, Volume 21, Issue 2, pp 207–240 | Cite as

A variational approach of the rank function

  • Jean-Baptiste Hiriart-Urruty
  • Hai Yen LeEmail author
Invited Paper


In the same spirit as the one of the paper (Hiriart-Urruty and Malick in J. Optim. Theory Appl. 153(3):551–577, 2012) on positive semidefinite matrices, we survey several useful properties of the rank function (of a matrix) and add some new ones. Since the so-called rank minimization problems are the subject of intense studies, we adopt the viewpoint of variational analysis, that is the one considering all the properties useful for optimizing, approximating or regularizing the rank function.


Rank of matrix Optimization Nonsmooth analysis Moreau–Yosida regularization Generalized subdifferentials 

Mathematics Subject Classification

49-02 65K10 



We thank the various readers or referees who, after the first circulation of this survey paper (July 2012), provided comments, improvements and additional references.

Last but not least, we would like to thank Professor M. Goberna (University of Alicante) for his instrumental role in the publication of this survey paper.


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Copyright information

© Sociedad de Estadística e Investigación Operativa 2013

Authors and Affiliations

  1. 1.Institute of MathematicsPaul Sabatier UniversityToulouseFrance

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