Analysis of nonsmooth vector-valued functions associated with infinite-dimensional second-order cones

Given a Hilbert space HH, the infinite-dimensional Lorentz/second-order cone KK is introduced. For any x∈Hx∈H, a spectral decomposition is introduced, and for any function f:R→Rf:R→R, we define a corresponding vector-valued function fH(x)fH(x) on Hilbert space HH by applying ff to the spectral values of the spectral decomposition of x∈Hx∈H with respect to KK. We show that this vector-valued function inherits from ff the properties of continuity, Lipschitz continuity, differentiability, smoothness, as well as s-semismoothness. These results can be helpful for designing and analyzing solution methods for solving infinite-dimensional second-order cone programs and complementarity problems.