Abstract

Let F:S(m)→R‾F:\mathbf{S}(m)\rightarrow\overline{\mathbb{R}} be a spectral function (i.e. S(m)\mathbf{S}(m) is the space of m×mm\times m real symmetric matrices, ∀O∈O(m),∀X∈S(m), F(OXtO)=F(X)\forall O\in\mathbf{O}(m),\forall X\in\mathbf{S}(m),\ F(OX{^tO})=F(X), where O(m)\mathbf{O}(m) is the orthogonal group and tO{^tO} is the transpose of OO). We associate to it the symmetric function sF:Rm→R‾s_F:\mathbb{R}^m\rightarrow\overline{\mathbb{R}} 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 Fréchet subgradients of a spectral function FF and the Fréchet subgradients of the function sFs_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.

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Keywords

  • Spectral function
  • eigenvalues
  • eigenvalue optimization
  • perturbation theory
  • Clarke subgradient
  • nonsmooth analysis

Mathematics Subject Classification

90C31, 15A18; 49K40, 26B05

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Suggested citation

M. Ciligot-Travain, S. Traoré. “On Subgradients of Spectral Functions.” Journal of Convex Analysis 9 (2002), No. 2, 401–414. https://doi.org/10.68381/jca09023

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Published by Heldermann Verlag, 2002. Rights now held by Banach Press LLC.