Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces

The split feasibility problem (SFP) (Censor and Elfving 1994 Numer. Algorithms 8 221-39) is to find a point x* with the property that x* is an element of C and Ax* is an element of Q, where C and Q are the nonempty closed convex subsets of the real Hilbert spaces H(1) and H(2), respectively, and A is a bounded linear operator from H(1) to H(2). The SFP models inverse problems arising from phase retrieval problems (Censor and Elfving 1994 Numer. Algorithms 8 221-39) and the intensity-modulated radiation therapy (Censor et al 2005 Inverse Problems 21 2071-84). In this paper we discuss iterative methods for solving the SFP in the setting of infinite-dimensional Hilbert spaces. The CQ algorithm of Byrne (2002 Inverse Problems 18 441-53, 2004 Inverse Problems 20 10320) is indeed a special case of the gradient-projection algorithm in convex minimization and has weak convergence in general in infinite-dimensional setting. We will mainly use fixed point algorithms to study the SFP. A relaxed CQ algorithm is introduced which only involves projections onto half-spaces so that the algorithm is implementable. Both regularization and iterative algorithms are also introduced to find the minimum-norm solution of the SFP.