Convergence Analysis of Modified Hybrid Steepest-Descent Methods with Variable Parameters for Variational Inequalities

Zeng, Lei; Wong, Ngai–Ching; Yao, Jen‐Chih

doi:10.1007/s10957-006-9068-x

Cited by 59 publications

(44 citation statements)

References 16 publications

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“…It is worth pointing out that, related iterative methods for solving fixed point problems, variational inequalities and optimization problems can be found in [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems

Ceng

Guu

Yao

2012

Fixed Point Theory Appl

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Recently, Colao et al. (J Math Anal Appl 344:340-352, 2008) introduced a hybrid viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space. In this paper, by combining Colao, Marino and Xu's hybrid viscosity approximation method and Yamada's hybrid steepest-descent method, we propose a hybrid iterative method for finding a common element of the set GMEP of solutions of a generalized mixed equilibrium problem and the setFix (S i ) of fixed points of a finite family of nonexpansivein a real Hilbert space. We prove the strong convergence of the proposed iterative algorithm to an element ofFix (S i ) ∩ GMEP , which is the unique solution of a variational inequality.

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Section: Introductionmentioning

confidence: 99%

Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems

Ceng

Guu

Yao

2012

Fixed Point Theory Appl

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“…Hence, it would be reasonable to identify the unique minimizer of an appropriate objective function over the hierarchical fixed point constraint. Very recently, related iterative methods and their convergence analysis for solving hierarchical fixed point problems, hierarchical optimization problems and hierarchical variational inequality problems can be found in [9][10][11][12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Strong convergence of relaxed hybrid steepest-descent methods for triple hierarchical constrained optimization

Zeng

Wong

Yao

2012

Fixed Point Theory Appl

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Up to now, a large number of practical problems such as signal processing and network resource allocation have been formulated as the monotone variational inequality over the fixed point set of a nonexpansive mapping, and iterative algorithms for solving these problems have been proposed. The purpose of this article is to investigate a monotone variational inequality with variational inequality constraint over the fixed point set of one or finitely many nonexpansive mappings, which is called the triple-hierarchical constrained optimization. Two relaxed hybrid steepest-descent algorithms for solving the triple-hierarchical constrained optimization are proposed. Strong convergence for them is proven. Applications of these results to constrained generalized pseudoinverse are included.

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“…Subsequently, Xu and Kim [9] and Zeng et al [10] proved the convergence of hybrid steepest-descent method. Noor introduced iterations after analysis of several three-step iterative methods [18].…”

Section: It Is Known That Xmentioning

confidence: 99%

“…To prove the convergence theorem, we first introduce several lemmas and the main results reported by others [10,11,21].…”

Section: Preliminariesmentioning

confidence: 99%

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Efficient implementation of a modified and relaxed hybrid steepest-descent method for a type of variational inequality

2012

J Inequal Appl

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To reduce the difficulty and complexity in computing the projection from a real Hilbert space onto a nonempty closed convex subset, researchers have provided a hybrid steepest-descent method for solving VI(F, K) and a subsequent three-step relaxed version of this method. In a previous study, the latter was used to develop a modified and relaxed hybrid steepest-descent (MRHSD) method. However, choosing an efficient and implementable nonexpansive mapping is still a difficult problem. We first establish the strong convergence of the MRHSD method for variational inequalities under different conditions that simplify the proof, which differs from previous studies. Second, we design an efficient implementation of the MRHSD method for a type of variational inequality problem based on the approximate projection contraction method. Finally, we design a set of practical numerical experiments. The results demonstrate that this is an efficient implementation of the MRHSD method.

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Convergence Analysis of Modified Hybrid Steepest-Descent Methods with Variable Parameters for Variational Inequalities

Cited by 59 publications

References 16 publications

Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems

Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems

Strong convergence of relaxed hybrid steepest-descent methods for triple hierarchical constrained optimization

Efficient implementation of a modified and relaxed hybrid steepest-descent method for a type of variational inequality

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