Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space

Censor, Yair; Gibali, Aviv; Reich, Simeon

doi:10.1080/10556788.2010.551536

Cited by 310 publications

(104 citation statements)

References 19 publications

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“…In this paper, we generalized the results of [14] from Hilbert spaces to Banach spaces. Consequently, we improve and extend the corresponding results in [2,[5][6][7]. Furthermore, our results are different from the ones studied in [7] as described in Remarks 3.7 and 3.8.…”

Section: Discussionmentioning

confidence: 51%

Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Banach spaces

Liu¹

2017

J. Nonlinear Sci. Appl.

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In this paper, we combine the subgradient extragradient method with the Halpern method for finding a solution of a variational inequality involving a monotone Lipschitz mapping in Banach spaces. By using the generalized projection operator and the Lyapunov functional introduced by Alber, we prove a strong convergence theorem. We also consider the problem of finding a common element of the set of solutions of a variational inequality problem and the set of fixed points of a relatively nonexpansive mapping. Our results improve some well-known results in Banach spaces or Hilbert spaces. c 2017 all rights reserved.Keywords: Subgradient extragradient method, Halpern method, generalized projection operator, monotone mapping, variational inequality, relatively nonexpansive mapping. 2010 MSC: 47H09, 47H05, 47H06, 47J25, 47J05.

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Section: Discussionmentioning

confidence: 51%

Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Banach spaces

Liu¹

2017

J. Nonlinear Sci. Appl.

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“…The following algorithm can be considered as an extension of the results in [10,11] to equilibrium problems. …”

Section: Resultsmentioning

confidence: 99%

“…In this paper, motivated and inspired by the results in [10,11], we propose three subgradient extragradient algorithms for solving EPs for pseudomonotone bifunctions. In these algorithms, we replaced strongly convex optimization program (4) on feasible set C by that one on a specific half-space whose the bounding hyperplane supported on the feasible set C. It seems to be more easily performed than on the feasible set.…”

Section: Introductionmentioning

confidence: 99%

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Weak and Strong Convergence of Subgradient Extragradient Methods for Pseudomonotone Equilibrium Problems

Hieu¹

2016

Communications of the Korean Mathematical Society

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Abstract. In this paper, we introduce three subgradient extragradient algorithms for solving pseudomonotone equilibrium problems. The paper originates from the subgradient extragradient algorithm for variational inequalities and the extragradient method for pseudomonotone equilibrium problems in which we have to solve two optimization programs onto feasible set. The main idea of the proposed algorithms is that at every iterative step, we have replaced the second optimization program by that one on a specific half-space which can be performed more easily. The weakly and strongly convergent theorems are established under widely used assumptions for bifunctions.

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“…The method (1.4) has been further modified and extended to obtain strong convergence results in Hilbert spaces and Banach spaces, see [5,13,16] and the reference therein. In [13], the authors combined the method (1.4) with the Halpern method to propose the Halpern subgradient extragradient method for the strong convergence in Hilbert spaces.…”

Section: Aymentioning

confidence: 99%

Weak convergence of a modified subgradient extragradient algorithm for monotone variational inequalities in Banach spaces

Liu¹,

Kong²

2017

J. Nonlinear Sci. Appl.

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Applying the generalized projection operator, we introduce a modified subgradient extragradient algorithm in Banach spaces for a variational inequality involving a monotone Lipschitz continuous mapping which is more general than an inversestrongly-monotone mapping. Weak convergence of the iterative algorithm is also proved. An advantage of the algorithm is the computation of only one value of the inequality mapping and one projection onto the admissible set per one iteration.

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Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space

Cited by 310 publications

References 19 publications

Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Banach spaces

Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Banach spaces

Weak and Strong Convergence of Subgradient Extragradient Methods for Pseudomonotone Equilibrium Problems

Weak convergence of a modified subgradient extragradient algorithm for monotone variational inequalities in Banach spaces

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