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Journal of Convex Analysis 09 (2002), No. 2, 401--414
Copyright Heldermann Verlag 2002

On Subgradients of Spectral Functions

Marc Ciligot-Travain
Dép. des Mathématiques, Université d'Avignon, 33 rue Pasteur, 84000 Avignon, France

Sado Traore
U.P.B. / E.S.I., Bobo-Dioulasso 01, Burkina Faso


\newcommand{\Or}[1]{\mathbf{O}(#1)} \newcommand{\R}{\mathbb{R}} \newcommand{\Ret}{\overline{\mathbb{R}}} \newcommand{\Sy}[1]{\mathbf{S}(#1)} Let $F:\Sy{m}\rightarrow\Ret$ be a {\em spectral function} (i.e.\ $\Sy{m}$ is the space of $m\times m$ real symmetric matrices, $\forall O\in\Or{m},\forall X\in\Sy{m},\ F(OX{^tO})=F(X)$, where $\Or{m}$ is the orthogonal group and ${^tO}$ is the transpose of $O$). We associate to it the symmetric function $s_F:\R^m\rightarrow\Ret$ by restricting it to the subspace of diagonal matrices. In this work, on the one hand, we give a new, natural proof of the formula which binds the Fr\'echet subgradients of a spectral function $F$ and the Fr\'echet subgradients of the function $s_F$ (identical formulas follow for the subgradients and the horizon subgradients); on the other hand we deduce from the previous results and from convexity arguments that, in the general case, a similar formula holds for the Clarke subgradients.

Keywords: Spectral function, eigenvalues, eigenvalue optimization, perturbation theory, Clarke subgradient, nonsmooth analysis.

MSC: 90C31, 15A18; 49K40, 26B05

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