An iterative regularization method for the solution of the split feasibility problem in Banach spaces

The split feasibility problem (SFP) consists of finding a common point in the intersection of finitely many convex sets, where some of the sets arise by imposing convex constraints in the range of linear operators. We are concerned with its solution in Banach spaces. To this end we generalize the CQ algorithm of Byrne with Bregman and metric projections to obtain an iterative solution method. In case the sets projected onto are contaminated with noise we show that a discrepancy principle renders this algorithm a regularization method. We measure the distance between convex sets by local versions of the Hausdorff distance, which in contrast to the standard Hausdorff distance allow us to measure the distance between unbounded sets. Hereby we prove a uniform continuity result for both kind of projections. The performance of the algorithm is demonstrated with some numerical experiments.