Iterative oblique projection onto convex sets and the split feasibility problem

Abstract

Let C andQ be nonempty closedconvex sets in RNand RM,respectively, and Aan Mby Nreal matrix. The split feasibility problem (SFP) is to find x ∊ C withAx ∊ Q,if such xexist. An iterative method for solving the SFP, called the CQ algorithm,has the following iterative step: xk+1 = P C (xk + γAT (P Q − I)Axk), where γ ∊ (0, 2∖L) withL the largest eigenvalueof the matrix ATAand PC andPQ denote the orthogonalprojections onto C andQ, respectively; thatis, PCx minimizes||c − x||,over all c ∊ C.The CQalgorithm converges to a solution of the SFP, or, more generally, to a minimizer of ||PQAc − Ac||over c inC, whenever such exist.The CQalgorithm involves only the orthogonal projections ontoC andQ,which we shall assume are easily calculated, and involves no matrix inverses. IfAis normalized so that each row has length one, thenL doesnot exceed the maximum number of nonzero entries in any column ofA, which provides ahelpful estimate of Lfor sparse matrices.Particular cases of the CQalgorithm are the Landweber and projected Landweber methods forobtaining exact or approximate solutions of the linear equationsAx = b;the algebraic reconstruction technique of Gordon, Bender and Hermanis a particular case of a block-iterative version of the CQalgorithm.One application of the CQ algorithm that is the subject of ongoingwork is dynamic emission tomographic image reconstruction, in whichthe vectorx is theconcatenation of several images corresponding to successive discrete times. The matrixA andthe set Qcan then be selected to impose constraints on the behaviour over time of theintensities at fixed voxels, as well as to require consistency (or near consistency)with measured data.