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Zbl 1065.49013
Chen, Jein-Shan; Chen, Xin; Tseng, Paul
Analysis of nonsmooth vector-valued functions associated with second-order cones. (English)
[J] Math. Program. 101, No. 1 (B), 95-117 (2004). ISSN 0025-5610; ISSN 1436-4646/e

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.

MSC 2000:: *49J52 Nonsmooth analysis (other weak concepts of optimality)
90C33 Complementarity problems
90C22 Semidefinite programming

Keywords: second-order cone; vector-valued function; nonsmooth analysis; semismooth function; complementarity

Cited in: Zbl 1071.49014

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