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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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References

  1. Higham, N.J.: The symmetric procrustes problem. BIT Numer. Math. 28, 133–143 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. Andersson, L.E., Elfving, T.: A constrained procrustes problem. SIAM J. Matrix Anal. Appl. 18, 124–139 (2006)

    Article  MathSciNet  Google Scholar 

  3. Henk Don, F.J.: On the symmetric solution of a linear matrix equation. Linear Algebra Appl. 93, 1–7 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  4. Liao, A.P., Lei, Y.: Least-squares solutions of matrix inverse problem for bi-symmetric matrices with a submatrix constraint. Numer. Linear Algebra Appl. 14, 425–444 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bai, Z.J.: The inverse eigenproblem of centrosymmetric matrices with a submatrix constraint and its approximation. SIAM J. Matrix Anal. Appl. 26, 1100–1114 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Trench, W.F.: Inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry. Linear Algebra Appl. 380, 199–211 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Trench, W.F.: Minimization problems for \((R, S)\)-symmetric and \((R, S)\)-skew symmetric matrices. Linear Algebra Appl. 389, 23–31 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Wu, L., Cain, B.: The Re-nonnegative definite solutions to the matrix inverse problem \(AX=B\). Linear Algebra Appl. 236, 137–146 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Peng, Z.Y., Hu, X.Y.: The reflexive and anti-reflexive solutions of the matrix equation \(AX=B\). Linear Algebra Appl. 375, 147–155 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Meng, C.J., Hu, X.Y., Zhang, L.: The skew symmetric orthogonal solutions of the matrix equation \(AX=B\). Linear Algebra Appl. 402, 303–318 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Escalante, R., Raydan, M.: Dykstra’s algorithm for constrained least-squares rectangular matrix problems. Comput. Math. Appl. 6, 73–79 (1998)

    Article  MathSciNet  Google Scholar 

  12. Bouhamidi, A., Jbilou, K., Raydan, M.: Convex constrained optimization for large-scale generalized Sylvester equations. Comput. Optim. Appl. 48, 233–253 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Peng, Z.Y., Wang, L., Peng, J.J.: The solutions of matrix equation \(AX=B\) over a matrix inequality constraint. SIAM J. Matrix Anal. Appl. 33, 554–568 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, J.F., Wen, L., Peng, Z.Y.: A hybrid algorithm for solving minimization problem over \((R, S)\)-symmetric matrices with the matrix inequality constraint. Linear Multilinear Algebra 63, 1049–1072 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ng, M.K., Wang, F., Yuan, X.M.: Inexact alternating direction methods for image recovery. SIAM J. Sci. Comput. 33, 1643–1668 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bai, Z.J., Chen, M.X., Yuan, X.M.: Applications of the alternating direction method of multipliers to the semidefinite inverse quadratic eigenvalue problem with a partial eigenstructure. Inverse Prob. 29, 075011 (2013)

    Article  MathSciNet  Google Scholar 

  17. Zhao, Z., Bai, Z.J., Chen, G.Z.: On the alternating direction method of multipliers for nonnegative inverse eigenvalue problems with partial eigendata. J. Comput. Appl. Math. 239, 114–134 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Xiao, Y.H., Song, H.N.: An inexact alternating directions algorithm for constrained total variation regularized compressive sensing problems. J. Math. Imaging Vis. 44, 114–127 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chan, R.H., Yang, J.F., Yuan, X.M.: Alternating direction method for image inpainting in wavelet domains. SIAM J. Imaging Sci. 4, 807–826 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Yuan, X.M.: Alternating direction method for covariance selection models. J. Sci. Comput. 51, 261–273 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Bouhamidi, A., Jbilou, K.: A Kronecker approximation with a convex constrained optimization method for blind image restoration. Optim. Lett. 6, 1251–1264 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Birgin, E.G., Martínez, J.M., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 10, 1196–1211 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Paige, C.C., Saunders, A.: LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Softw. 8, 43–71 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  24. Peng, Z.Y.: Solutions of symmetry-constrained least-squares problems. Numer. Linear Algebra Appl. 15, 373–389 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Li, S.K., Huang, T.Z.: LSQR iterative method for generalized coupled Sylvester matrix equations. Appl. Math. Model. 36, 3545–3554 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. He, B.S.: Inexact implicit methods for monotone general variational inequalities. Math. Program. 86, 199–217 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. He, B.S., Liao, L.Z., Han, D., Yang, H.: A new inexact alternating directions method for monotone variational inequalities. Math. Program. 92, 103–118 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gu, G.Y., He, B.S., Yang, J.F.: Inexact alternating-direction-based contraction methods for separable linearly constrained convex optimization. J. Optim. Theory Appl. 163, 105–129 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Bnouhachem, A., Benazza, H., Khalfaoui, M.: An inexact alternating direction method for solving a class of structured variational inequalities. Appl. Math. Comput. 219, 7837–7846 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Birgin, E.G., Martínez, J.M.: Augmented Lagrangian method with nonmonotone penalty parameters for constrained optimization. Comput. Optim. Appl. 51, 941–965 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Birgin, E.G., Fernandez, D., Martnez, J.M.: The boundedness of penalty parameters in an augmented Lagrangian method with constrained subproblems. Optim. Methods Softw. 27, 1001–1024 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Bouhamidi, A., Enkhbat, R., Jbilou, K.: Conditional gradient Tikhonov method for a convex optimization problem in image restoration. J. Comput. Appl. Math. 255, 580–592 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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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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