## Abstract and Applied Analysis

### The Vector-Valued Functions Associated with Circular Cones

#### Abstract

The circular cone is a pointed closed convex cone having hyperspherical sections orthogonal to its axis of revolution about which the cone is invariant to rotation, which includes second-order cone as a special case when the rotation angle is 45 degrees. Let ${\mathcal{L}}_{\theta }$ denote the circular cone in ${\mathbb{R}}^{n}$. For a function $f$ from $\mathbb{R}$ to $\mathbb{R}$, one can define a corresponding vector-valued function ${f}^{{\mathcal{L}}_{\theta }}$ on ${\mathbb{R}}^{n}$ by applying $f$ to the spectral values of the spectral decomposition of ${x\in \mathbb{R}}^{n}$ with respect to ${\mathcal{L}}_{\theta }$. In this paper, we study properties that this vector-valued function inherits from $f$, including Hölder continuity, $B$-subdifferentiability, $\rho$-order semismoothness, and positive homogeneity. These results will play crucial role in designing solution methods for optimization problem involved in circular cone constraints.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 603542, 21 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412606019

Digital Object Identifier
doi:10.1155/2014/603542

Mathematical Reviews number (MathSciNet)
MR3228078

Zentralblatt MATH identifier
07022706

#### Citation

Zhou, Jinchuan; Chen, Jein-Shan. The Vector-Valued Functions Associated with Circular Cones. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 603542, 21 pages. doi:10.1155/2014/603542. https://projecteuclid.org/euclid.aaa/1412606019

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