Smooth and nonsmooth analyses of vector-valued functions associated with circular cones

Let L-theta be the circular cone in R-n which includes a second-order cone as a special case. For any function f from R to R, one can define a corresponding vector-valued function f(c)(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 L-theta. We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, directional differentiability, Frechet differentiability, continuous differentiability, as well as semismoothness. These results will play a crucial role in designing solution methods for optimization problem associated with the circular cone. (C) 2013 Elsevier Ltd. All rights reserved.