scholarly journals Self adaptive inertial relaxed $ CQ $ algorithms for solving split feasibility problem with multiple output sets

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Guash Haile Taddele ◽  
Poom Kumam ◽  
Habib ur Rehman ◽  
Anteneh Getachew Gebrie
Keyword(s):  
Hilbert Spaces ◽  
Convex Sets ◽  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
Weak And Strong Convergence ◽  
Step Size ◽  
The Split Feasibility Problem ◽  
Adaptive Step ◽  
Split Feasibility ◽  
Self Adaptive

<p style='text-indent:20px;'>In this paper, we propose two new self-adaptive inertial relaxed <inline-formula><tex-math id="M2">\begin{document}$ CQ $\end{document}</tex-math></inline-formula> algorithms for solving the split feasibility problem with multiple output sets in the framework of real Hilbert spaces. The proposed algorithms involve computing projections onto half-spaces instead of onto the closed convex sets, and the advantage of the self-adaptive step size introduced in our algorithms is that it does not require the computation of operator norm. We establish and prove weak and strong convergence theorems for the iterative sequences generated by the introduced algorithms for solving the aforementioned problem. Moreover, we apply the new results to solve some other problems. Finally, we present some numerical examples to illustrate the implementation of our algorithms and compared them to some existing results.</p>

scholarly journals Modified Relaxed CQ Iterative Algorithms for the Split Feasibility Problem

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 119
Author(s):  
Xinglong Wang ◽  
Jing Zhao ◽  
Dingfang Hou
Keyword(s):  
Hilbert Spaces ◽  
Convex Sets ◽  
Iterative Algorithms ◽  
Intensity Modulated Radiation Therapy ◽  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
Weak And Strong Convergence ◽  
Cq Algorithm ◽  
The Split Feasibility Problem ◽  
Split Feasibility

The split feasibility problem models inverse problems arising from phase retrievals problems and intensity-modulated radiation therapy. For solving the split feasibility problem, Xu proposed a relaxed CQ algorithm that only involves projections onto half-spaces. In this paper, we use the dual variable to propose a new relaxed CQ iterative algorithm that generalizes Xu’s relaxed CQ algorithm in real Hilbert spaces. By using projections onto half-spaces instead of those onto closed convex sets, the proposed algorithm is implementable. Moreover, we present modified relaxed CQ algorithm with viscosity approximation method. Under suitable conditions, global weak and strong convergence of the proposed algorithms are proved. Some numerical experiments are also presented to illustrate the effectiveness of the proposed algorithms. Our results improve and extend the corresponding results of Xu and some others.

scholarly journals New Inertial Relaxed CQ Algorithms for Solving Split Feasibility Problems in Hilbert Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Haiying Li ◽  
Yulian Wu ◽  
Fenghui Wang
Keyword(s):  
Hilbert Spaces ◽  
Split Feasibility Problem ◽  
Linear Operators ◽  
Feasibility Problem ◽  
Variable Step Size ◽  
Step Size ◽  
Bounded Linear ◽  
Variable Step ◽  
The Split Feasibility Problem ◽  
Split Feasibility

The split feasibility problem SFP has received much attention due to its various applications in signal processing and image reconstruction. In this paper, we propose two inertial relaxed C Q algorithms for solving the split feasibility problem in real Hilbert spaces according to the previous experience of applying inertial technology to the algorithm. These algorithms involve metric projections onto half-spaces, and we construct new variable step size, which has an exact form and does not need to know a prior information norm of bounded linear operators. Furthermore, we also establish weak and strong convergence of the proposed algorithms under certain mild conditions and present a numerical experiment to illustrate the performance of the proposed algorithms.

scholarly journals An Inertial Accelerated Algorithm for Solving Split Feasibility Problem with Multiple Output Sets

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Huijuan Jia ◽  
Shufen Liu ◽  
Yazheng Dang
Keyword(s):  
Strong Convergence ◽  
Numerical Experiments ◽  
Convex Sets ◽  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
Matrix Inverse ◽  
The Split Feasibility Problem ◽  
Inertial Technique ◽  
Split Feasibility ◽  
Self Adaptive

The paper proposes an inertial accelerated algorithm for solving split feasibility problem with multiple output sets. To improve the feasibility, the algorithm involves computing of projections onto relaxed sets (half spaces) instead of computing onto the closed convex sets, and it does not require calculating matrix inverse. To accelerate the convergence, the algorithm adopts self-adaptive rules and incorporates inertial technique. The strong convergence is shown under some suitable conditions. In addition, some newly derived results are presented for solving the split feasibility problem and split feasibility problem with multiple output sets. Finally, numerical experiments illustrate that the algorithm converges more quickly than some existing algorithms. Our results extend and improve some methods in the literature.

scholarly journals An Explicit Method for the Split Feasibility Problem with Self-Adaptive Step Sizes

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Youli Yu
Keyword(s):  
Iterative Method ◽  
Strong Convergence ◽  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
Strong Convergence Theorem ◽  
Explicit Method ◽  
The Split Feasibility Problem ◽  
Adaptive Step ◽  
Split Feasibility ◽  
Self Adaptive

An explicit iterative method with self-adaptive step-sizes for solving the split feasibility problem is presented. Strong convergence theorem is provided.

scholarly journals An Extrapolated Iterative Algorithm for Multiple-Set Split Feasibility Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Yazheng Dang ◽  
Yan Gao
Keyword(s):  
Iterative Algorithm ◽  
Numerical Experiments ◽  
Convex Sets ◽  
Lipschitz Constant ◽  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
Step Size ◽  
Fixed Constant ◽  
The Split Feasibility Problem ◽  
Split Feasibility

The multiple-set split feasibility problem (MSSFP), as a generalization of the split feasibility problem, is to find a point in the intersection of a family of closed convex sets in one space such that its image under a linear transformation will be in the intersection of another family of closed convex sets in the image space. Censor et al. (2005) proposed a method for solving the multiple-set split feasibility problem (MSSFP), whose efficiency depends heavily on the step size, a fixed constant related to the Lipschitz constant of∇p(x)which may be slow. In this paper, we present an accelerated algorithm by introducing an extrapolated factor to solve the multiple-set split feasibility problem. The framework encompasses the algorithm presented by Censor et al. (2005). The convergence of the method is investigated, and numerical experiments are provided to illustrate the benefits of the extrapolation.

scholarly journals Half-Space Relaxation Projection Method for Solving Multiple-Set Split Feasibility Problem

2020 ◽  
Vol 25 (3) ◽  
pp. 47
Author(s):  
Guash Haile Taddele ◽  
Poom Kumam ◽  
Anteneh Getachew Gebrie ◽  
Kanokwan Sitthithakerngkiet
Keyword(s):  
Convex Sets ◽  
Split Feasibility Problem ◽  
Finite Family ◽  
Feasibility Problem ◽  
Extended Form ◽  
Numerical Examples ◽  
Cq Algorithm ◽  
The Split Feasibility Problem ◽  
Convergent Algorithm ◽  
Split Feasibility

In this paper, we study an iterative method for solving the multiple-set split feasibility problem: find a point in the intersection of a finite family of closed convex sets in one space such that its image under a linear transformation belongs to the intersection of another finite family of closed convex sets in the image space. In our result, we obtain a strongly convergent algorithm by relaxing the closed convex sets to half-spaces, using the projection onto those half-spaces and by introducing the extended form of selecting step sizes used in a relaxed CQ algorithm for solving the split feasibility problem. We also give several numerical examples for illustrating the efficiency and implementation of our algorithm in comparison with existing algorithms in the literature.

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