scholarly journals The Split Feasibility Problem and Its Solution Algorithm

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Biao Qu ◽  
Binghua Liu
Keyword(s):  
Signal Processing ◽  
Real World ◽  
Lipschitz Constant ◽  
Split Feasibility Problem ◽  
Solution Algorithm ◽  
Feasibility Problem ◽  
Step Size ◽  
Convergence Results ◽  
The Split Feasibility Problem ◽  
Split Feasibility

The split feasibility problem arises in many fields in the real world, such as signal processing, image reconstruction, and medical care. In this paper, we present a solution algorithm called memory gradient projection method for solving the split feasibility problem, which employs a parameter and two previous iterations to get the next iteration, and its step size can be calculated directly. It not only improves the flexibility of the algorithm, but also avoids computing the largest eigenvalue of the related matrix or estimating the Lipschitz constant in each iteration. Theoretical convergence results are established under some suitable conditions.

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 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.

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 Regularized Methods for the Split Feasibility Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Yonghong Yao ◽  
Wu Jigang ◽  
Yeong-Cheng Liou
Keyword(s):  
Signal Processing ◽  
Convergence Analysis ◽  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
Minimum Norm ◽  
Minimum Norm Solution ◽  
Image Reconstructions ◽  
Regularized Methods ◽  
The Split Feasibility Problem ◽  
Split Feasibility

Many applied problems such as image reconstructions and signal processing can be formulated as the split feasibility problem (SFP). Some algorithms have been introduced in the literature for solving the (SFP). In this paper, we will continue to consider the convergence analysis of the regularized methods for the (SFP). Two regularized methods are presented in the present paper. Under some different control conditions, we prove that the suggested algorithms strongly converge to the minimum norm solution of the (SFP).

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 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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