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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)


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

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  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].


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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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Correspondence to 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.

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