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Dang Van Hieu

Publications and source records attributed to Dang Van Hieu.

18 recordsLinked to original sources
arXiv2019

Modified golden ratio algorithms for solving equilibrium problems

In this paper an explicit algorithm is proposed for solving an equilibrium problem whose associated bifunction is pseudomonotone and satisfies a Lipschitz-type condition. Contrary to many algorithms, our algorithm is done without using explicitly the Lipschitz constants of bifunction although its convergence is obtained under such that condition. The introduced method is a form of proximal-like method whose steplengths are explicitly generated at each iteration without using any linesearch procedure. First we prove the convergence of the algorithm, and after we establish its $R$-linear rate of convergence under the assumption of strong pseudomonotonicity of the bifunction. Afterwards several numerical results are displayed to illustrate and to compare the behavior of the new algorithm with other ones.

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arXiv2019

Golden ratio algorithms with new stepsize rules for variational inequalities

In this paper, we introduce two golden ratio algorithms with new stepsize rules for solving pseudomonotone and Lipschitz variational inequalities in finite dimensional Hilbert spaces. The presented stepsize rules allow the resulting algorithms to work without the prior knowledge of the Lipschitz constant of operator. The first algorithm uses a sequence of stepsizes which is previously chosen, diminishing and non-summable. While the stepsizes in the second one are updated at each iteration and by a simple computation. A special point is that the sequence of stepsizes generated by the second algorithm is separated from zero. The convergence as well as the convergence rate of the proposed algorithms are established under some standard conditions. Also, we give several numerical results to show the behavior of the algorithms in comparisons with other algorithms.

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arXiv2019

Projection methods for solving split equilibrium problems

The paper considers a split inverse problem involving component equilibrium problems in Hilbert spaces. This problem therefore is called the split equilibrium problem (SEP). It is known that almost solution methods for solving problem (SEP) are designed from two fundamental methods as the proximal point method and the extended extragradient method (or the two-step proximal-like method). Unlike previous results, in this paper we introduce a new algorithm, which is only based on the projection method, for finding solution approximations of problem (SEP), and then establish that the resulting algorithm is weakly convergent under mild conditions. Several of numerical results are reported to illustrate the convergence of the proposed algorithm and also to compare with others.

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arXiv2018

New inertial regularized algorithm for solving strongly pseudomonotone equilibrium problems

The article introduces a new algorithm for solving a class ofequilibrium problems involving strongly pseudomonotone bifunctions with Lipschitz-type condition. We describe how to incorporate the proximal-like regularized technique with inertial effects. The main novelty of the algorithm is that it can be done without previously knowing the information on the strongly pseudomonotone and Lipschitz-type constants of cost bifunction. A reasonable explain for this is that the algorithm uses a sequence of stepsizes which is diminishing and non-summable. Theorem of strong convergence is proved. In the case, when the information on the modulus of strong pseudomonotonicity and Lispchitz-type constant is known,the rate of linear convergence of the algorithm has been established. Several of numerical experiments are performed to illustrate the convergence of the algorithm and also compare it with other algorithms.

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arXiv2016

Parallel extragradient - viscosity methods for equilibrium problems and fixed point problems

In this paper, we propose two parallel extragradient - viscosity methods for finding a particular element in the common solution set of a system of equilibrium problems and finitely many fixed point problems. This particular point is the unique solution of a variational inequality problem on the common solution set. The main idea of the paper is to combine three methods including the extragradient method, the Mann iteration method, the hybrid steepest-descent method with the parallel splitting-up technique to design the algorithms which improve the performance over some existing methods. The strongly convergent theorems are established under the widely used assumptions for equilibrium bifunctions.

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arXiv2016

Parallel hybrid methods for generalized equilibrium problems and asymptotically strictly pseudocontractive mappings

In this paper, we propose two novel parallel hybrid methods for finding a common element of the set of solutions of a finite family of generalized equilibrium problems for monotone bifunctions $\left\{f_i\right\}_{i=1}^N$ and $α$ - inverse strongly monotone operators $\left\{A_i\right\}_{i=1}^N$ and the set of common fixed points of a finite family of (asymptotically) $κ$- strictly pseudocontractive mappings $\left\{S_j\right\}_{j=1}^M$ in Hilbert spaces. The strong convergence theorems are established under the standard assumptions imposed on equilibrium bifunctions and operators. A numerical example is presented to illustrate the efficiency of the proposed parallel methods.

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arXiv2015

Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings

In this paper we propose and analyze three parallel hybrid extragradient methods for finding a common element of the set of solutions of equilibrium problems involving pseudomonotone bifunctions and the set of fixed points of nonexpansive mappings in a real Hilbert space. Based on parallel computation we can reduce the overall computational effort under widely used conditions on the bifunctions and the nonexpansive mappings.A simple numerical example is given to illustrate the proposed parallel algorithms.

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arXiv2015

Parallel extragradient-proximal methods for split equilibrium problems

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 hybrid (outer approximation) method. The weak and strong convergence theorems for iterative sequences generated by the algorithms are established under widely used assumptions for equilibrium bifunctions.

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arXiv2015

Parallel projection methods for variational inequalities involving common fixed point problems

In this paper, we introduce two novel parallel projection methods for finding a solution of a system of variational inequalities which is also a common fixed point of a family of (asymptotically) $κ$ - strict pseudocontractive mappings. A technical extension in the proposed algorithms helps in computing practical numerical experiments when the number of subproblems is large. Some numerical examples are implemented to demonstrate the efficiency of parallel computations.

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arXiv2015

An extension of hybrid method without extrapolation step to equilibrium problems

In this paper, we introduce a new hybrid algorithm for solving equilibrium problems. The algorithm combines the extragradient method and the hybrid (outer approximation) method. In this algorithm, only an optimization program is solved at each iteration without the extra-steps like as in the extragradient method and the Armijo linesearch method. A specially constructed half-space in the hybrid method is the reason for the absence of an optimization program in our algorithm. The strong convergence theorem is established and several numerical experiments are implemented to illustrate the convergence of the algorithm and compare it with others.

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arXiv2015

Hybrid algorithms without the extra-steps for equilibrium problems

In this paper, we introduce some new hybrid algorithms for finding a solution of a system of equilibrium problems. In these algorithms, by constructing specially cutting-halfspaces, we avoid using the extra-steps as in the extragradient method and the Armijo linesearch method which are inherently costly when the feasible set has a complex structure. The strong convergence of the algorithms is established.

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arXiv2015

A novel hybrid method for equilibrium problems and fixed point problems

The paper proposes a novel hybrid method for solving equilibrium problems and fixed point problems. By constructing specially cutting-halfspaces, in this algorithm, only an optimization program is solved at each iteration without the extra-steps as in some previously known methods. The strongly convergence theorem is established and some numerical examples are presented to illustrate its convergence.

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arXiv2015

The common solutions to pseudomonotone equilibrium problems

In this paper, we propose two iterative methods for finding a common solution of a finite family of equilibrium problems for pseudomonotone bifunctions. The first is a parallel hybrid extragradient-cutting algorithm which is extended from the previously known one for variational inequalities to equilibrium problems. The second is a new cyclic hybrid extragradient-cutting algorithm. In the cyclic algorithm, using the known techniques, we can perform and develop practical numerical experiments.

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arXiv2015

Parallel hybrid iterative methods for variational inequalities, equilibrium problems and common fixed point problems

In this paper we propose two strongly convergent parallel hybrid iterative methods for finding a common element of the set of fixed points of a family of quasi $ϕ$-asymptotically nonexpansive mappings $\{F(S_j)\}_{j=1}^N$, the set of solutions of variational inequalities $\{VI(A_i,C)\}_{i=1}^M$ and the set of solutions of equilibrium problems $\{EP(f_k)\}_{k=1}^K$ in uniformly smooth and 2-uniformly convex Banach spaces. A numerical experiment is given to verify the efficiency of the proposed parallel algorithms.

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arXiv2015

A parallel hybrid method for equilibrium problems, variational inequalities and nonexpansive mappings in Hilbert space

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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