scholarly journals Linear Convergence of an Iterative Algorithm for Solving the Multiple-Sets Split Feasibility Problem

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 644 ◽  
Author(s):  
Tingting Tian ◽  
Luoyi Shi ◽  
Rudong Chen
Keyword(s):  
Iterative Algorithm ◽  
Orthogonal Projection ◽  
Sufficient Conditions ◽  
Linear Convergence ◽  
Split Feasibility Problem ◽  
Projection Algorithm ◽  
Feasibility Problem ◽  
Step Size ◽  
Bounded Linear ◽  
Split Feasibility

In this paper, we propose the simultaneous sub-gradient projection algorithm with the dynamic step size (SSPA for short) for solving the multiple-sets split feasibility problem (MSSFP for short) and investigate its linear convergence. We involve a notion of bounded linear regularity for the MSSFP and construct several sufficient conditions to prove the linear convergence for the SSPA. In particular, the SSPA is an easily calculated algorithm that uses orthogonal projection onto half-spaces. Furthermore, some numerical results are provided to verify the effectiveness of our proposed algorithm.

scholarly journals Convergence rate analysis of an iterative algorithm for solving the multiple-sets split equality problem

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Shijie Sun ◽  
Meiling Feng ◽  
Luoyi Shi
Keyword(s):  
Convergence Rate ◽  
Iterative Algorithm ◽  
Sufficient Conditions ◽  
Linear Convergence ◽  
Regularity Property ◽  
Step Size ◽  
Bounded Linear ◽  
Split Equality Problem ◽  
Linear Convergence Rate ◽  
Equality Problem

Abstract This paper considers an iterative algorithm of solving the multiple-sets split equality problem (MSSEP) whose step size is independent of the norm of the related operators, and investigates its sublinear and linear convergence rate. In particular, we present a notion of bounded Hölder regularity property for the MSSEP, which is a generalization of the well-known concept of bounded linear regularity property, and give several sufficient conditions to ensure it. Then we use this property to conclude the sublinear and linear convergence rate of the algorithm. In the end, some numerical experiments are provided to verify the validity of our consequences.

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 An Iterative Algorithm for the Split Equality and Multiple-Sets Split Equality Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Luoyi Shi ◽  
Ru Dong Chen ◽  
Yu Jing Wu
Keyword(s):  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Split Feasibility Problem ◽  
Linear Operators ◽  
Feasibility Problem ◽  
Bounded Linear ◽  
Split Equality Problem ◽  
Infinite Dimensional ◽  
Equality Problem ◽  
Split Feasibility

The multiple-sets split equality problem (MSSEP) requires finding a pointx∈∩i=1NCi,y∈∩j=1MQjsuch thatAx=By, whereNandMare positive integers,{C1,C2,…,CN}and{Q1,Q2,…,QM}are closed convex subsets of Hilbert spacesH1,H2, respectively, andA:H1→H3,B:H2→H3are two bounded linear operators. WhenN=M=1, the MSSEP is called the split equality problem (SEP). If  B=I, then the MSSEP and SEP reduce to the well-known multiple-sets split feasibility problem (MSSFP) and split feasibility problem (SFP), respectively. One of the purposes of this paper is to introduce an iterative algorithm to solve the SEP and MSSEP in the framework of infinite-dimensional Hilbert spaces under some more mild conditions for the iterative coefficient.

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 On Two Projection Algorithms for the Multiple-Sets Split Feasibility Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Qiao-Li Dong ◽  
Songnian He
Keyword(s):  
Adaptive Algorithm ◽  
Split Feasibility Problem ◽  
Rates Of Convergence ◽  
Projection Algorithm ◽  
Feasibility Problem ◽  
Projection Algorithms ◽  
Numerical Tests ◽  
Split Feasibility ◽  
Self Adaptive

We present a projection algorithm which modifies the method proposed by Censor and Elfving (1994) and also introduce a self-adaptive algorithm for the multiple-sets split feasibility problem (MSFP). The global rates of convergence are firstly investigated and the sequences generated by two algorithms are proved to converge to a solution of the MSFP. The efficiency of the proposed algorithms is illustrated by some numerical tests.

scholarly journals A Perturbed Projection Algorithm with Inertial Technique for Split Feasibility Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Yazheng Dang ◽  
Yan Gao ◽  
Yanli Han
Keyword(s):  
Convergence Theorem ◽  
Convex Set ◽  
Split Feasibility Problem ◽  
Projection Algorithm ◽  
Feasibility Problem ◽  
Asymptotic Convergence ◽  
Closed Convex Set ◽  
The Split Feasibility Problem ◽  
Inertial Technique ◽  
Split Feasibility

This paper deals with the split feasibility problem that requires to find a point closest to a closed convex set in one space such that its image under a linear transformation will be closest to another closed convex set in the image space. By combining perturbed strategy with inertial technique, we construct an inertial perturbed projection algorithm for solving the split feasibility problem. Under some suitable conditions, we show the asymptotic convergence. The results improve and extend the algorithms presented in Byrne (2002) and in Zhao and Yang (2005) and the related convergence theorem.

scholarly journals New Iterative Algorithm for Solving Constrained Convex Minimization Problem and Split Feasibility Problem

2021 ◽  
Vol 1 (2) ◽  
pp. 106-132
Author(s):  
Austine Efut Ofem ◽  
Unwana Effiong Udofia ◽  
Donatus Ikechi Igbokwe
Keyword(s):  
Fixed Points ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
Almost Contraction ◽  
Contraction Mappings ◽  
Constrained Convex Minimization Problem ◽  
Split Feasibility

The purpose of this paper is to introduce a new iterative algorithm to approximate the fixed points of almost contraction mappings and generalized α-nonexpansive mappings. Also, we show that our proposed iterative algorithm converges weakly and strongly to the fixed points of almost contraction mappings and generalized α-nonexpansive mappings. Furthermore, it is proved analytically that our new iterative algorithm converges faster than one of the leading iterative algorithms in the literature for almost contraction mappings. Some numerical examples are also provided and used to show that our new iterative algorithm has better rate of convergence than all of S, Picard-S, Thakur and M iterative algorithms for almost contraction mappings and generalized α-nonexpansive mappings. Again, we show that the proposed iterative algorithm is stable with respect to T and data dependent for almost contraction mappings. Some applications of our main results and new iterative algorithm are considered. The results in this article are improvements, generalizations and extensions of several relevant results existing in the literature.

scholarly journals Bounded Perturbation Resilience of Two Modified Relaxed CQ Algorithms for the Multiple-Sets Split Feasibility Problem

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 197
Author(s):  
Yingying Li ◽  
Yaxuan Zhang
Keyword(s):  
Hilbert Spaces ◽  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
Step Size ◽  
Bounded Perturbation ◽  
Bounded Perturbation Resilience ◽  
Perturbation Resilience ◽  
Lasso Problem ◽  
Bounded Perturbations ◽  
Split Feasibility

In this paper, we present some modified relaxed CQ algorithms with different kinds of step size and perturbation to solve the Multiple-sets Split Feasibility Problem (MSSFP). Under mild assumptions, we establish weak convergence and prove the bounded perturbation resilience of the proposed algorithms in Hilbert spaces. Treating appropriate inertial terms as bounded perturbations, we construct the inertial acceleration versions of the corresponding algorithms. Finally, for the LASSO problem and three experimental examples, numerical computations are given to demonstrate the efficiency of the proposed algorithms and the validity of the inertial perturbation.

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