The relaxed CQ algorithm solving the split feasibility problem

Let C and Q be nonempty closed convex sets in R N and R M , respectively, and A is an M by N real matrix. The split feasibility problem (SFP) is to find x C with Ax Q , if such x exists. Byrne proposed an iterative method called the CQ algorithm that involves only the orthogonal projections onto C and Q and does not need to compute the matrix inverse, which may be the main advantage compared with other algorithms. The CQ algorithm is as follows: where γ (0, 2/ L ), with L being the largest eigenvalue of the matrix A T A and P C and P Q denote the orthogonal projections onto C and Q , respectively. Byrne ( 2002 Inverse Problems 18 441–53 , 2004 Inverse Problems 20 103–20 ) proved the convergence of the CQ algorithm for arbitrary nonzero matrix A . In his algorithm, Byrne assumed that the projections P C and P Q are easily calculated. In this paper, a modification of the CQ algorithm, called the relaxed CQ algorithm, is given, in which we replace P C and P Q by and , respectively, and the latter are easy to implement. Under mild assumptions, the convergence of the proposed algorithm is established. Then another algorithm for SFP is given; with the help of the CQ algorithm and its relaxed version, it is easy to obtain the convergence of this algorithm and corresponding relaxed scheme.