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Analysis of nonsmooth vector-valued functions associated with second-order cones. (English) Zbl 1065.49013
Summary: Let 𝒦 n be the Lorentz/second-order cone in n . For any function f from to , one can define a corresponding function f soc (x) on n by applying f to the spectral values of the spectral decomposition of x n with respect to 𝒦 n . We show that this vector-valued function inherits from f the properties of continuity, (local) Lipschitz continuity, directional differentiability, FrĂ©chet differentiability, continuous differentiability, as well as (𝒫-order) semismoothness. These results are useful for designing and analyzing smoothing methods and nonsmooth methods for solving second-order cone programs and complementarity problems.

MSC:
49J52Nonsmooth analysis (other weak concepts of optimality)
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
90C22Semidefinite programming