On Subgradients of Spectral Functions
Marc Ciligot–Travain, Sado Traoré · Journal of convex analysis · 2002
Let F:\mathbf{S}(m)\rightarrow\overline{\mathbb{R}} F : S ( m ) → R ‾ 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.