REGULARIZATION FOR THE SPLIT FEASIBILITY PROBLEM

The split feasibility problem (SFP) models inverse problems that arise from phase retrievals and medical treatments such as the intensity-modulated radiation therapy. It is formulated as the problem of finding 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 closed convex subsets of R-n and R-m, respectively, and A is an m x n matrix. The SFP is usually ill-posed and regularization is therefore needed. In this paper we use the l(p)-norm to regularize the SFP; hence we have a differentiable regularizer if 1 < p < infinity and a nondifferentiable regularizer if p = 1. Various properties for the l(p)-norm regularization of the SFP are obtained, one of which says that the l(p)-norm of the solution of the regularized SFP tends to the least l(p)-norm of the solution set of the SFP and of the closed convex set C as the regularization parameter tends to zero and the infinity, respectively.