An implicit iterative scheme for monotone variational inequalities and fixed point problems

Ceng, Lu-Chuan; Cubiotti, Paolo; Yao, Jen‐Chih

doi:10.1016/j.na.2007.08.023

Cited by 23 publications

(14 citation statements)

References 16 publications

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“…Lemma 2.4 (see [10], Lemma 2.1). Let {a k }, {b k } and {c k } be three sequences of nonnegative real numbers satisfying the inequality…”

Section: Preliminariesmentioning

confidence: 99%

“…The authors showed that under certain conditions over {α k } and {r k }, sequences {x k } and {u k } strongly converge to z = P r Sol(f,C)∩F ix(S) g(z) . Recently, iterative methods for finding a common element of the set of solutions of equilibrium problems and the set of fixed points of a nonexpansive mapping have further developed by many authors (see [3,8,10,22,21] and the references quoted therein).…”

Section: Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

Fixed Point Solution Methods for Solving Equilibrium Problems

Ánh¹,

Hien²

2014

Bulletin of the Korean Mathematical Society

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Abstract. In this paper, we propose new iteration methods for finding a common point of the solution set of a pseudomonotone equilibrium problem and the solution set of a monotone equilibrium problem. The methods are based on both the extragradient-type method and the viscosity approximation method. We obtain weak convergence theorems for the sequences generated by these methods in a real Hilbert space.

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“…Lemma 2.4 (see [10], Lemma 2.1). Let {a k }, {b k } and {c k } be three sequences of nonnegative real numbers satisfying the inequality…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Fixed Point Solution Methods for Solving Equilibrium Problems

Ánh¹,

Hien²

2014

Bulletin of the Korean Mathematical Society

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“…Subsequently, motivated by Nadezhkina and Takahashi [23], Ceng and Yao [26] introduced another iterative process for finding a common element of the common fixed-point set of N nonexpansive self-mappings of 0 and the set of solutions to the variational inequality problem for monotone, k-Lipschitz-continuous mapping by the combination of extragradient and approximate proximal methods.…”

Section: (X) -mentioning

confidence: 99%

“…In this paper inspired by Bonnel, Iusem and Svaiter [1], Nadezhkina and Takahashi [23], and Ceng and Yao [22,26], we introduce and study both the absolute hybrid approximate proximal method and the relative hybrid approximate proximal method for solving the general vector-valued optimization problem considered at the beginning of this section. Let {an} be a bounded sequence of positive real numbers.…”

Section: (X) -mentioning

confidence: 99%

Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization

Ceng

Mordukhovich

Yao

2010

J Optim Theory Appl

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Abstract. This paper studies the general vector optimization problem of finding weakly efficient points for mappings in a Banach space Y, with respect to the partial order induced by a closed, convex, and pointed cone C C Y with nonempty interior. In order to find a solution of this problem, we introduce an auxiliary variational inequality problem for monotone, Lipschitz-continuous mapping. The approximate proximal method in vector optimization is extended to develop a hybrid approximate proximal method for the general vector optimization problem by the combination of extragradient method for finding a solution to the variational inequality problem and approximate proximal point method for finding a root of a maximal monotone operator. In this hybrid approximate proximal method, the subproblems consist of finding approximate solutions to the variational inequality problem for monotone, Lipschitz-continuous mapping, and finding weakly efficient points for suitable regularizations of the original mapping. We present both an absolute and a relative version in which the subproblems are solved only approximately. Weak convergence of the generated sequence to a weak efficient point is established under quite mild conditions. In addition, we also discuss an extension to Bregman-function-based hybrid approximate proximal algorithms for finding weakly efficient points for mappings.

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“…In this paper, motivated by research work going on this direction; see, for example, [1,[3][4][5][6][7][8][9][10][11][12][13][14][15][16][18][19][20][21][22][23][24][25], we continue to study the iterative algorithm (ϒ ). We improve Iiduka and Takahashi [9] from Hilbert spaces to Banach spaces.…”

Section: Introductionmentioning

confidence: 99%