
Journal of Convex Analysis 27 (2020), No. 1, 335360 Copyright Heldermann Verlag 2020 The Chain Rule for VUDecompositions 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, 13083859 Campinas, Brazil sagastiz@unicamp.br In Variational Analysis, VUtheory 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, maxofquadratic functions and LASSOtype regularizations. Keywords: Chain rule, fast track, finitemax function, manifold, nonsmooth analysis, partly smooth function, primaldual gradient, strong transversality, ULagrangian, VUalgorithm, VUdecomposition. MSC: 58C05, 90C47; 49A52, 65K10, 90C25. [ Fulltextpdf (196 KB)] for subscribers only. 