Journal of Convex Analysis 27 (2020), No. 1, 335--360
Copyright Heldermann Verlag 2020
The Chain Rule for VU-Decompositions of Nonsmooth Functions
Warren Hare
Dept. of Mathematics, University of British Columbia, Okanagan, Kelowna, Canada
warren.hare@ubc.ca
Chayne Planiden
Dept. of Mathematics and Applied Statistics, University of Wollongong, New South Wales, Australia
chayne@uow.edu.au
Claudia Sagastizábal
IMECC -- UNICAMP, 13083-859 Campinas, Brazil
sagastiz@unicamp.br
In Variational Analysis, VU-theory provides a set of tools that is helpful for understanding and exploiting the structure of nonsmooth functions. The theory takes advantage of the fact that at any point, the space can be separated into two orthogonal subspaces: one that describes the direction of nonsmoothness of the function, and the other on which the function behaves smoothly and has a gradient. For a composite function, this work establishes a chain rule that facilitates the computation of such gradients and characterizes the smooth subspace under reasonable conditions. From the chain rule presented, formulae for the separation, smooth perturbation and sum of functions are provided. Several nonsmooth examples are explored, including norm functions, max-of-quadratic functions and LASSO-type regularizations.
Keywords: Chain rule, fast track, finite-max function, manifold, nonsmooth analysis, partly smooth function, primal-dual gradient, strong transversality, U-Lagrangian, VU-algorithm, VU-decomposition.
MSC: 58C05, 90C47; 49A52, 65K10, 90C25.
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