The regularized CQ algorithm without a priori knowledge of operator norm for solving the split feasibility problem

摘要

The split feasibility problem (SFP) is finding a point x ∈ C such that Ax ∈ Q, where C and Q are nonempty closed convex subsets of Hilbert spaces H1 and H2, and A:H1 → H2 is a bounded linear operator. Byrne’s CQ algorithm is an effective algorithm to solve the SFP, but it needs to compute  ∥ A ∥ , and sometimes  ∥ A ∥  is difficult to work out. López introduced a choice of stepsize λn, λn=ρnf(xn)∥∇f(xn)∥2, 0  ρn  4. However, he only obtained weak convergence theorems. In order to overcome the drawbacks, in this paper, we first provide a regularized CQ algorithm without computing  ∥ A ∥  to find the minimum-norm solution of the SFP and then obtain a strong convergence theorem.