A modified Korpelevich's method convergent to the minimum-norm solution of a variational inequality

Yao, Yonghong; Marino, Giuseppe; Muglia, Luigi

doi:10.1080/02331934.2012.674947

Cited by 79 publications

(40 citation statements)

References 20 publications

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“…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. The results obtained in this paper improve and extend the results of Nadezhkina and Takahashi [10], Yao et al [21] and Yao et al [22] and some other results in this direction.…”

Section: (12)supporting

confidence: 74%

“…Theorem 3.1 extends Theorem 3.1 of Nadezhkina and Takahashi [10], Yao et al [21] and Therem 1 of Yao et al [22] in the sense that our scheme provides strong convergence to a common solution of variational inequality problem for a Lipschitz monotone mappings.…”

Section: )supporting

confidence: 59%

“…Then, they proved that the sequence {x n } converges strongly to the minimum-norm point of V I(C, A). One may also see related results in [21]. More recently, Yao et al [22] investigated the problem of finding a solution of variational inequality V I(C, A) for α-inverse strongly monotone mapping A by considering the following iterative algorithm:…”

Section: (12)mentioning

confidence: 99%

“…So to obtain strong convergence the original method was modified by several authors. For example in [2,6,21] it is proved that some very interesting Korpelevich-type algorithms strongly converge to a solution of V I(C, A). Recently, Yao et al [21] suggested the following modified Korpelevich's method for a solution of a variational inequality V I(C, A) for α-inverse strongly monotone mapping A in infinite-dimensional Hilbert spaces.…”

Section: (12)mentioning

confidence: 99%

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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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Section: (12)supporting

confidence: 74%

Section: )supporting

confidence: 59%

Section: (12)mentioning

confidence: 99%

Section: (12)mentioning

confidence: 99%

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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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“…Therefore, several authors studied to obtain strong convergence by modifying the original method of Korpelevich. For example, in [4,9,31], it is proved that some very interesting Korpelevich-type algorithms strongly converge to a solution of VIP.…”

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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Modified parallel projection methods for the multivalued lexicographic variational inequalities using proximal operator in Hilbert spaces

Ánh

Thi

2019

Math Methods in App Sciences

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In this paper, building upon projection methods and parallel splitting‐up techniques with using proximal operators, we propose new algorithms for solving the multivalued lexicographic variational inequalities in a real Hilbert space. First, the strong convergence theorem is shown with Lipschitz continuity of the cost mapping, but it must satisfy a strongly monotone condition. Second, the convergent results are also established to the multivalued lexicographic variational inequalities involving a finite system of demicontractive mappings under mild assumptions imposed on parameters. Finally, some numerical examples are developed to illustrate the behavior of our algorithms with respect to existing algorithms.

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A modified Korpelevich's method convergent to the minimum-norm solution of a variational inequality

Cited by 79 publications

References 20 publications

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

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

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

Modified parallel projection methods for the multivalued lexicographic variational inequalities using proximal operator in Hilbert spaces

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