest descent algorithm for solving a variational inequality problem VI(Fix(T), F), were F: \mathcal{H}\rightarrow \mathcal{H} is a boundedly Lipschitz continuous (i.e., Lipschitz continuous on any bounded subset of \mathcal{H}) and strongly monotone operator and T: \mathcal{H}\rightarrow \mathcal{H} is a nonexpansive mapping with a nonempty fixed point set Fix(T). The strong convergence of our proposed algorithm is proved and the convergence rate estimation is also obtained. The advantage of our algorithm is that it does not require a priori knowledge of the Lipschitz constant of F on any bounded subset of \mathcal{H} and also the strong monotone coefficient.

 

How to Cite this Article:
Songnian He, Lili Liu, Xiaolong Qin, A self-adaptive hybrid steepest descent algorithm for solving a class of variational inequalities, Journal of Nonlinear Functional Analysis, Vol. 2018 (2018), Article ID 49, pp. 1-9.