Twice Epi-Differentiability of Orthogonally Invariant Matrix Functions and Application

TL;DR

This work analyzes second-order variational properties of orthogonally invariant matrix functions $F=f\circ\sigma$, where $f$ is absolutely symmetric and $\sigma$ denotes singular values. It develops a parabolic epi-differentiability framework and a chain rule for subderivatives, enabling precise characterization of first- and second-order variability (including the nuclear norm) and leading to second-order optimality conditions for matrix optimization problems. A central achievement is the explicit computation of the second subderivative for convex $F$ and the establishment of sufficient conditions for twice epi-differentiability, with concrete results for the nuclear norm. These contributions advance variational analysis in matrix settings and have implications for problems in matrix completion, rank minimization, and related optimization tasks.

Abstract

In this paper, our focus lies on the study of the second-order variational analysis of orthogonally invariant matrix functions. It is well-known that an orthogonally invariant matrix function is an extended-real-value function defined on ${\mathbb M}_{m,n}\,(n \leqslant m)$ of the form $f \circ σ$ for an absolutely symmetric function $f \colon \R^n \rightarrow [-\infty,+\infty]$ and the singular values $σ\colon {\mathbb M}_{m,n} \rightarrow \R^{n}$. We establish several second-order properties of orthogonally invariant matrix functions, such as parabolic epi-differentiability, parabolic regularity, and twice epi-differentiability when their associated absolutely symmetric functions enjoy some properties. Specifically, we show that the nuclear norm of a real $m \times n$ matrix is twice epi-differentiable and we derive an explicit expression of its second-order epi-derivative. Moreover, for a convex orthogonally invariant matrix function, we calculate its second subderivative and present sufficient conditions for twice epi-differentiability. This enables us to establish second-order optimality conditions for a class of matrix optimization problems.

Theorems & Definitions (33)

• Proposition 2.1
• Lemma 2.2
• Lemma 2.3
• Proposition 2.4
• Proposition 2.5
• Corollary 2.6
• Proposition 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4