On Properties of Univariate Max Functions at Local Maximizers
Abstract
More than three decades ago, Boyd and Balakrishnan established a regularity result for the twonorm 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.07754
 Bibcode:
 2021arXiv210807754M
 Keywords:

 Mathematics  Optimization and Control;
 49J52;
 65F99
 EPrint:
 Initial preprint