# Semismoothness of spectral functions

Qi, Houduo and Yang, Xiaoqi Qi (2004) Semismoothness of spectral functions. SIAM Journal on Matrix Analysis and Applications, 25, (3), 784-803.

Any spectral function can be written as a composition of a symmetric function $f: \rn \mapsto \Re$ and the eigenvalue function $\lambda(\cdot): \s \mapsto \rn$, often denoted by $(f \circ \lambda)$, 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) $(f \circ \lambda)$ is directionally differentiable if f is semidifferentiable, (b) $(f \circ \lambda)$ is LC 1 if and only if f is LC 1, and (c) $(f \circ \lambda)$ is SC 1 if and only if f is SC 1. Result (a) is complementary to a known (negative) fact that $(f \circ \lambda)$ might not be directionally differentiable if f is directionally differentiable only. Results (b) and (c) are particularly useful for the solution of LC 1 and SC 1 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.