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

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