An Adaptive Relaxed Algorithm For The Split Feasibility

The split feasibility problem(SFP) has received much attention due to its applications in signal processing and image reconstruction and many other applied fields. Many effective algorithms have been proposed. The split feasibility problem(SFP) was first proposed in Euclidean space, in recent years, for the purpose of generality, some authors study the split feasibility problem(SFP) in Hilbert space or Banach space, and a number of algorithms are proposed. For instance, in 2012 López et al. proposed a relaxed CQ algorithm with a new adaptive way of determining the stepsize for solving the SFP in Hilbert space. This algorithm can be implemented easily due to the following two reasons. First, this algorithm has the adaptive way of determining the stepsize and has no need to know a priori the norm of the bounded linear operator. Second, this algorithm computes the projection onto level set of a convex function by computing the projection onto a series of half-spaces containing a level set. However, their algorithm has only weak convergence. In this paper, we introduce a new relaxation algorithm such that the strong convergence is guaranteed in infinite-dimensional Hilbert spaces. Our result extends and improves some others.This paper mainly includes the following three aspects:First, we consider the split feasibility problem in Hilbert space, we propose a new adaptive relaxation algorithm, and the strong convergence is guaranteed.Second, we discuss the multiple-sets split feasibility problem(MSFP) in Hilbert space, we will promote the above algorithm to multiple sets split feasibility problems,and we prove the strong convergence of the algorithm, and some numerical results show the effectiveness of the algorithms.Third, we study the multiple-sets split feasibility problem(MSFP) in Hilbert space, we propose a new adaptive relaxation algorithm, which strong convergence will be proved.