JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2016, Vol. 51 ›› Issue (4): 86-89.doi: 10.6040/j.issn.1671-9352.0.2015.384

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Equivalence between the affine matrix rank minimization problem and the unconstrained matrix rank minimization problem

CUI An-gang, LI Hai-yang*   

  1. School of Science, Xian Polytechnic University, Xian 710048, Shaanxi, China
  • Received:2015-08-04 Online:2016-04-20 Published:2016-04-08

Abstract: Equivalence between the affine matrix rank minimization problem and the unconstrained matrix rank minimization problem is proved; i.e., there is a λ0>0 such that, for any λ∈(00), the affine matrix rank minimization problem and the unconstrained matrix rank minimization problem have the same optimal solution. Based on this, it is feasible to use the optimal solution to the unconstrained penalty function matrix rank minimization problem to approximate the optimal solution to the affine matrix rank minimization problem.

Key words: the unconstrained penalty function matrix rank minimization problem, the unconstrained matrix rank minimization problem, the affine matrix rank minimization problem

CLC Number: 

  • O242.2
[1] CANDES E J, RECHT B. Exact matrix completion via convex optimization[J]. Foundations of Computational Mathematics, 2009, 9:717-772.
[2] JANNACH D, ZANKER M, FELFERNIG A, FRIEDRICH G. Recommender systems: an introduction[M]. New York: Cambridge University Press, 2012: 38-40.
[3] FAZEL M, HINDI H, BOYD S. A rank minimization heuristic with application to minimum order approximation[C] // Proceedings of the American Control Conference. Arlington: ACC Press, 2001, 6:4734-4739.
[4] FAZEL M, HINDI H, BOYD S. Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices[C] // Proceedings of the American Control Conference. Denver: ACC Press, 2003, 3:2156-2162.
[5] JI Senshan, ZHOU Zirui, ANTHONY M S, et al. Beyond convex relaxation: a polynomial-time non-convex optimization approach to network localization[C] // Proceedings of the 32nd IEEE International Conference on Computer Communications. Hungary: ICC Press, 2013, 10(11):2499-2507.
[6] RECHT B, FAZEL M, PARRILO P A. Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization[J]. SIAM Review, 2012, 52(3):471-501.
[7] CAI Jianfeng, CANDES E J, SHEN Zuowei. A singular value thresholding algorithm for matrix completion[J]. SIAM Journal on Optimization, 2010, 20(4):1956-1982.
[8] CANDES E J, TAO T. The power of convex relaxation: near-optimal matrix completion[J]. IEEE Transactions on Information Theory, 2010, 56(5):2053-2080.
[9] RAGHUNANDAN H Keshavan, ANDREA Montanari, SEWOONG Oh. Matrix completion from a few entries[J]. IEEE Transactions on Information Theory, 2010, 56(6): 2980-2998.
[10] LAI Mingjun, XU Yangyang, YIN Wotao. Improved iteratively reweighted least squares for unconstrained smoothed lq minimization[J]. SIAM Journal on Numerical Analysis, 2013, 51(2):927-957.
[11] MA Shiqian, GOLDFARB D, CHEN Lifeng. Fixed point and bregman iterative methods for matrix rank minimization[J]. Mathematical Programming Computation, 2011, 128(12):321-353.
[12] MOHAN K, FAZEL M. Iterative reweighted algorithms for matrix rank minimization[J]. The Journal of Machine Learning Research, 2012, 13(1):3441-3473.
[13] WEN Zaiwen, YIN Wotao, ZHANG Yin. Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm[J]. Mathematical Programming Computation, 2012, 4(4):333-361.
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