Alternative proofs for some results of vector-valued functions associated with second-order cone
Jein-Shan Chen · 2005
Let K n be the Lorentz/second-order cone in IR n . For any function f from IR to IR, one can deflne a corresponding vector-valued function f soc (x) on IR n by applying f to the spectral values of the spectral decomposition of x 2 IR n with respect to K n . It was shown by J.-S. Chen, X. Chen and P. Tseng that this vector-valued function inher- its from f the properties of continuity, Lipschitz continuity, directional difierentiability, Frechet difierentiability, continuous difierentiability, as well as semismoothness. It was also proved by D. Sun and J. Sun that the vector-valued Fischer-Burmeister function as- sociated 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 in- tuitive way to prove all the above results by using the simple structure of second-order cone and spectral decomposition.