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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

Adrian Lewis
School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853, U.S.A.

Jérôme Malick
CNRS, Laboratoire J. Kunztmann, Grenoble, France

Hristo Sendov
Dept. of Statistical and Actuarial Sciences, The University of Western Ontario, London, Ontario, Canada


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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