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

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.

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Notes

  1. 1.

    Σ “pseudo-diagonal” means that Σ ij = 0 for i ≠ j.

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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).

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Correspondence to Hai Yen Le.

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Dedicated to Lionel Thibault at the occasion of his 65th birthday and the honorary degree (doctorate honoris causa) conferred by The University of Santiago, Chile.

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Hiriart-Urruty, JB., Le, H.Y. The Viscosity Subdifferential of the Rank Function via the Corresponding Subdifferential of its Moreau Envelopes. Acta Math Vietnam 40, 735–746 (2015). https://doi.org/10.1007/s40306-015-0118-z

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Keywords

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

Mathematics Subject Classification (2010)

  • 15A
  • 46N10
  • 65K10
  • 90C