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Analysis of nonsmooth vector-valued functions associated with second-order cones. (English) Zbl 1065.49013
Summary: Let $$\mathcal K^n$$ be the Lorentz/second-order cone in $$\mathbb R^n$$. For any function $$f$$ from $$\mathbb R$$ to $$\mathbb R$$, one can define a corresponding function $$f^{\text{soc}}(x)$$ on $$\mathbb R^n$$ by applying $$f$$ to the spectral values of the spectral decomposition of $$x\in \mathbb R^n$$ with respect to $$\mathcal K^n$$. 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 ($$\mathcal P$$-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:
 49J52 Nonsmooth analysis 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) 90C22 Semidefinite programming
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