Subdifferentials of convex matrix-valued functions

TL;DR

This paper investigates the complex structure of subdifferentials for convex matrix-valued functions, highlighting differences from scalar cases and providing computation rules and properties, especially for functions on the real line.

Contribution

It develops subdifferential calculus rules for convex matrix-valued functions, revealing key differences from scalar convex analysis and characterizing subdifferentials on the real line.

Findings

Subdifferentials of matrix-valued functions can be complex and differ from scalar cases.

Standard convex analysis rules, like sum rules, do not always hold for matrix functions.

Nonsmooth convex matrix functions have nonsmooth diagonal elements.

Abstract

Subdifferentials (in the sense of convex analysis) of matrix-valued functions defined on $\mathbb{R}^d$ that are convex with respect to the L\"{o}wner partial order can have a complicated structure and might be very difficult to compute even in simple cases. The aim of this paper is to study subdifferential calculus for such functions and properties of their subdifferentials. We show that many standard results from convex analysis no longer hold true in the matrix-valued case. For example, in this case the subdifferential of the sum is not equal to the sum of subdifferentials, the Clarke subdifferential is not equal to the subdifferential in the sense of convex analysis, etc. Nonetheless, it is possible to provide simple rules for computing nonempty subsets of subdifferentials (in particular, individual subgradients) of convex matrix-valued functions in the general case and to completely describe subdifferentials of such functions defined on the real line. As a by-product of our analysis, we derive some interesting properties of convex matrix-valued functions, e.g. we show that if such function is nonsmooth, then its diagonal elements must be nonsmooth as well.