Several solution methods for the split feasibility problem

Abstract

Let C and Q be nonempty, closed convex sets in Rn and Rm respectively, and A be an m × n real matrix. The split feasibility problem (SFP) is to find x ∊ C with Ax ∊ Q, if such points exist. Byrne proposed the following CQ algorithm to solve the SFP: where γ ∊ (0, 2/ρ(ATA)) with ρ(ATA) the spectral radius of the matrix ATA and PC and PQ denote the orthogonal projections onto C and Q, respectively. However, in some cases, it is difficult or even impossible to compute PC and PQ exactly. In this paper, based on the CQ algorithm, we present several algorithms to solve the SFP. Compared with the CQ algorithm, our algorithms are more practical and easier to implement. They can be regarded as improvements of the CQ algorithm.