ALTERNATIVE PROOFS FOR SOME RESULTS OF VECTOR-VALUED FUNCTIONS ASSOCIATED WITH SECOND-ORDER CONE

Let k(n) be the Lorentz/second-order cone in R-n. For any function f from R to R, one can define a corresponding vector-valued function f(BOC)(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 k(n). 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, Frechet 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.