Semismoothness of Spectral Functions

Any spectral function can be written as a composition of a symmetric function f : IR 7→ IR and the eigenvalue function λ(·) : S 7→ IR, often denoted by (f ◦ λ), where S is the subspace of n×n symmetric matrices. In this paper, we present some nonsmooth analysis for such spectral functions. Our main results are: (a) (f ◦ λ) is directionally differentiable if f is semidifferentiable; (b) (f ◦ λ) is LC if and only if f is LC; and (c) (f ◦ λ) is SC if and only if f is SC. Result (a) is complementary to a known (negative) fact that (f ◦ λ) might not be directionally differentiable if f is directionally differentiable only. Results (b) and (c) are particularly useful for the solution of LC and SC minimization problems which often can be solved by fast (generalized) Newton methods. Our analysis makes use of recent results on continuously differentiable spectral functions as well as on nonsmooth symmetric-matrix-valued functions.