Advertisement

Tangent and Normal Cones for Low-Rank Matrices

Chapter
  • 486 Downloads
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 

Notes

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).

References

  1. 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.MathSciNetCrossRefGoogle Scholar
  2. 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.MathSciNetCrossRefGoogle Scholar
  3. 3.
    D. Drusvyatskiy and A. S. Lewis, Optimality, identifiability, and sensitivity, Math. Program., 147 (2014), 467–498.MathSciNetCrossRefGoogle Scholar
  4. 4.
    M. Guignard, Generalized Kuhn–Tucker conditions for mathematical programming problems in a Banach space, SIAM J. Control, 7 (1969), 232–241.MathSciNetCrossRefGoogle Scholar
  5. 5.
    J. Harris, Algebraic geometry. A First Course, Springer-Verlag, New York, 1992.Google Scholar
  6. 6.
    U. Helmke and M. A. Shayman, Critical points of matrix least squares distance functions, Linear Algebra Appl., 215 (1995), 1–19.MathSciNetCrossRefGoogle Scholar
  7. 7.
    D. R. Luke, Prox-regularity of rank constraints sets and implications for algorithms, J. Math. Imaging Vision, 47 (2013), 231–238.MathSciNetCrossRefGoogle Scholar
  8. 8.
    B. S. Mordukhovich, Variational analysis and generalized differentiation. I, Springer-Verlag, Berlin, 2006.Google Scholar
  9. 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.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Hausdorff Center for Mathematics & Institute for Numerical SimulationUniversity of BonnBonnGermany
  2. 2.Institute for Numerical and Applied MathematicsUniversity of GöttingenGöttingenGermany
  3. 3.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

Personalised recommendations