A new hybrid extragradient algorithm for solving the equilibrium and variational inequality problems

Chamnarnpan, Tanom; Phiangsungnoen, Supak

doi:10.1007/s13370-013-0187-x

Cited by 9 publications

(7 citation statements)

References 18 publications

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“…The last inequality follows from Equations (11) and (12) and lim i→∞ At n i − As n i = 0 (this follows from the fact that A is Lipschitz on H, and lim i→∞ t n i − s n i = 0.). Let {η i } be a decreasing sequence of positive numbers tending to 0 and for each i, N i a smallest positive integer such that…”

Section: Lemmamentioning

confidence: 99%

“…The Variational Inequality (VI) problem Equation (1) is a fundamental problem in optimization theory which is applied in many areas of study, such as transportation problems, equilibrium, economics, engineering and so on (Refs. [1][2][3][4][5][6][7][8][9][10][11][12][13][14]). There are many approaches to the VI problem, the basic one being the regularization and projection method.…”

Section: Introductionmentioning

confidence: 99%

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Inertial Iterative Schemes with Variable Step Sizes for Variational Inequality Problem Involving Pseudomonotone Operator

2020

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Two inertial subgradient extragradient algorithms for solving variational inequality problems involving pseudomonotone operator are proposed in this article. The iterative schemes use self-adaptive step sizes which do not require the prior knowledge of the Lipschitz constant of the underlying operator. Furthermore, under mild assumptions, we show the weak and strong convergence of the sequences generated by the proposed algorithms. The strong convergence in the second algorithm follows from the use of viscosity method. Numerical experiments both in finite- and infinite-dimensional spaces are reported to illustrate the inertial effect and the computational performance of the proposed algorithms in comparison with the existing state of the art algorithms.

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Section: Lemmamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Inertial Iterative Schemes with Variable Step Sizes for Variational Inequality Problem Involving Pseudomonotone Operator

2020

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“…where E is a nonempty closed convex subset of a real Hilbert space H and B : H −→ H is an operator. Problem 49 is an important problem in optimization theory with several applications in different areas of study such as control problem, economics, equilibrium, transportation problems, engineering and many more (see [31,2,21,22,3,23,19,25,11]). It has been shown that problem (49) is equivalent to problem (1), where A 2 is equals to zero and A 1 is a subdifferential of an indicator function on E (A 1 maximal monotone).…”

Section: Application To Variational Inequality Problem a Variational Inequality (Vi)mentioning

confidence: 99%

An efficient iterative method for solving split variational inclusion problem with applications

Abubakar

Garba

Abdullahi

et al. 2022

JIMO

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<p style='text-indent:20px;'>A new strong convergence iterative method for solving a split variational inclusion problem involving a bounded linear operator and two maximally monotone mappings is proposed in this article. The study considers an iterative scheme comprised of inertial extrapolation step together with the Mann-type step. A strong convergence theorem of the iterates generated by the proposed iterative scheme is given under suitable conditions. In addition, methods for solving variational inequality problems and split convex feasibility problems are derived from the proposed method. Applications of solving Nash-equilibrium problems and image restoration problems are solved using the derived methods to demonstrate the implementation of the proposed methods. Numerical comparisons with some existing iterative methods are also presented.</p>

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“…Because the monotone inclusion problem (3) is the core of image processing and many mathematical problems [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], many researchers have proposed and developed iterative methods for solving inclusion problems (3). The forward-backward splitting method, constructed and studied by Lions and Mercier [23] in 1979, is the most popular algorithm for solving the (3) problem.…”

Section: Introductionmentioning

confidence: 99%

Modified Tseng’s Method with Inertial Viscosity Type for Solving Inclusion Problems and Its Application to Image Restoration Problems

Kaewyong

Sitthithakerngkiet

2021

Mathematics

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In this paper, we study a monotone inclusion problem in the framework of Hilbert spaces. (1) We introduce a new modified Tseng’s method that combines inertial and viscosity techniques. Our aim is to obtain an algorithm with better performance that can be applied to a broader class of mappings. (2) We prove a strong convergence theorem to approximate a solution to the monotone inclusion problem under some mild conditions. (3) We present a modified version of the proposed iterative scheme for solving convex minimization problems. (4) We present numerical examples that satisfy the image restoration problem and illustrate our proposed algorithm’s computational performance.

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A new hybrid extragradient algorithm for solving the equilibrium and variational inequality problems

Cited by 9 publications

References 18 publications

Inertial Iterative Schemes with Variable Step Sizes for Variational Inequality Problem Involving Pseudomonotone Operator

Inertial Iterative Schemes with Variable Step Sizes for Variational Inequality Problem Involving Pseudomonotone Operator

An efficient iterative method for solving split variational inclusion problem with applications

Modified Tseng’s Method with Inertial Viscosity Type for Solving Inclusion Problems and Its Application to Image Restoration Problems

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