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 is an element of C with Ax is an element of 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: x(k+1) = P-C(x(k) + gammaA(T)(P-Q - I)Ax(k)), k = 0, 1.... where gamma is an element of (0, 2/L), with L being the largest eigenvalue of the matrix A(T) A and Pc and PQ 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 P-Ck. and P-Qk, 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 C-Q algorithm and its relaxed version, it is easy to obtain the convergence of this algorithm and corresponding relaxed scheme.