Abstract
We give an explicit formula for the generalized subdifferentials; i.e. the proximal subdifferential, the Fréchet subdifferential, the limitting subdifferential and the Clarke subdifferential of the counting function. Then, thanks to theorems of A.S. Lewis and H.S. Sendov, we obtain the corresponding generalized subdifferentials of the rank function.
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Clarke F.H.: Optimisation and Nonsmooth Analysis. Wiley, New York (1983)
Hiriart-Urruty, J.-B.: When only global optimization matters. J. Global Optim. (2012). doi:10.1007/s10898-011-9826-7 http://www.springerlink.com/content/33t63w1407n12lm0/
Le, H.Y.: Convexifying the counting function on \({\mathbb{R}^p}\) for convexifying the rank function on \({\mathcal{M}_{m,n}(\mathbb{R})}\) . J. Convex Anal. (2011). http://www.heldermann.de/JCA/JCA19/JCA192/jca19028.htm
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)
Mordukhovich B.S.: Variational Analysis And Generalized Differentiation I. Springer, Berlin (2006)
Rockafellar R.T., Wets R.J.-B.: Variational Analysis. Springer, Berlin (1998)
Schirotzek W.: Nonsmooth Analysis. Springer, Berlin (2007)
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Le, H.Y. Generalized subdifferentials of the rank function. Optim Lett 7, 731–743 (2013). https://doi.org/10.1007/s11590-012-0456-x
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DOI: https://doi.org/10.1007/s11590-012-0456-x