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.

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 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 Three-Step Projective Methods for Solving the Split Feasibility Problems

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 712 ◽  
Author(s):  
Suthep Suantai ◽  
Nontawat Eiamniran ◽  
Nattawut Pholasa ◽  
Prasit Cholamjiak
Keyword(s):  
Hilbert Spaces ◽  
Iteration Process ◽  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
Adaptive Technique ◽  
Convergence Results ◽  
Projective Methods ◽  
Cq Algorithm ◽  
The Split Feasibility Problem ◽  
Split Feasibility

In this paper, we focus on studying the split feasibility problem (SFP) in Hilbert spaces. Based on the CQ algorithm involving the self-adaptive technique, we introduce a three-step iteration process for approximating the solution of SFP. Then, the convergence results are established under mild conditions. Numerical experiments are provided to show the efficiency in signal processing. Some comparisons to various methods are also provided in this paper.

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 Partially Projective Algorithm for the Split Feasibility Problem with Visualization of the Solution Set

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 608
Author(s):  
Andreea Bejenaru ◽  
Mihai Postolache
Keyword(s):  
Numerical Simulation ◽  
Numerical Modeling ◽  
Visual Image ◽  
Split Feasibility Problem ◽  
Solution Set ◽  
Feasibility Problem ◽  
Cq Algorithm ◽  
Step Algorithm ◽  
The Split Feasibility Problem ◽  
Split Feasibility

This paper introduces a new three-step algorithm to solve the split feasibility problem. The main advantage is that one of the projective operators interferes only in the final step, resulting in less computations at each iteration. An example is provided to support the theoretical approach. The numerical simulation reveals that the newly introduced procedure has increased performance compared to other existing methods, including the classic CQ algorithm. An interesting outcome of the numerical modeling is an approximate visual image of the solution set.

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