Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space

Censor, Yair; Gibali, Aviv; Reich, Simeon

doi:10.1080/02331934.2010.539689

Cited by 314 publications

(111 citation statements)

References 9 publications

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“…A popular algorithm for solving this problem is extragradient method introduced by Korpelevich [22]. This method has been improved by several researchers; see e.g., [9,12,14,27] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Strong convergence of implicit and explicit iterations for a class of variational inequalities in Banach spaces

Ceng¹,

Wen²

2017

J. Nonlinear Sci. Appl.

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In this paper, we introduce and analyze implicit and explicit iteration methods for solving a variational inequality problem over the set of common fixed points of an infinite family of nonexpansive mappings on a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Strong convergence results are given. Our results improve and extend the corresponding results in the literature.

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Section: Introductionmentioning

confidence: 99%

Strong convergence of implicit and explicit iterations for a class of variational inequalities in Banach spaces

Ceng¹,

Wen²

2017

J. Nonlinear Sci. Appl.

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“…Then, they proved that the sequence {x n } converges weakly to an element of VI(C, A). However, we realize that Korpelevichs modified method (1.9) has only weak convergence in the infinitedimensional Hilbert spaces (you may also see, e.g., [6,7]). Therefore, several authors studied to obtain strong convergence by modifying the original method of Korpelevich.…”

Section: Introductionmentioning

confidence: 99%

“…There are several iterative methods for solving VIP (see, e.g., [4,5,7,11,18,30,33,35]). The basic idea consists of extending the projected gradient method for solving the problem of minimizing f (x) subject to x ∈ C given by 6) where {α n } is a positive real sequence satisfying certain conditions and P C is the metric projection onto C. For convergence properties of this method for the case in which f : R 2 → R is convex and differentiable function, one may see [2]. An immediate extension of method (1.6) to VIP is the projected gradient method for optimization problems, substituting the operator A for the gradient, so that we generate a sequence {x n } through:…”

Section: Introductionmentioning

confidence: 99%

An algorithm for finding a common point of the solutions of fixed point and variational inequality problems in Banach spaces

Tufa

Zegeye

2015

Arab. J. Math.

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Let C be a nonempty, closed and convex subset of a 2-uniformly convex and uniformly smooth real Banach space E. Let T : C → C be relatively nonexpansive mapping and let A i : C → E * be L i -Lipschitz monotone mappings, for i = 1, 2. In this paper, we introduce and study an iterative process for finding a common point of the fixed point set of a relatively nonexpansive mapping and the solution set of variational inequality problems for A 1 and A 2 . Under some mild assumptions, we show that the proposed algorithm converges strongly to a point inOur theorems improve and unify most of the results that have been proved for this important class of nonlinear operators. Mathematics Subject Classification

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“…Then, they proved that the sequences {x n }, {y n } converge weakly to the minimumnorm point of V I(C, A). We remark that Korpelevich's modified method (1.9) has only weak convergence in the infinite-dimensional Hilbert spaces (see Censor et al [5] and [4]). So to obtain strong convergence the original method was modified by several authors.…”

Section: (12)mentioning

confidence: 99%

Construction of a common solution of a finite family of variational inequality problems for monotone mappings

Alghamdi¹,

Shahzad²,

Zegeye³

2016

J. Nonlinear Sci. Appl.

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Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let A i : C → H, for i = 1, 2, be two L i -Lipschitz monotone mappings and let f : C → C be a contraction mapping. It is our purpose in this paper to introduce an iterative process for finding a point in V I(C, A 1 ) ∩ V I(C, A 2 ) under appropriate conditions. As a consequence, we obtain a convergence theorem for approximating a common solution of a finite family of variational inequality problems for Lipschitz monotone mappings. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators. c 2016 All rights reserved.

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Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space

Cited by 314 publications

References 9 publications

Strong convergence of implicit and explicit iterations for a class of variational inequalities in Banach spaces

Strong convergence of implicit and explicit iterations for a class of variational inequalities in Banach spaces

An algorithm for finding a common point of the solutions of fixed point and variational inequality problems in Banach spaces

Construction of a common solution of a finite family of variational inequality problems for monotone mappings

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