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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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