Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces

Abstract

The split feasibility problem (SFP) (Censor and Elfving 1994 Numer. Algorithms 8 221-39) is to find a point x * with the property that x * ∈ C and Ax * ∈ Q, where C and Q are the nonempty closed convex subsets of the real Hilbert spaces H 1 and H 2 , respectively, and A is a bounded linear operator from H 1 to H 2 . The SFP models inverse problems arising from phase retrieval problems (Censor and Elfving 1994 Numer. Algorithms 8 221-39) and the intensity-modulated radiation therapy (Censor et al 2005 Inverse Problems 21 2071-84). In this paper we discuss iterative methods for solving the SFP in the setting of infinite-dimensional Hilbert spaces. The CQ algorithm of Byrne (2002 Inverse Problems 18 2004 Inverse Problems 20 103-20) is indeed a special case of the gradient-projection algorithm in convex minimization and has weak convergence in general in infinite-dimensional setting. We will mainly use fixed point algorithms to study the SFP. A relaxed CQ algorithm is introduced which only involves projections onto half-spaces so that the algorithm is implementable. Both regularization and iterative algorithms are also introduced to find the minimum-norm solution of the SFP.

Viscosity approximation methods for the multiple-set split equality common fixed-point problems of demicontractive mappings

“…where P C and P Q are the metric projections from H 1 onto C and from H 2 onto Q, respectively, γ is a positive constant and A * denotes the adjoint of A (see [15,Proposition 3.2] for the details). This implies that the SFP (1.1) can be solved by using fixed point algorithms.…”

Strong and weak convergence theorems for split equilibrium problems and fixed point problems in Banach spaces

“…For more results on split feasibility problems, the readers refer to [3,4,6,9,29,34,35]. Now, by Theorem 3.1 and Theorem 4.2, we give the following results on split feasibility problems in Banach spaces: Theorem 5.1.…”

“…Recently, split feasibility problems [3,4,6,9,29,34,35], split variational inequality problems [10,21] and split equilibrium problems [2,17,31] have been investigated by many authors. However, most of the results on these kinds of these problems are investigated only in Hilbert spaces, only a few works are considered in Banach spaces.…”

Strong convergence theorems for a nonexpansive mapping and its applications for solving the split feasibility problem

“…As we know, the SFP has received so much attention due to its applications in intensity-modulated radiation therapy, signal processing, and image reconstruction, see Byrne [1,2], Censor [4][5][6], Ceng [3], Fan et al [7], Xu [20,21], Kraikaew and Saejung [9], Moudafi [10], Qu et al [12][13][14], Qin and Yao [11], Yang et al [16,22,27,28], Yao et al [23][24][25][26], and so on.…”