Fast self-adaptive regularization iterative algorithm for solving split feasibility problem

Split feasibility problem (SFP) is to find a point which belongs to one convex set in one space, such that its image under a linear transformation belongs to another convex set in the image space. This paper deals with a unified regularized SFP for the convex case. We first construct a self-adaptive regularization iterative algorithm by using Armijo-like search for the SFP and show it converges at a subliner rate of $ O(1/k) $, where $ k $ represents the number of iterations. More interestingly, inspired by the acceleration technique introduced by Nesterov[12], we present a fast Armijo-like regularization iterative algorithm and show it converges at rate of $ O(1/k^{2}) $. The efficiency of the algorithms is demonstrated by some random data and image debluring problems.