Modified Relaxed CQ Iterative Algorithms for the Split Feasibility Problem

Xinglong Wang,Jing Zhao,Dingfang Hou

doi:10.3390/math7020119

Abstract

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.

Highlights

Throughout this paper, we always assume that H is a real Hilbert space with inner product h·, ·i and norm k · k
The SFP (SFP) in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and medical image reconstruction [2], with particular progress in intensity-modulated radiation therapy [3,4]
In the setting of finite-dimensional spaces, relaxed projection method was followed by Yang [26], who introduced the following relaxed CQ algorithms for solving the SFP (Equation (1)) where the closed convex subsets C and Q are level sets of convex functions: un+1 = PCn (un − μA T ( I − PQn ) Aun ), n ≥ 1, (15)

Summary

Introduction

Throughout this paper, we always assume that H is a real Hilbert space with inner product h·, ·i and norm k · k. For solving the SFP (Equation (1)), we note that the CQ algorithm and many related iterative algorithms (see [19,20,21,22,23,24]) only involves the computations of the projections PC and PQ onto the sets C and Q, respectively, and is implementable in the case where PC and PQ have closed-form expressions. In the setting of finite-dimensional spaces, relaxed projection method was followed by Yang [26], who introduced the following relaxed CQ algorithms for solving the SFP (Equation (1)) where the closed convex subsets C and Q are level sets of convex functions: un+1 = PCn (un − μA T ( I − PQn ) Aun ), n ≥ 1,. We give some numerical experiments to illustrate the efficiency of the proposed iterative methods

Preliminaries

Weak Convergence Theorems

Strong Convergence Theorems

Numerical Results