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.

Subgradient algorithm for split hierarchical optimization problems

2018 ◽  
Vol 34 (3) ◽  
pp. 391-399
Author(s):  
NIMIT NIMANA ◽  
◽  
NARIN PETROT ◽  
◽  
Keyword(s):  
Hilbert Spaces ◽  
Optimization Problems ◽  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
Hierarchical Optimization ◽  
Convergence Results ◽  
Hierarchical Optimization Problems ◽  
Subgradient Algorithm ◽  
The Split Feasibility Problem ◽  
Split Feasibility

In this paper we emphasize a split type problem of some integrating ideas of the split feasibility problem and the hierarchical optimization problem. Working on real Hilbert spaces, we propose a subgradient algorithm for approximating a solution of the introduced problem. We discuss its convergence results and present a numerical example.

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 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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