Semismoothness of spectral functions

Abstract

Any spectral function can be written as a composition of a symmetric function f : ℝⁿ ↦ ℝ and the eigenvalue function λ(·): S ↦ℝⁿ, 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 LC¹ if and only if f is LC¹ , and (c) (f◦λ) is SC¹ if and only if f is SC¹ . 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 LC¹ and SC¹ 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.