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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
Σ “pseudo-diagonal” means that Σ ij = 0 for i ≠ j.
Clarke, F.H.: Functional analysis, calculus of variations and optimal control. Springer (2013)
Fazel, M.: Matrix rank minimization with applications.PhD thesis,Stanford University (2002)
Hiriart-Urruty, J.-B.: Bases, outils et Principes pour L’analyse Variationnelle. Springer (2012)
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)
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)
Jourani, A.: Limit superior of subdifferentials of uniformly convergent functions. Positivity 3, 33–47 (1999)
Le, H.Y.: The generalized subdifferentials of the rank function. Optimization Letters 7 (4), 731–743 (2013)
Lewis, A.S., Sendov, H.S.: Nonsmooth analysis of singular values. Part I: Theory. Set-Valued Anal. 13, 213–241 (2005)
Lewis, A.S., Sendov, H.S.: Nonsmooth analysis of singular values. Part II: Applications. Set-Valued Anal. 13, 243–264 (2005)
Penot, J.-P.: Calculus without derivatives. Graduate Texts in Mathematics 266. Springer (2013)
Schirotzek, W.: Nonsmooth analysis. Springer (2007)
Rockafellar, R.T., Wets, R.J.-B.: Variational analysis. Springer (1998)
Zhao, Y.-B.: Approximation theory of matrix rank minimization and its application to quadratic equations. Linear Algebra Appl. 437(1), 77–93 (2012)
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).
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.
About this article
Cite this article
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
- Rank function
- Singular values of a matrix
- Moreau envelopes
- Generalized subdifferentials
- Viscosity subdifferential