A note on the
                    <i>CQ</i>
                    algorithm for the split feasibility problem

Qu, Bo; Xiu, Naihua

doi:10.1088/0266-5611/21/5/009

Cited by 263 publications

(123 citation statements)

References 15 publications

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“…As we know, the SFP has received so much attention due to its applications in intensity-modulated radiation therapy, signal processing, and image reconstruction, see Byrne [1,2], Censor [4][5][6], Ceng [3], Fan et al [7], Xu [20,21], Kraikaew and Saejung [9], Moudafi [10], Qu et al [12][13][14], Qin and Yao [11], Yang et al [16,22,27,28], Yao et al [23][24][25][26], and so on.…”

Section: Introductionmentioning

confidence: 99%

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Strong convergence theorems for a nonexpansive mapping and its applications for solving the split feasibility problem

Fan¹,

Yao²

2017

J. Nonlinear Sci. Appl.

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The aim of this paper is to propose some novel algorithms and their strong convergence theorems for solving the split feasibility problem, and we obtain the corresponding strong convergence results under mild conditions. The split feasibility problem was proposed by [Y. Censor, Y. Elfving, Numer. Algorithms, 8 (1994), 221-239]. So far a lot of algorithms have been given for solving this problem due to its applications in intensity-modulated radiation therapy, signal processing, and image reconstruction. But most of these algorithms are of weak convergence. In this paper, we propose the new algorithms which can provide useful guidelines for solving the relevant problem, such as the split common fixed point problem (SCFP), multi-set split feasibility problem and so on.

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Section: Introductionmentioning

confidence: 99%

“…In 2004, Yang [22] constructed a relaxed CQ algorithm for solving a special case of the SFP, in which he replaced them by projections onto halfspaces C k and Q k . In 2005, Qu and Xiu [13] modified Yang's relaxed CQ algorithm and the CQ algorithm by adopting the Armijo-like searches to get the stepsize.…”

Section: Introductionmentioning

confidence: 99%

Strong convergence theorems for a nonexpansive mapping and its applications for solving the split feasibility problem

Fan¹,

Yao²

2017

J. Nonlinear Sci. Appl.

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“…In [7,9,10], it has been shown that the SPF (1.1) can also be used to model the intensity-modulated radiation therapy. Various iterative algorithms have been studied to solve the SFP (1.1), see, e.g., [8,14,16,18,22,25,27,[30][31][32] and the references therein. In particular, Jung [16] introduced iterative algorithms based on the Yamada's hybrid steepest descent method [28] for solving SFP (1.1).…”

Section: Introductionmentioning

confidence: 99%

Hybrid iterative algorithms for the split common fixed point problems

Jung¹

2017

J. Nonlinear Sci. Appl.

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In this paper, we introduce two iterative algorithms (one implicit algorithm and one explicit algorithm) based on the hybrid steepest descent method for solving the split common fixed point problems. We establish the strong convergence of the sequences generated by the proposed algorithms to a solution of the split common fixed point problems, which is also a solution of a certain variational inequality. In particular, the minimum norm solution of the split common fixed point problems is obtained. As applications, variational problems and equilibrium problems are considered. c 2017 All rights reserved.Keywords: Split common fixed point problem, firmly nonexpansive mapping, nonexpansive mapping, variational inequality, minimum-norm, ρ-Lipschitzian and η-strongly monotone operator, bounded linear operator, variational problems, equilibrium problems, iterative algorithms. 2010 MSC: 47J05, 49J40, 49J52, 47J20, 47H10.

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“…In recent years, a large number of scholars have done a lot of meaningful research on the split feasibility problem (SFP), because the problem in signal processing and linear constrained optimization problems such as the feasible solution plays an important role (Censor, Y., et al, 2006;Byrne, C., 2002;Byrne, C., 2004.;Yang, Q., 2004.;Qu, B. & Xiu, N., 2005.;Xu, H. K., 2006.;Xu, H. K., 2010).…”

Section: Introductionmentioning

confidence: 99%

A Viscosity Approximation Method for the Split Feasibility Problems in Hilbert Space

Yang¹

2017

JMR

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In this paper, the most basic idea is to apply the viscosity approximation method to study the split feasibility problem (SFP), we will be in the infinite-dimensional Hilbert space to study the problem . We defined x 0 ∈ C as arbitrary andfor n ≥ 0, where {α n } ⊂ (0, 1). Under the proper control conditions of some parameters, we show that the sequence {x n } converges strongly to a solution of SFP. The results in this paper extend and further improve the relevant conclusions in Deepho (Deepho, J. & Kumam, P., 2015).

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A note on the CQ algorithm for the split feasibility problem

Abstract: Let C and Q be nonempty closed convex sets in N and M , respectively, and A an M × N real matrix. The split feasibility problem (SFP) is to find x ∈ C with Ax ∈ Q, if such x exist. Byrne (2002 Inverse Problems 18 441-53) proposed a CQ algorithm with the following iterative scheme:

Cited by 263 publications

References 15 publications

Strong convergence theorems for a nonexpansive mapping and its applications for solving the split feasibility problem

Strong convergence theorems for a nonexpansive mapping and its applications for solving the split feasibility problem

Hybrid iterative algorithms for the split common fixed point problems

A Viscosity Approximation Method for the Split Feasibility Problems in Hilbert Space

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