The author discusses iterative methods for solving the split feasibility problem (SFP), i.e., to find a point in a infinite-dimensional Hilbert space such that and . Here and are nonempty closed convex subsets of , and is a bounded linear operator from into another infinite-dimensional Hilbert space . Such problems have been considered as models in phase retrievals, medical image reconstruction and recently also in the intensity-modulated radiation therapy.
In this paper the SFP problem is reformulated as a minimization problem to enable approximations via gradient-projection methods. Equivalently, the SFP problem is also considered as a fixed point equation so that iterative methods with adaptive relaxation factors can be used to approximate the solution as a fixed point. Both weak convergence to the original solution and strong convergence to the regularized solution are established.