A note on the CQ algorithm for the split feasibility problem

2005 ◽  
Vol 21 (5) ◽  
pp. 1655-1665 ◽  
Author(s):  
Biao Qu ◽  
Naihua Xiu
Keyword(s):  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
Cq Algorithm ◽  
The Split Feasibility Problem ◽  
Split Feasibility

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 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 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 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.

scholarly journals A New Hybrid CQ Algorithm for the Split Feasibility Problem in Hilbert Spaces and Its Applications to Compressed Sensing

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 789 ◽  
Author(s):  
Suthep Suantai ◽  
Suparat Kesornprom ◽  
Prasit Cholamjiak
Keyword(s):  
Compressed Sensing ◽  
Line Search ◽  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
Step Size ◽  
Cq Algorithm ◽  
Line Search Techniques ◽  
Weak Convergence Theorem ◽  
The Split Feasibility Problem ◽  
Split Feasibility

In this paper, we focus on studying the split feasibility problem (SFP), which has many applications in signal processing and image reconstruction. A popular technique is to employ the iterative method which is so called the relaxed CQ algorithm. However, the speed of convergence usually depends on the way of selecting the step size of such algorithms. We aim to suggest a new hybrid CQ algorithm for the SFP by using the self adaptive and the line-search techniques. There is no computation on the inverse and the spectral radius of a matrix. We then prove the weak convergence theorem under mild conditions. Numerical experiments are included to illustrate its performance in compressed sensing. Some comparisons are also given to show the efficiency with other CQ methods in the literature.

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