On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space

Vuong, Phan Tu; Strodiot, Jean Jacques; Nguyen, Van Hien

doi:10.1080/02331934.2012.759327

Cited by 67 publications

(43 citation statements)

References 31 publications

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“…Assumptions (A2) is satisfied thanks to the well known fact that (see e.g. Vuong et al 2015) if f (x, .) is lower semicontinuous, differentiable convex function, then the diagonal subdifferential ∂ f 2 (x, x) maps a bounded set into a bounded set.…”

Section: A Practical Model and Computational Resultsmentioning

confidence: 99%

An algorithm for a class of split feasibility problems: application to a model in electricity production

2016

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We propose a projection algorithm for solving split feasibility problems involving paramonotone equilibria and convex optimization. The proposed algorithm can be considered as a combination of the projection ones for equilibrium and convex optimization problems. We apply the algorithm for finding an equilibrium point with minimal environmental cost for a model in electricity production. Numerical results for the model are reported.

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Section: A Practical Model and Computational Resultsmentioning

confidence: 99%

An algorithm for a class of split feasibility problems: application to a model in electricity production

2016

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“…In [19], a hybrid pseudoviscosity approximation algorithm is considered in the framework of Hilbert spaces. Finally, other works have also been done in that direction, but for a finite number of fixed-point problems [21][22][23][24], or with other types of projection [25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

“…Then we can prove exactly as in Step 4 of the proof of [23,Theorem 4.4] that the sequences {y n i } and {w n i } are bounded. So there exist subsequences of {x n i }, {y n i }, and {w n i }, again denoted {x n i }, {y n i }, and {w n i }, that converge weakly tox,ȳ, andw, respectively.…”

mentioning

confidence: 97%

Hybrid Methods for Solving Simultaneously an Equilibrium Problem and Countably Many Fixed Point Problems in a Hilbert Space

Nguyen

Strodiot

Nguyen

2013

J Optim Theory Appl

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This paper presents a framework of iterative methods for finding a common solution to an equilibrium problem and a countable number of fixed point problems defined in a Hilbert space. A general strong convergence theorem is established under mild conditions. Two hybrid methods are derived from the proposed framework in coupling the fixed point iterations with the iterations of the proximal point method or the extragradient method, which are well-known methods for solving equilibrium problems. The strategy is to obtain the strong convergence from the weak convergence of the iterates without additional assumptions on the problem data. To achieve this aim, the solution set of the problem is outer approximated by a sequence of polyhedral subsets.

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“…Hypothesis (A3) was used by the authors in [33]. If U is compact and linear then the bifunction f satisfies condition (A3), in addition, if U is self-adjoint and positive semidefinite then f satisfies Condition 1, for example, U is a linear integral operator with the kernel being symmetric and continuous in L 2 [a, b].…”

Section: Conditionmentioning

confidence: 99%

“…The advantages of the extragradient method are that it is used for the class of pseudomonotone bifunctions and two optimization programs are solved at each iteration which seems to be numerically easier than non-linear inequality (1.5) in the proximal method, see for instance [28,31,33] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Parallel Extragradient-Proximal Methods for Split Equilibrium Problems

Hieu

2016

Mathematical Modelling and Analysis

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In this paper, we introduce two parallel extragradient-proximal methods for solving split equilibrium problems. The algorithms combine the extragradient method, the proximal method and the shrinking projection method. The weak and strong convergence theorems for iterative sequences generated by the algorithms are established under widely used assumptions for equilibrium bifunctions. We also present an application to split variational inequality problems and a numerical example to illustrate the convergence of the proposed algorithms.Keywords: equilibrium problem, split equilibrium problem, extragradient method, proximal method, parallel algorithm.

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On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space

Cited by 67 publications

References 31 publications

An algorithm for a class of split feasibility problems: application to a model in electricity production

An algorithm for a class of split feasibility problems: application to a model in electricity production

Hybrid Methods for Solving Simultaneously an Equilibrium Problem and Countably Many Fixed Point Problems in a Hilbert Space

Parallel Extragradient-Proximal Methods for Split Equilibrium Problems

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