Tangent and Normal Cones for Low-Rank Matrices

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


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.


Matrix optimization Low rank constraint Optimality conditions 

Mathematics Subject Classification (2000)

Primary 15B99 49J52; Secondary 65K10 



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).


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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

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