Acta Mathematica Vietnamica

, Volume 40, Issue 4, pp 735–746 | Cite as

The Viscosity Subdifferential of the Rank Function via the Corresponding Subdifferential of its Moreau Envelopes

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Abstract

We derive the so-called viscosity subdifferential of the rank function via a limiting process applied to the Moreau envelopes of the rank function. Before that, we obtain the explicit expressions of all the generalized subdifferentials of the Moreau envelopes of the rank function.

Keywords

Rank function Singular values of a matrix Moreau envelopes Generalized subdifferentials Viscosity subdifferential 

Mathematics Subject Classification (2010)

15A 46N10 65K10 90C 

Notes

Acknowledgments

We would like to thank Prof. A. Jourani (University of Bourgogne, Dijon) for drawing our attention to this possible way of getting at the Fréchet generalized subdifferential of the rank function (ALEL meeting in Castro-Urdiales, June 2011).

References

  1. 1.
    Clarke, F.H.: Functional analysis, calculus of variations and optimal control. Springer (2013)Google Scholar
  2. 2.
    Fazel, M.: Matrix rank minimization with applications.PhD thesis,Stanford University (2002)Google Scholar
  3. 3.
    Hiriart-Urruty, J.-B.: Bases, outils et Principes pour L’analyse Variationnelle. Springer (2012)Google Scholar
  4. 4.
    Hiriart-Urruty, J.-B., Le, H.Y.: From Eckart-Young approximation to Moreau envelopes and vice versa. RAIRO - Oper. Res. 47(3), 299–310 (2013)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Hiriart-Urruty, J.-B., Le, H.Y.: A variational approach of the rank function. TOP J. Span. Soc. Stat. Oper. Res. 21 (2), 207–240 (2013)MATHMathSciNetGoogle Scholar
  6. 6.
    Jourani, A.: Limit superior of subdifferentials of uniformly convergent functions. Positivity 3, 33–47 (1999)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Le, H.Y.: The generalized subdifferentials of the rank function. Optimization Letters 7 (4), 731–743 (2013)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Lewis, A.S., Sendov, H.S.: Nonsmooth analysis of singular values. Part I: Theory. Set-Valued Anal. 13, 213–241 (2005)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Lewis, A.S., Sendov, H.S.: Nonsmooth analysis of singular values. Part II: Applications. Set-Valued Anal. 13, 243–264 (2005)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Penot, J.-P.: Calculus without derivatives. Graduate Texts in Mathematics 266. Springer (2013)Google Scholar
  11. 11.
    Schirotzek, W.: Nonsmooth analysis. Springer (2007)Google Scholar
  12. 12.
    Rockafellar, R.T., Wets, R.J.-B.: Variational analysis. Springer (1998)Google Scholar
  13. 13.
    Zhao, Y.-B.: Approximation theory of matrix rank minimization and its application to quadratic equations. Linear Algebra Appl. 437(1), 77–93 (2012)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Authors and Affiliations

  1. 1.Institute of MathematicsPaul Sabatier UniversityToulouseFrance
  2. 2.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam

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