Modified projection methods for the split feasibility problem and the multiple-sets split feasibility problem

The split feasibility problem is to find x∈C with Ax∈Q, if such points exist, where A is a given M×N real matrix, C and Q are nonempty closed convex sets in RN and RM, respectively. Byrne (2002) [2] proposed a well-known CQ algorithm for solving this problem. In this paper, we propose a modification for the CQ algorithm, which computes the stepsize adaptively, and performs an additional projection step onto some simple closed convex set X⊆RN in each iteration. We further give a relaxation scheme for this modification to make it more easily implemented. Convergence results of both algorithms are analyzed, and preliminary numerical results are reported. We also extend these modified algorithms to solve the multiple-sets split feasibility problem, which is to find a point closest to the intersection of a family of closed convex sets in one space such that its image under a linear transformation will be closest to the intersection of another family of closed convex sets in the image space.