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Note on the modified relaxation CQ algorithm for the split feasibility problem

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Abstract

In this paper we are concerned with the Split Feasibility Problem (SFP) in which there are given two Hilbert spaces \(H_1\) and \(H_2\), nonempty, closed and convex sets \(C\subseteq H_1\) and \(Q\subseteq H_2\), and a bounded and linear operator \(A:H_1 \rightarrow H_2\). The SFP is then to find a point \(x^*\in C\) such that its image under A belongs to Q, meaning that \(Ax^*\in Q\). This reformulation was employed successfully for solving many inverse problems: for example, in intensity-modulated radiation therapy treatment planning. One of the typical classes of methods that have been used to solve the SFP is the class of projection method. This note focuses on the modified relaxation CQ algorithm with the Armijo-line search rule for solving the SFP. Under common and standard assumptions, we show that the proposed method weakly converges to a solution of the SFP. Numerical examples illustrating our method’s efficiency are presented for solving the LASSO problem in which the goal is to recover a sparse signal from a limited number of observations.

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Acknowledgements

This work was supported by a Visiting Scholarship of Academy of Mathematics and Systems Science, the Chinese Academy of Sciences (AM201622C04), the National Natural Science Foundations of China (11401293,11661056), the Natural Science Foundations of Jiangxi Province (20151BAB211010, 20142BAB211016), the China Postdoctoral Science Foundation (2015M571989) and the Jiangxi Province Postdoctoral Science Foundation (2015KY51). The first author’s work is supported by the EU FP7 IRSES Program STREVCOMS, Grant No. PIRSES-GA-2013-612669.

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Authors and Affiliations

  1. Department of Mathematics, ORT Braude College, 2161002, Karmiel, Israel

    Aviv Gibali

  2. Department of Mathematics, NanChang University, Nanchang, 330031, People’s Republic of China

    Li-Wei Liu & Yu-Chao Tang

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  1. Aviv Gibali
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  2. Li-Wei Liu
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  3. Yu-Chao Tang
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Correspondence to Yu-Chao Tang.

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Gibali, A., Liu, LW. & Tang, YC. Note on the modified relaxation CQ algorithm for the split feasibility problem. Optim Lett 12, 817–830 (2018). https://doi.org/10.1007/s11590-017-1148-3

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