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

Let [InlineMediaObject not available: see fulltext.] be the Lorentz/second-order cone in [InlineMediaObject not available: see fulltext.]. For any function f from [InlineMediaObject not available: see fulltext.] to [InlineMediaObject not available: see fulltext.], one can define a corresponding function f ^soc(x) on [InlineMediaObject not available: see fulltext.] by applying f to the spectral values of the spectral decomposition of x [InlineMediaObject not available: see fulltext.] with respect to [InlineMediaObject not available: see fulltext.]. 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.