Applications of Fixed-Point and Optimization Methods to the Multiple-Set Split Feasibility Problem

The multiple-set split feasibility problem requires finding a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. It can be a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator’s range. It generalizes the convex feasibility problem as well as the two-set split feasibility problem. In this paper, we will review and report some recent results on iterative approaches to the multiple-set split feasibility problem.

Download Full-text

A new iterative method for the split feasibility problem

Carpathian Journal of Mathematics ◽

10.37193/cjm.2018.03.05 ◽

2018 ◽

Vol 34 (3) ◽

pp. 313-320

Author(s):

QIAO-LI DONG ◽

◽

DAN JIANG ◽

Keyword(s):

Inverse Problems ◽

Linear Operator ◽

Iterative Method ◽

Projection Method ◽

Numerical Experiments ◽

Split Feasibility Problem ◽

Feasibility Problem ◽

The Split Feasibility Problem ◽

Split Feasibility ◽

New Iterative Method

The split feasibility problem (SFP) has many applications, which can be a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator’s range. In this paper, we introduce a new projection method to solve the SFP and prove its convergence under standard assumptions. Our results improve previously known corresponding methods and results of this area. The preliminary numerical experiments illustrates the advantage of our proposed methods.

Download Full-text

Iterative Methods for Finding Solutions of a Class of Split Feasibility Problems over Fixed Point Sets in Hilbert Spaces

Mathematics ◽

10.3390/math7111012 ◽

2019 ◽

Vol 7 (11) ◽

pp. 1012

Author(s):

Suthep Suantai ◽

Narin Petrot ◽

Montira Suwannaprapa

Keyword(s):

Fixed Point ◽

Hilbert Spaces ◽

Nonexpansive Mappings ◽

Split Feasibility Problem ◽

Inverse Image ◽

Feasibility Problem ◽

Point Sets ◽

Fixed Point Set ◽

Split Feasibility ◽

Fixed Point Sets

We consider the split feasibility problem in Hilbert spaces when the hard constraint is common solutions of zeros of the sum of monotone operators and fixed point sets of a finite family of nonexpansive mappings, while the soft constraint is the inverse image of a fixed point set of a nonexpansive mapping. We introduce iterative algorithms for the weak and strong convergence theorems of the constructed sequences. Some numerical experiments of the introduced algorithm are also discussed.

Download Full-text

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

Mathematical and Computational Applications ◽

10.3390/mca25030047 ◽

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.

Download Full-text

Solving a Split Feasibility Problem by the Strong Convergence of Two Projection Algorithms in Hilbert Spaces

Journal of Function Spaces ◽

10.1155/2021/5562694 ◽

2021 ◽

Vol 2021 ◽

pp. 1-11

Author(s):

Hasanen A. Hammad ◽

Habib ur Rehman ◽

Yaé Ulrich Gaba

Keyword(s):

Strong Convergence ◽

Hilbert Spaces ◽

Split Feasibility Problem ◽

Convex Feasibility Problem ◽

Feasibility Problem ◽

Projection Algorithms ◽

Convergence Theorems ◽

Numerical Examples ◽

Convex Feasibility ◽

Split Feasibility

The goal of this manuscript is to establish strong convergence theorems for inertial shrinking projection and CQ algorithms to solve a split convex feasibility problem in real Hilbert spaces. Finally, numerical examples were obtained to discuss the performance and effectiveness of our algorithms and compare the proposed algorithms with the previous shrinking projection, hybrid projection, and inertial forward-backward methods.

Download Full-text

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

Journal of Mathematics ◽

10.1155/2021/6252984 ◽

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.

Download Full-text

Multiple-sets split feasibility problem and split equality fixed point problem for firmly quasi-nonexpansive or nonexpansive mappings

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02656-1 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Tongxin Xu ◽

Luoyi Shi

Keyword(s):

Fixed Point ◽

Hilbert Spaces ◽

Nonexpansive Mappings ◽

Split Feasibility Problem ◽

Fixed Point Problem ◽

Feasibility Problem ◽

Point Problem ◽

Projection Operators ◽

Nonexpansive Operators ◽

Split Feasibility

AbstractIn this paper, we propose a new iterative algorithm for solving the multiple-sets split feasibility problem (MSSFP for short) and the split equality fixed point problem (SEFPP for short) with firmly quasi-nonexpansive operators or nonexpansive operators in real Hilbert spaces. Under mild conditions, we prove strong convergence theorems for the algorithm by using the projection method and the properties of projection operators. The result improves and extends the corresponding ones announced by some others in the earlier and recent literature.

Download Full-text

Modified Relaxed CQ Iterative Algorithms for the Split Feasibility Problem

Mathematics ◽

10.3390/math7020119 ◽

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.

Download Full-text