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

Let LθLθ be the circular cone in RnRn which includes a second-order cone as a special case. For any function ff from RR to RR, one can define a corresponding vector-valued function fc(x) on RnRn by applying ff to the spectral values of the spectral decomposition of x∈Rnx∈Rn with respect to LθLθ. We show that this vector-valued function inherits from ff the properties of continuity, Lipschitz continuity, directional differentiability, Fréchet 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.