Parallel methods for regularizing systems of equations involving accretive operators

Anh, Pham Ky; Buong, Nguyen; Hieu, Dang Van

doi:10.1080/00036811.2013.872777

Cited by 26 publications

(13 citation statements)

References 19 publications

(14 reference statements)

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“…Let us denote the solution set of VIP (2) by V I (A, K ). Problem 1 is a generalization of many other problems including: convex feasibility problems, common fixed point problems, common minimizer problems, common saddle-point problems, hierarchical variational inequality problems, variational inequality problems over the intersection of convex sets, etc., see [3][4][5]7,12,21,22]. In this paper, we focus on projection methods, which together with regularization ones are fundamental methods for solving VIPs with monotone and Lipschitz continuous mappings.…”

Section: Introductionmentioning

confidence: 99%

Modified hybrid projection methods for finding common solutions to variational inequality problems

2016

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In this paper we propose several modified hybrid projection methods for solving common solutions to variational inequality problems involving monotone and Lipschitz continuous operators. Based on differently constructed half-spaces, the proposed methods reduce the number of projections onto feasible sets as well as the number of values of operators needed to be computed. Strong convergence theorems are established under standard assumptions imposed on the operators. An extension of the proposed algorithm to a system of generalized equilibrium problems is considered and numerical experiments are also presented.

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Section: Introductionmentioning

confidence: 99%

Modified hybrid projection methods for finding common solutions to variational inequality problems

2016

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“…Several sequential and parallel iterative regularization methods [1,2,6,8,10] have been proposed for finding a solution of the system (32). Using Theorem 3.1, we also obtain the following result:…”

Section: Resultsmentioning

confidence: 99%

“…Let C be a nonempty closed convex subset of H. Let A : C → H be a (nonlinear) operator. The variational inequality problem is to find p * ∈ C such that (1) Ap * , p − p * ≥ 0, ∀p ∈ C.…”

Section: Introductionmentioning

confidence: 99%

A Parallel Hybrid Method for Equilibrium Problems, Variational Inequalities and Nonexpansive Mappings in Hilbert Space

Hieu¹

2015

Journal of the Korean Mathematical Society

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Abstract. In this paper, a novel parallel hybrid iterative method is proposed for finding a common element of the set of solutions of a system of equilibrium problems, the set of solutions of variational inequalities for inverse strongly monotone mappings and the set of fixed points of a finite family of nonexpansive mappings in Hilbert space. Strong convergence theorem is proved for the sequence generated by the scheme. Finally, a parallel iterative algorithm for two finite families of variational inequalities and nonexpansive mappings is established.

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“…CSEP (1) is very general in the sense that it includes, as special cases, many mathematical models: common solutions to variational inequalities, convex feasibility problems, common fixed point problems, see for instance [2,8,10,11,14,21,34,37]. These problems have been widely studied both theoretically and algorithmically over the past decades due to their applications to other fields [5,10,15,29].…”

Section: Introductionmentioning

confidence: 99%

Cyclic subgradient extragradient methods for equilibrium problems

Hieu

2016

Arab. J. Math.

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In this paper, we introduce a cyclic subgradient extragradient algorithm and its modified form for finding a solution of a system of equilibrium problems for a class of pseudomonotone and Lipschitz-type continuous bifunctions. The main idea of these algorithms originates from several previously known results for variational inequalities. The proposed algorithms are extensions of the subgradient extragradient method for variational inequalities to equilibrium problems and the hybrid (outer approximation) method. The paper can help in the design and analysis of practical algorithms and gives us a generalization of the most convex feasibility problems. Mathematics Subject Classification

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Parallel methods for regularizing systems of equations involving accretive operators

Abstract: In this paper, two parallel methods for solving systems of accretive operator equations in Banach spaces are studied. The convergence analysis of the methods in both free-noise and noisy data cases is provided.

Cited by 26 publications

References 19 publications

Modified hybrid projection methods for finding common solutions to variational inequality problems

Modified hybrid projection methods for finding common solutions to variational inequality problems

A Parallel Hybrid Method for Equilibrium Problems, Variational Inequalities and Nonexpansive Mappings in Hilbert Space

Cyclic subgradient extragradient methods for equilibrium problems

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