Research On Linear Convergence Of Iterative Algorithm For Splitting Feasibility Problem

In 1994,Censor and Elfving was first proposed split feasibility problem in finite-dimensional Euclidean space.This problem was a mathematical model abstracted from the fields of medical image reconstruction and phase restoration.In2005,Censor et al.extended the split feasibility problem to the multiple-sets split feasibility problem.The split equality problem,as another important extension of the split feasibility problem,was presented by Moudafi in 2013.In fact,various algorithms have been used in studies extensively to find a solution to such inverse problems.However,the convergence results of existed algorithms are not good enough and the convergence rate of these algorithms has not been rarely explicitly estimated.Consequently,it is worth further research.In this paper,we mainly studied the multiple-sets split feasibility problem and the split equality problem in Hilbert spaces.Firstly,we proposed the simultaneous sub-gradient projection algorithm with the dynamic step size for solving the multiple-sets split feasibility problem.In particular,we involved a notion of bounded linear regularity for the multiple-sets split feasibility problem and constructed several sufficient conditions to prove the linear convergence for the simultaneous sub-gradient projection algorithm with the dynamic step size.Then,we considered the relaxed gradient projection algorithm to solve the split equality problem and creatively investigated its linear convergence.Furthermore,we concluded the linear convergence rate of the relaxed gradient projection algorithm.Finally,some numerical experiments are given to test the validity of our results.