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^TA 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.