The vector-valued functions associated with circular cones
Summary: 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.
scientific article; zbMATH DE number 7022706
scientific article