Some Iterative Methods For Extended Multiple-Sets Split Feasibility Problem

The multiple-sets split feasibility problem which has been proposed by Censor, Elfving is formulated aswhere N,M≥1 are integers, {C₁,…,C_N} are closed convex subsets of H₁, {Q₁,…,Q_M} are closed convex subsets of H₂ and A : H₁→H₂ is a bounded linear operator.This paper extended the single operator A to A₁,A₂,…,A_M as the Hilbert spacs H₁ and H₂ are infinite-dimensional. The extended multiple-set split feasibility problem is formulated asThis master dissertation mainly consists of five parts: In chapter one, as the introduction,we review the background of the split feasibility problem and CQ algorithm which is designed to solve the question; In chapter two, we introduce some basic lemmas relatedwith nonexpansive operators and these lemmas will be used in the later parts of the paper, then we write down two generalizations of the Krasnoselski-Mann theorem.The left three chapters are the mainly parts of the paper, and in chapter three, we extend the multiple-sets split feasibility problem and propose an iterative alogorithm which is formulated as Then we prove that the sequence {x^k} converges weakly to the minimal point of the functioninΩas some obligations of the coefficientsα_i,β_j,γ_k are satisfied. As the special cases of the above algorithm, two corollaries are brought forward and proved to be true that the convergence is still satisfied whenγ_k is replaced by a constantγor the projection operator P_Ωis omitted.In chapter four, to accelerate the convergence, we present a block-iterative version of the multiple-set split feasibility problem and the algorithm is formulated asThen we prove that the {x^k} converges weakly to the common minimal point of the functionin maps C₁,C₂,…,C_N, and the limit point is the solution of the multiple-sets split feasibility problem in the consistent case. We later multiply a suitable coefficient to the mapping of the above iterative equation to have the strong convergence, and the algorithm is formulated asThen we prove that the sequence {x^k} converges strongly to a solution of the multiple-setssplit feasibility problem that is closest to u from the solution set of the multiple-sets split feasibility problem.In chapter five, we directly present another iterative formula asby the result of one paper and discussed the convergence of the sequence {x^k} as some restrictions ofω_k andσ_k are satisfied.