Modified Halfspace-Relaxation Projection Methods for Solving the Split Feasibility Problem

This paper presents modified halfspace-relaxation projection (HRP) methods for solving the split feasibility problem (SFP). Incorporating with the techniques of identifying the optimal step length with positive lower bounds, the new methods improve the efficiencies of the HRP method (Qu and Xiu (2008)). Some numerical results are reported to verify the computational preference. 1. Introduction Let and be nonempty closed convex sets in and , respectively, and an real matrix. The problem, to find with if such exists, was called the split feasibility problem (SFP) by Censor and Elfving [1]. In this paper, we consider an equivalent reformulation [2] of the SFP: where For convenience, we only consider the Euclidean norm. It is obvious that is convex. If and , then solves the SFP. Throughout we assume that the solution set of the SFP is nonempty. And thus the solution set of (1.1), denoted by , is nonempty. In addition, in this paper, we always assume that the set is given by where is a convex (not necessarily differentiable) function. This representation of is general enough, because any system of inequalities , where are convex and is an arbitrary index set, can be reformulated as the single inequality with . For any , at least one subgradient can be calculated, where is a subgradient of at and is defined as follows: Qu and Xiu [2] proposed a halfspace-relaxation projection method to solve the convex optimization problem (1.1). Starting from any , the HRP method iteratively updates according to the formulae: where is an element in , and is the smallest nonnegative integer such that The notation denotes the projection of onto under the Euclidean norm, that is, Here the halfspace contains the given closed convex set and is related to the current iterative point . From the expressions of , the projection onto is simple to be computed (for details, see Proposition 3.3). The idea to construct the halfspace and replace by is from the halfspace-relaxation projection technique presented by Fukushima [3]. This technique is often used to design algorithms (see, e.g., [2, 4, 5]) to solve the SFP. The drawback of the HRP method in [2] is that the step length defined in (1.9) may be very small since . Note that the reformulation (1.1) is equivalent to a monotone variational inequality (VI): where The forward-backward splitting method [6] and the extragradient method [7, 8] are considerably simple projection-type methods in the literature. They are applicable for solving monotone variational inequalities, especially for (1.11). For given , let Under the assumption the