Abstract
In the setting of Hilbert space, a modified subgradient extragradient method is proposed for solving Lipschitz-continuous and monotone variational inequalities defined on a level set of a convex function. Our iterative process is relaxed and self-adaptive, that is, in each iteration, calculating two metric projections onto some half-spaces containing the domain is involved only and the step size can be selected in some adaptive ways. A weak convergence theorem for our algorithm is proved. We also prove that our method has O(frac{1}{n}) convergence rate.
Highlights
Let H be a real Hilbert space with inner product ·, · and norm ·
The main purpose of this paper is to propose an improved subgradient extragradient method for solving the Lipschitz-continuous and monotone variational inequalities defined on a level set of a convex function [ ], that is, C := {x ∈ H | c(x) ≤ } and c : H → R
6 Results and discussion Since the modified subgradient extragradient method proposed in this paper is relaxed and self-adaptive, it is implemented
Summary
Let H be a real Hilbert space with inner product ·, · and norm ·. In order to overcome this weakness, Censor et al developed the subgradient extragradient method in Euclidean space [ ], in which the second projection in In [ , ], Censor et al studied the subgradient extragradient method for solving the VIP in Hilbert spaces. They proved the weak convergence theorem under the assumption that f is a Lipschitzian continuous and monotone mapping. The main purpose of this paper is to propose an improved subgradient extragradient method for solving the Lipschitz-continuous and monotone variational inequalities defined on a level set of a convex function [ ], that is, C := {x ∈ H | c(x) ≤ } and c : H → R is a convex function. We our algorithm convergence rate in the last section
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