Journal of Convex Analysis 15 (2008), No. 3, 547--560
Copyright Heldermann Verlag 2008
Prox-Regularity of Spectral Functions and Spectral Sets
Aris Daniilidis
Dep. de Matemàtiques C1/308, Universitat Autònoma de Barcelona, 08193 Bellaterra - Cerdanyola del Vallès, Spain
arisd@mat.uab.es
Adrian Lewis
School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853, U.S.A.
aslewis@orie.cornell.edu
Jérôme Malick
CNRS, Laboratoire J. Kunztmann, Grenoble, France
jerome.malick@inria.fr
Hristo Sendov
Dept. of Statistical and Actuarial Sciences, The University of Western Ontario, London, Ontario, Canada
hssendov@stats.uwo.ca
[Abstract-pdf]
Important properties such as differentiability and convexity of symmetric functions in $\mathbb{R}^{n}$ can be transferred to the corresponding spectral functions and vice-versa. Continuing to built on this line of research, we hereby prove that a spectral function $F\colon {\bf S}^n \rightarrow \mathbb{R\cup \{+\infty \}}$ is prox-regular if and only if the underlying symmetric function $f\colon\mathbb{R}^{n}\rightarrow \mathbb{R\cup \{+\infty \}}$ is prox-regular. Relevant properties of symmetric sets are also discussed.
Keywords: Spectral function, prox-regular function, eigenvalue optimization, invariant function, permutation theory.
MSC: 15A18, 49J52; 47A75, 90C22
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