Mathematical Problems in Engineering

Volume 2018, Article ID 4123168, 7 pages

https://doi.org/10.1155/2018/4123168

## The Split Feasibility Problem and Its Solution Algorithm

Institute of Operations Research, Qufu Normal University, Rizhao, Shandong 276826, China

Correspondence should be addressed to Biao Qu; moc.361@100oaibuq

Received 20 August 2017; Revised 7 November 2017; Accepted 5 December 2017; Published 8 January 2018

Academic Editor: Shuming Wang

Copyright © 2018 Biao Qu and Binghua Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The split feasibility problem arises in many fields in the real world, such as signal processing, image reconstruction, and medical care. In this paper, we present a solution algorithm called memory gradient projection method for solving the split feasibility problem, which employs a parameter and two previous iterations to get the next iteration, and its step size can be calculated directly. It not only improves the flexibility of the algorithm, but also avoids computing the largest eigenvalue of the related matrix or estimating the Lipschitz constant in each iteration. Theoretical convergence results are established under some suitable conditions.

#### 1. Introduction

The split feasibility problem (SFP) was first put forward in [1] by Censor and Elfving. It requires finding a point in a nonempty closed convex subset in one space such that its image under a certain operator is in another nonempty closed convex subset in the image space. Its precise mathematical formulation is as follows: Given closed convex set and closed convex sets in - and -dimensional Euclidean space, respectively, the split feasibility problem is to find a vector for which where is a given real matrix.

The SFP arises in many fields in the real world, such as signal processing, image reconstruction, and medical care; for details see [1–3] and the references therein. For example, a number of image reconstruction problems can be formulated as split feasibility problems. The vector represents a vectorized image, with the entries of , the intensity levels at each voxel or pixel. The set can be selected to incorporate such features as nonnegativity of the entries of , while the matrix can describe linear functional or projection measurements we have made, as well as other linear combinations of entries of on which we wish to impose constraints. The set then can be the product of the vector of measured data with other convex sets, such as nonnegative cones, that serve to describe the constraints to be imposed [4]. Here we give a discretized model of SFP in image reconstruction problem of X-ray tomography [5, 6]. In image reconstruction, we consider a portion of the form of a two-dimensional cross section in which the attenuation intensities of X-ray are not identical for different tissue. This attenuation effect can be seen as a function with nonnegative variable, called the image or images. We hope to get information about the physiological state of the organization by measuring the data. The fundamental model is formulated in the following way: a Cartesian grid of square picture elements, called pixels, is introduced into the region of interest so that it covers the whole picture that has to be reconstructed. The pixels are numbered in some agreed manner, say from 1 (top left corner pixel) to (bottom right corner pixel) (see Figure 1).