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

Let L_θ be the circular cone in ^Rn 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 ^fc(x) on ^Rn by applying f to the spectral values of the spectral decomposition of x ⋯ Rⁿ with respect to ^Lθ. We show that this vector-valued function inherits from f 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.