On properties of univariate max functions at local maximizers

Tim Mitchell, Michael L. Overton

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Pages (from-to)2527-2541
JournalOptimization Letters
Volume16
Issue number9
DOIs
StatePublished - Dec 2022

Keywords

  • Eigenvalues of Hermitian matrix families
  • H-infinity norm
  • Numerical radius
  • Optimization of passive systems
  • Univariate max functions

ASJC Scopus subject areas

  • Control and Optimization

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