Please use this identifier to cite or link to this item: https://hdl.handle.net/10419/308732 
Year of Publication: 
2022
Citation: 
[Journal:] Optimization Letters [ISSN:] 1862-4480 [Volume:] 16 [Issue:] 9 [Publisher:] Springer [Place:] Berlin, Heidelberg [Year:] 2022 [Pages:] 2527-2541
Publisher: 
Springer, Berlin, Heidelberg
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=3and the maximizer is isolated.
Subjects: 
Univariate max functions
Eigenvalues of Hermitian matrix families
H-infinity norm
Numerical radius
Optimization of passive systems
Persistent Identifier of the first edition: 
Creative Commons License: 
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Document Type: 
Article
Document Version: 
Published Version

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