Abstract

Spectral functions of symmetric matrices – those depending on matrices only through their eigenvalues – appear often in optimization. A cornerstone variational analytic tool for studying such functions is a formula relating their subdifferentials to the subdifferentials of their diagonal restrictions. This paper presents a new, short, and revealing derivation of this result. The argument has a direct analogue for spectral functions of Hermitian matrices, and for singular value functions of rectangular matrices.

Author information

Contact details are reproduced from the original publication and may be historical.

Dmitriy Drusvyatskiy

Mathematics Department, University of Washington, Seattle, WA 98195, U.S.A.

ddrusv@uw.edu

Courtney Paquette

Mathematics Department, University of Washington, Seattle, WA 98195, U.S.A.

yumiko88@uw.edu

Keywords

  • Eigenvalues
  • singular values
  • nonsmooth analysis
  • proximal mapping
  • subdifferential
  • Hessian
  • quadratic growth
  • group actions

Mathematics Subject Classification

14A18, 49J52, 46G05, 26B05, 49J53

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Suggested citation

D. Drusvyatskiy, C. Paquette. “Variational Analysis of Spectral Functions Simplified.” Journal of Convex Analysis 25 (2018), No. 1, 119–134. https://doi.org/10.68381/jca25008

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Published by Heldermann Verlag, 2018. Rights now held by Banach Press LLC.