Analysis of nonsmooth vector-valued functions associated with second-order cones

Let K-n be the Lorentz/second-order cone in R-n. For any function f from R to R, one can define a corresponding function f(soc)(x) on R-n by applying f to the spectral values of the spectral decomposition of x is an element of R-n with respect to K-n. We show that this vector-valued function inherits from f the properties of continuity, (local) Lipschitz continuity, directional differentiability, Frechet differentiability, continuous differentiability, as well as (rho-order) semismoothness. These results are useful for designing and analyzing smoothing methods and nonsmooth methods for solving second-order cone programs and complementarity problems.