The ball-relaxed CQ algorithms for the split feasibility problem

作     者：Yu, Hai Zhan, Wanrong Wang, Fenghui 

作者机构：Luoyang Normal Univ Dept Math Luoyang 471022 Peoples R China 

出 版 物：《OPTIMIZATION》 (最优化)

年 卷 期：2018年第67卷第10期

页      面：1687-1699页

核心收录：

学科分类：12[管理学] 1201[管理学-管理科学与工程(可授管理学、工学学位)] 07[理学] 070105[理学-运筹学与控制论] 0701[理学-数学] 

基　　金：National Natural Science Foundation of China [11301253, 11571005] Program for Science and Technology Innovation Talents in the Universities of Henan Province [15HASTIT013] Innovation Scientists and Technicians Troop Construction Projects of Henan Province [CXTD20150027] Foundation of He'nan Educational Committee [16A520064, 15A520087] 

主　　题：Split feasibility problem CQ algorithm variable stepsize projection 

摘      要：The split feasibility problem (SFP) is to find x is an element of C so that Ax is an element of Q, where C and Q are non-empty closed convex subsets in Hilbert spaces H-1 and H-2, respectively, and A is a linear bounded operator from H-1 to H-2. Byrne proposed an iterative method called the CQ algorithm that involves the orthogonal projections onto C and Q. However, the projections onto C and Q might be hard to be implemented in general. In this paper, we propose a ball-relaxed projection method for the SFP. Instead of half spaces, we replace C and Q in the proposed algorithm by two properly chosen closed balls C-k(b) and Q(k)(b). Since the projection onto the closed ball has closed form, the proposed algorithm is thus easy to be implemented. Under some mild conditions, we establish the weak convergence of the proposed algorithm to a solution of the SFP. As an application, we obtain new algorithms for solving the split equality problem. Preliminary numerical experiments show the efficiency of the proposed method.