Strong Convergence of the Halpern Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Spaces

Kraikaew, Rapeepan; Saejung, Satit

doi:10.1007/s10957-013-0494-2

Cited by 227 publications

(99 citation statements)

References 21 publications

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“…If E = H, then a relatively nonexpansive mapping S reduces to a quasi-nonexpansive mapping S which satisfies I − S is demiclosed at zero. Hence, taking E = H and λ n ≡ τ satisfying τL < 1, then Theorem 4.1 reduces to Theorem 4.1 of [14]. Therefore, Theorem 4.1 absolutely generalizes Theorem 4.1 of [14] from Hilbert spaces to Banach spaces.…”

Section: The Modified Subgradient Extragradient Algorithmmentioning

confidence: 85%

“…Taking E = H and λ n ≡ τ satisfying τL < 1, then Algorithm (3.1) reduces to Algorithm (1.4) and Theorem 3.5 reduces to Theorem 3.1 of [14]. Therefore, Theorem 3.5 absolutely generalizes Theorem 3.1 of [14] from Hilbert spaces to Banach spaces. Furthermore, we change the parameter from a fixed constant τ to a changeable sequence {λ n }.…”

Section: Lemma 33mentioning

confidence: 94%

“…Inspired by Kraikaew and Saejung' results [14], we propose the Algorithm (3.1) to extend the Algorithm (1.4) from Hilbert spaces to Banach spaces and prove a strong convergence theorem (i.e., Theorem 3.5), which is quite different from the scheme proposed by Nakajo [20]. In fact, we do not need to calculate the generalized projections onto the constructible sets C n and Q n as in [20].…”

Section: The Halpern Subgradient Extragradient Algorithmmentioning

confidence: 99%

“…Inspired by the second main result of Kraikaew et al [14], we present a modified subgradient extragradient algorithm in Banach spaces for finding a solution of the variational inequality (1.1) which is also a fixed point of a given relatively nonexpansive mapping. Our algorithm is as follows.…”

Section: The Modified Subgradient Extragradient Algorithmmentioning

confidence: 99%

“…Under some mild assumptions, Censor et al in [7] proved the method (1.3) converges weakly to a solution of variational inequality (1.1) in a Hilbert space. In order to obtain the strong convergence, Kraikaew and Saejung in [14] combined the subgradient extragradient method (1.3) with the Halpern method introduced in [9] and proposed the following iterative algorithm: 4) where {α n } is a sequence in [0, 1] satisfying lim n→∞ α n = 0 and ∞ n=1 α n = ∞. They proved the method (1.4) converges strongly to a solution of variational inequality (1.1) in a Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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: The Modified Subgradient Extragradient Algorithmmentioning

confidence: 85%

Section: Lemma 33mentioning

confidence: 94%

Section: The Halpern Subgradient Extragradient Algorithmmentioning

confidence: 99%

Section: The Modified Subgradient Extragradient Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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Strong convergence of new iterative projection methods with regularization for solving monotone variational inequalities in Hilbert spaces

Hieu

Mưu

Duong

et al. 2020

Math Methods in App Sciences

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In this paper, we introduce two new numerical methods for solving a variational inequality problem involving a monotone and Lipschitz continuous operator in a Hilbert space. We describe how to incorporate a regularization term depending on a parameter in the projection method and then establish the strong convergence of the resulting iterative regularization projection methods. Unlike known hybrid methods, the strong convergence of the new methods comes from the regularization technique. The first method is designed to work in the case where the Lipschitz constant of cost operator is known, whereas the second one is more easily implemented without this requirement. The reason is because the second method has used a simple computable stepsize rule. The variable stepsizes are generated by the second method at each iteration and based on the previous iterates. These stepsizes are found with only one cheap computation without line-search procedure. Several numerical experiments are implemented to show the computational effectiveness of the new methods over existing methods.

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An inertial subgradient extragradient method of variational inequality problems involving quasi‐nonexpansive operators with applications

Suparatulatorn

Charoensawan

Khemphet

2021

Math Methods in App Sciences

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In this work, a new algorithm is suggested to solve the variational inequality problems for Lipschitz continuous and monotone operators and the fixed point problems for quasi-nonexpansive operators. This algorithm is constructed based on the inertial subgradient extragradient method. In addition, a strong convergence theorem for this algorithm is obtained under some extra conditions. Furthermore, an application to a signal recovery in compressed sensing problem is shown as a numerical example of the algorithm. KEYWORDScompressed sensing, fixed point, inertial method, subgradient extragradient method, variational inequality MSC CLASSIFICATION 47J20; 90C25; 90C30; 90C52where Fix( ) represents the fixed point set of  .

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Strong Convergence of the Halpern Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Spaces

Cited by 227 publications

References 21 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

Strong convergence of new iterative projection methods with regularization for solving monotone variational inequalities in Hilbert spaces

An inertial subgradient extragradient method of variational inequality problems involving quasi‐nonexpansive operators with applications

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