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Abstract
In this paper, we investigate the monotone variational inequality in Hilbert spaces. Based on Censor’s subgradient extragradient method, we propose two modified subgradient extragradient algorithms with self-adaptive and inertial techniques for finding the solution of the monotone variational inequality in real Hilbert spaces. Strong convergence analysis of the proposed algorithms have been obtained under some mild conditions.
Highlights
Let H be a real Hilbert space and S ∈ H be a nonempty closed convex subset
We investigate the following variational inequality problem (VIPs): find a point u‡ ∈ S, s.t.〈fu‡, x − u‡〉 ≥ 0, ∀x ∈ S. (1)
Ere are a variety of methods to solve the VIPs, such as regularization method and projection method [31,32,33,34,35,36,37,38,39]
Summary
Let H be a real Hilbert space and S ∈ H be a nonempty closed convex subset. Let f: H ⟶ H be an operator. We investigate the following variational inequality problem (VIPs): find a point u‡ ∈ S, s.t.〈fu‡, x − u‡〉 ≥ 0, ∀x ∈ S. In order to obtain a convergent result, this algorithm requires that the operator f is strongly monotone. In order to overcome this difficulty, Censor et al [43] constructed a half space with subdifferentiation and proposed subgradient extragradient method which is defined by Journal of Mathematics. Motivated and inspired by the above work, in this paper, we continue to investigate iterative algorithms for solving the monotone variational inequality in Hilbert spaces. We construct two modified subgradient extragradient algorithms for finding the solution of the monotone variational inequality. We prove that the proposed algorithms converge strongly to a solution of the monotone variational inequality.
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