Variational Analysis of the Orthogonally Invariant Norm Cone of Symmetric Matrices

TL;DR

This paper provides a comprehensive variational analysis of the orthogonally invariant norm cone of symmetric matrices, including formulas for tangent, normal, and second-order tangent sets, and characterizes projection derivatives.

Contribution

It introduces explicit formulas for variational objects related to orthogonally invariant norm cones, facilitating optimization and analysis involving symmetric matrices.

Findings

Formulas for tangent, normal, and second-order tangent sets are derived.

Differentiability properties of the projection operator are established.

Results are specialized to Schatten p-norm cones and second-order cones.

Abstract

A large number matrix optimization problems are described by orthogonally invariant norms. This paper is devoted to the study of variational analysis of the orthogonally invariant norm cone of symmetric matrices. For a general orthogonally invariant norm cone of symmetric matrices, formulas for the tangent cone, normal cone and second-order tangent set are established. The differentiability properties of the projection operator onto the orthogonally invariant norm cone are developed, including formulas for the directional derivative and the B-subdifferential. Importantly, the directional derivative is characterized by the second-order derivative of the corresponding symmetric function, which is convenient for computation. All these results are specified to the Schatten $p$-norm cone, especially to the second-order cone of symmetric matrices.