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Nonsmooth Optimization and Its Applications pp 45–53Cite as

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Tangent and Normal Cones for Low-Rank Matrices

Part of the International Series of Numerical Mathematics book series (ISNM,volume 170)

Abstract

In (D. R. Luke, J. Math. Imaging Vision, 47 (2013), 231–238) the structure of the Mordukhovich normal cone to varieties of low-rank matrices at rank-deficient points has been determined. A simplified proof of that result is presented here. As a corollary we obtain the corresponding Clarke normal cone. The results are put into the context of first-order optimality conditions for low-rank matrix optimization problems.

Keywords

  • Matrix optimization
  • Low rank constraint
  • Optimality conditions

Mathematics Subject Classification (2000)

  • Primary 15B99
  • 49J52; Secondary 65K10

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Notes

  1. 1.

    Some inaccuracies in the statement of Theorem 3.1 in [7] are corrected here. Also, the “⊆” part is proven by a more direct argument compared to [7].

References

  1. F. Bernard, L. Thibault, and N. Zlateva, Prox-regular sets and epigraphs in uniformly convex Banach spaces: various regularities and other properties, Trans. Amer. Math. Soc., 363 (2011), 2211–2247.

    CrossRef  MathSciNet  Google Scholar 

  2. T. P. Cason, P.-A. Absil, and P. Van Dooren, Iterative methods for low rank approximation of graph similarity matrices, Linear Algebra Appl., 438 (2013), 1863–1882.

    CrossRef  MathSciNet  Google Scholar 

  3. D. Drusvyatskiy and A. S. Lewis, Optimality, identifiability, and sensitivity, Math. Program., 147 (2014), 467–498.

    CrossRef  MathSciNet  Google Scholar 

  4. M. Guignard, Generalized Kuhn–Tucker conditions for mathematical programming problems in a Banach space, SIAM J. Control, 7 (1969), 232–241.

    CrossRef  MathSciNet  Google Scholar 

  5. J. Harris, Algebraic geometry. A First Course, Springer-Verlag, New York, 1992.

    Google Scholar 

  6. U. Helmke and M. A. Shayman, Critical points of matrix least squares distance functions, Linear Algebra Appl., 215 (1995), 1–19.

    CrossRef  MathSciNet  Google Scholar 

  7. D. R. Luke, Prox-regularity of rank constraints sets and implications for algorithms, J. Math. Imaging Vision, 47 (2013), 231–238.

    CrossRef  MathSciNet  Google Scholar 

  8. B. S. Mordukhovich, Variational analysis and generalized differentiation. I, Springer-Verlag, Berlin, 2006.

    Google Scholar 

  9. R. Schneider and A. Uschmajew, Convergence results for projected line-search methods on varieties of low-rank matrices via Łojasiewicz inequality, SIAM J. Optim., 25 (2015), 622–646.

    CrossRef  MathSciNet  Google Scholar 

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Acknowledgements

We thank B. Kutschan for bringing Harris’ book [5] as a reference for the tangent cone \(T^B_{\mathcal {M}_{\le k}}\) to our attention, and for pointing out that formula (2.5) is equivalent to (2.6).

Author information

Authors and Affiliations

  1. Hausdorff Center for Mathematics & Institute for Numerical Simulation, University of Bonn, Bonn, Germany

    Seyedehsomayeh Hosseini

  2. Institute for Numerical and Applied Mathematics, University of Göttingen, Göttingen, Germany

    D. Russell Luke

  3. Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

    André Uschmajew

Authors
  1. Seyedehsomayeh Hosseini
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  2. D. Russell Luke
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  3. André Uschmajew
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Corresponding author

Correspondence to André Uschmajew .

Editor information

Editors and Affiliations

  1. IAV GmbH, Gifhorn, Germany

    Seyedehsomayeh Hosseini

  2. Department of Mathematics, Wayne State University, Detroit, MI, USA

    Boris S. Mordukhovich

  3. Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

    André Uschmajew

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Hosseini, S., Luke, D.R., Uschmajew, A. (2019). Tangent and Normal Cones for Low-Rank Matrices. In: Hosseini, S., Mordukhovich, B., Uschmajew, A. (eds) Nonsmooth Optimization and Its Applications. International Series of Numerical Mathematics, vol 170. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-11370-4_3

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