Alternative proofs for some results of vector-valued functions associated with second-order cone

Let Kn be the Lorentz/second-order cone in IR. For any function f from IR to IR, one can define a corresponding vector-valued function f soc (x) on IR by applying f to the spectral values of the spectral decomposition of x ∈ IR with respect to Kn. It was shown by J.-S. Chen, X. Chen and P. Tseng that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, directional differentiability, Fréchet differentiability, continuous differentiability, as well as semismoothness. It was also proved by D. Sun and J. Sun that the vector-valued Fischer-Burmeister function associated with second-order cone is strongly semismooth. All proofs for the above results are based on a special relation between the vector-valued function and the matrix-valued function over symmetric matrices. In this paper, we provide a straightforward and intuitive way to prove all the above results by using the simple structure of second-order cone and spectral decomposition.