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

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