Journal of Convex Analysis 25 (2018), No. 1, 119--134
Copyright Heldermann Verlag 2018
Variational Analysis of Spectral Functions Simplified
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
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.
Keywords: Eigenvalues, singular values, nonsmooth analysis, proximal mapping, subdifferential, Hessian, quadratic growth, group actions.
MSC: 14A18, 49J52, 46G05, 26B05, 49J53
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