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.