Inertial relaxed CQ algorithms for solving a split feasibility problem in Hilbert spaces
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Inertial relaxed CQ algorithms for solving a split feasibility problem in Hilbert spaces D.R. Sahu1 · Y.J. Cho2,3 · Q.L. Dong4
· M.R. Kashyap1 · X.H. Li4
Received: 10 May 2019 / Accepted: 5 August 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The split feasibility problem is to find a point x ∗ with the property that x ∗ ∈ C and Ax ∗ ∈ Q, where C and Q are nonempty closed convex subsets of real Hilbert spaces X and Y , respectively, and A is a bounded linear operator from X to Y . The split feasibility problem models inverse problems arising from phase retrieval problems and the intensity-modulated radiation therapy. In this paper, we introduce a new inertial relaxed CQ algorithm for solving the split feasibility problem in real Hilbert spaces and establish weak convergence of the proposed CQ algorithm under certain mild conditions. Our result is a significant improvement of the recent results related to the split feasibility problem. Keywords Split feasibility problem · CQ algorithm · Inertial technique · Self-adaptive algorithm · Weak convergence Mathematics Subject Classification (2010) 65K05 · 65K10 · 49J52
1 Introduction The split feasibility problem in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [8] in 1994, for modeling inverse problem which arises from the phase retrievals and in medical image reconstruction [7]. The split feasibility problem can also be used to model the intensity-modulated radiation therapy [9]. Let X be a real Hilbert space with the inner product ·, ·, the induced norm · and C be a nonempty closed and convex subset of X. Let Q be a nonempty closed and convex subset a real Hilbert space Y . Let I denote the identity operator on X.
Q.L. Dong
[email protected]
Extended author information available on the last page of the article.
Numerical Algorithms
The split feasibility problem (SFP) is formulated as follows: Find x ∗ ∈ C such that Ax ∗ ∈ Q
(1.1)
if such points exist, where A : X → Y is a bounded linear operator. The solution set of the problem (SFP) (1.1) is denoted by , i.e., := {x ∗ ∈ C : Ax ∗ ∈ Q}. The original algorithm for solving the problem (SFP) (1.1) introduced by Censor and Elfving [8] involves the computation of the inverse A−1 (assuming the existence of the inverse of A) and since then it has been studied extensively and generalized in various ways for finite and infinite dimensional spaces. The problem (SFP) (1.1) has been applied successfully in many real-world problems such as for signal processing, image reconstruction and many more (for theory and application, the reader is referred to [4, 6, 10, 17, 20, 21, 27, 32, 35, 36] and the references therein). A more popular algorithm that solves the SFP (1.1) seems to be the CQ algorithm of Byrne [7] which generates a sequence {xn } by the iterative procedure: for any x1 ∈ X, xn+1 = PC (xn − γ A∗ (I − PQ )Axn ), ∀n ≥ 1,
(1.2)
where the step size γ is chosen in the open interval (0, 2/A2 ), while PC and PQ are the orthogonal proj
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