On Subgradients of Spectral Functions

Let

be a spectral function (i.e.

\mathbf{S}(m) S ( m )

is the space of

m\times m m × m

real symmetric matrices,

\forall O\in\mathbf{O}(m),\forall X\in\mathbf{S}(m),\ F(OX{^tO})=F(X) ∀ O ∈ O ( m ) , ∀ X ∈ S ( m ) ,   F ( O X t O ) = F ( X )

, where

\mathbf{O}(m) O ( m )

is the orthogonal group and

{^tO} t O

is the transpose of

O O

). We associate to it the symmetric function

s_F:\mathbb{R}^m\rightarrow\overline{\mathbb{R}} s F : R m → 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

F F

and the Fréchet subgradients of the function

s_F 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.