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An efficient method for solving a matrix least squares problem over a matrix inequality constraint

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Abstract

In this paper, we consider solving a class of matrix inequality constrained matrix least squares problems of the form

$$\begin{aligned} \begin{array}{rl} \text {min}&{}\dfrac{1}{2}\Vert \sum \limits _{i=1}^{t}A_iXB_i-C\Vert^2\\ \text {subject}\ \text {to}&{} L \le EXF\le U, \ \ X\in \mathcal {S}, \end{array} \end{aligned}$$

where \(\Vert {\cdot } \Vert \) is the Frobenius norm, matrices \(A_i\in \mathbb {R}^{l\times m}, B_i\in \mathbb {R}^{n\times s}\) \((i=1,\ldots , t), C\in \mathbb {R}^{l\times s}, E\in \mathbb {R}^{p\times m}, F\in \mathbb {R}^{n\times q}\) and \(L, U\in \mathbb {R}^{p\times q}\) are given. An inexact version of alternating direction method (ADM) with truly implementable inexactness criteria is proposed for solving this problem and its several reduced versions which are applicable in image restoration. Numerical experiments are performed to illustrate the feasibility and efficiency of the proposed algorithm, including when the algorithm is tested with randomly generated data and on some image restoration problems. Comparisons with some existing methods (with necessary modifications) are also given.

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Acknowledgments

The authors are grateful to the anonymous referees for valuable comments and suggestions which helped improve the exposition of this paper. Research supported by National Natural Science Foundation of China (11301107, 11261014, 11271144), Guangxi provincial Natural Science Foundation (2013GXNSFBA019009), China Postdoctoral Science Foundation (2014M552212), Guangdong provincial Natural Science Foundation (S2012010009985, S20130100112530) and Project of Department of Education of Guangdong Province (2013KJCX0053).

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, People’s Republic of China

    Jiao-fen Li & Wen Li

  2. School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin, People’s Republic of China

    Jiao-fen Li

  3. School of Mathematical Sciences, Fudan University, Shanghai, People’s Republic of China

    Ru Huang

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  1. Jiao-fen Li
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Correspondence to Jiao-fen Li.

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Li, Jf., Li, W. & Huang, R. An efficient method for solving a matrix least squares problem over a matrix inequality constraint. Comput Optim Appl 63, 393–423 (2016). https://doi.org/10.1007/s10589-015-9783-z

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  • DOI: https://doi.org/10.1007/s10589-015-9783-z

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