The Hölder continuity of vector-valued function associated with second-order cone

Let Kⁿ be the Lorentz/second-order cone in IRⁿ. For any function f from IR to IR, one can define a corresponding vector-valued function f^soc (x) on IRⁿ by applying f to the spectral values of the spectral decomposition of x ∈ IRⁿ with respect to Kⁿ. It was shown by J.-S. Chen, X. Chen and P. Tseng in [5] that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, directional differentiability, Fréchet differentiability, continuous differentiability, as well as semismoothness. In this note, we further show that the Holder continuity of this vector-valued function is also inherited from f. Such property will be useful in designing solution methods for second-order cone programming and second-order cone complementarity problem.