On the convergence analysis of the gradient-CQ algorithms for the split feasibility problem
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On the convergence analysis of the gradient-CQ algorithms for the split feasibility problem Suparat Kesornprom1 · Nattawut Pholasa1 · Prasit Cholamjiak1 Received: 15 July 2018 / Accepted: 25 July 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this work, based on the gradient method and the relaxed CQ algorithm introduced by L´opez et al. (Inverse Probl. 28, 085004, 2012), we introduce two gradient-CQ algorithms for solving the split feasibility problem in the framework of Hilbert spaces. The main advantage of the proposed method is not only that the variable stepsizes depending on the information from the current iterate not the operator norm are chosen but also that the metric projection onto half-spaces, which is convenient to be calculated, is taken into account. Then both weak and strong convergence are proved under some mild conditions. Finally, numerical experiments in signal processing reveal that the proposed algorithm is effective and outruns those of Yang, L´opez et al., Gibali et al., and others. Keywords Split feasibility problem · CQ algorithm · Hilbert space · Gradient method Mathematics Subject Classification (2010) 47H10 · 54H25
1 Introduction We study the split feasibility problem (SFP) which is described as the following form: find a point x ∗ ∈ C such that Ax ∗ ∈ Q (1) Prasit Cholamjiak
[email protected] Suparat Kesornprom [email protected] Nattawut Pholasa nattawut [email protected] 1
School of Science, University of Phayao, Phayao, 56000, Thailand
Numerical Algorithms
where C and Q are nonempty closed convex subsets of real Hilbert spaces H1 and H2 , respectively, and A : H1 → H2 is a bounded linear operator. This problem was first introduced, in Euclidean spaces, by Censor and Elfving [5]. The SFP relates to an inverse problem in intensity-modulated radiation therapy (IMRT) in the field of medical care and the LASSO problem in signal recovery and image processing. Throughout this work, we assume that SFP (1) is consistent and denote the solution set by S. Byrne [3, 4] introduced CQ algorithm which generates a sequence {xn } as follows: xn+1 = PC (xn − τn A∗ (I − PQ )Axn )
(2)
A∗
is the adjoint operator of A, PC and PQ where the stepsize τn ∈ are the metric projections onto C and Q, respectively. It is seen that if the metric projections onto C and Q are easily calculated, then the total cost of computation is not great. However, in some cases, it is impossible or needs too much work to exactly compute the metric projection. The determination of the stepsize depends on the operator norm which computation (or at least estimate) is not an easy task. In practical applications, the sets C and Q are usually the level sets of convex functions which are given by (0, 2/A2 ),
C = {x ∈ H1 : c(x) ≤ 0} and Q = {y ∈ H2 : q(y) ≤ 0}
(3)
where c : H1 → R and q : H2 → R are convex and subdifferential functions on H1 and H2 , respectively, and that ∂c and ∂q are bounded operators (i.e., bounded on bounded sets). Fukushima [7] proposed a relaxed proj
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