Iterative oblique projection onto convex sets and the split feasibility problem

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: x = PC(x + γ A (PQ − I )Ax), where γ ∈ (0, 2/L) with L the largest eigenvalue of the matrix AT A and PC and PQ denote the orthogonal projections onto C and Q, respectively; that is, PC x minimizes ‖c − x‖, over all c ∈ C . The CQ algorithm converges to a solution of the SFP, or, more generally, to a minimizer of ‖PQ Ac − Ac‖ over c in C , whenever such exist. The CQ algorithm involves only the orthogonal projections onto C and Q, which we shall assume are easily calculated, and involves no matrix inverses. If A is normalized so that each row has length one, then L does not exceed the maximum number of nonzero entries in any column of A, which provides a helpful estimate of L for sparse matrices. Particular cases of the CQ algorithm are the Landweber and projected Landweber methods for obtaining exact or approximate solutions of the linear equations Ax = b; the algebraic reconstruction technique of Gordon, Bender and Herman is a particular case of a block-iterative version of the CQ algorithm. One application of the CQ algorithm that is the subject of ongoing work is dynamic emission tomographic image reconstruction, in which the vector x is the concatenation of several images corresponding to successive discrete times. The matrix A and the set Q can then be selected to impose constraints on the behaviour over time of the intensities at fixed voxels, as well as to require consistency (or near consistency) with measured data. 0266-5611/02/020441+13$30.00 © 2002 IOP Publishing Ltd Printed in the UK 441