Abstract
Let Open image in new window be the Lorentz/second-order cone in Open image in new window. For any function f from Open image in new window to Open image in new window, one can define a corresponding function fsoc(x) on Open image in new window by applying f to the spectral values of the spectral decomposition of x∈Open image in new window with respect to Open image in new window. We show that this vector-valued function inherits from f the properties of continuity, (local) Lipschitz continuity, directional differentiability, Frechet differentiability, continuous differentiability, as well as (ρ-order) semismoothness. These results are useful for designing and analyzing smoothing methods and nonsmooth methods for solving second-order cone programs and complementarity problems.
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