On Properties of Univariate Max Functions at Local Maximizers

TL;DR

This paper explores the smoothness properties of univariate maximum functions at local maximizers, extending classical results and showing limitations in differentiability for more general max functions.

Contribution

It generalizes existing regularity results for specific matrix functions to broader classes of univariate max functions, highlighting where smoothness properties hold or fail.

Findings

Maximum eigenvalue functions are twice differentiable at local maximizers.

Pointwise maximum of differentiable functions may have non-differentiable points near maximizers.

Smoothness properties do not necessarily extend to arbitrary max functions, even with high differentiability.

Abstract

More than three decades ago, Boyd and Balakrishnan established a regularity result for the two-norm of a transfer function at maximizers. Their result extends easily to the statement that the maximum eigenvalue of a univariate real analytic Hermitian matrix family is twice continuously differentiable, with Lipschitz second derivative, at all local maximizers, a property that is useful in several applications that we describe. We also investigate whether this smoothness property extends to max functions more generally. We show that the pointwise maximum of a finite set of $q$-times continuously differentiable univariate functions must have zero derivative at a maximizer for $q=1$, but arbitrarily close to the maximizer, the derivative may not be defined, even when $q=3$ and the maximizer is isolated.