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. LetLθdenote the circular cone inRn. For a functionffromRtoR, one can define a corresponding vector-valued functionfLθonRnby applyingfto the spectral values of the spectral decomposition ofx∈Rnwith respect toLθ. In this paper, we study properties that this vector-valued function inherits fromf, including Hölder continuity,B-subdifferentiability,ρ-order semismoothness, and positive homogeneity. These results will play crucial role in designing solution methods for optimization problem involved in circular cone constraints.

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