Iterative oblique projection onto convex sets and the split feasibility problem

Abstract

Let C and Q be nonempty closed convex sets in RN and RM, respectively, and A an M by N real matrix. The split feasibility problem (SFP) is to find x ∊ C with Ax ∊ Q, if such x exist. An iterative method for solving the SFP, called the CQ algorithm, has the following iterative step: xk+1 = PC (xk + γAT (PQ − I)Axk), where γ ∊ (0, 2∖L) with L the largest eigenvalue of the matrix ATA and PC and PQ denote the orthogonal projections onto C and Q, respectively; that is, PCx minimizes ||c − x||, over all c ∊ C. The CQ algorithm converges to a solution of the SFP, or, more generally, to a minimizer of ||PQAc − Ac|| over c in C, whenever such exist.