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

Abstract

Abstract The split feasibility problem (SFP) is finding a point x ∈ C $x\in C$ such that A x ∈ Q $Ax\in Q$ , where C and Q are nonempty closed convex subsets of Hilbert spaces H 1 $H_{1}$ and H 2 $H_{2}$ , and A : H 1 → H 2 $A:H_{1}\rightarrow H_{2}$ is a bounded linear operator. Byrne’s CQ algorithm is an effective algorithm to solve the SFP, but it needs to compute ∥ A ∥ $\|A\|$ , and sometimes ∥ A ∥ $\|A\| $ is difficult to work out. López introduced a choice of stepsize λ n $\lambda_{n}$ , λ n = ρ n f ( x n ) ∥ ∇ f ( x n ) ∥ 2 $\lambda_{n}=\frac{\rho_{n}f(x_{n})}{\|\nabla f(x_{n})\| ^{2}}$ , 0 < ρ n < 4 $0<\rho_{n}<4$ . 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 ∥ $\|A\|$ to find the minimum-norm solution of the SFP and then obtain a strong convergence theorem.

Keywords

• split feasibility problem
• regularized CQ algorithm
• minimum-norm solution
• strong convergence
• operator norm