Evaluating an element of the Clarke generalized Jacobian of a composite piecewise differentiable function

TitleEvaluating an element of the Clarke generalized Jacobian of a composite piecewise differentiable function
Publication TypeJournal Article
Year of Publication2013
AuthorsKhan KA, Barton PI
JournalACM Transactions on Mathematical Software
Volume39
Abstract

Bundle methods for nonsmooth optimization and semismooth Newton methods for nonsmooth equation solving both require computation of elements of the (Clarke) generalized Jacobian, which provides slope information for locally Lipschitz continuous functions. Since the generalized Jacobian does not obey sharp calculus rules, this computation can be difficult. In this article, methods are developed for evaluating generalized Jacobian elements for a nonsmooth function that is expressed as a finite composition of known elemental piecewise differentiable functions. In principle, these elemental functions can include any piecewise differentiable function whose analytical directional derivatives are known. The methods are fully automatable, and are shown to be computationally tractable relative to the cost of a function evaluation. An implementation developed in C++ is discussed, and the methods are applied to several example problems for illustration.

URLhttp://dl.acm.org/citation.cfm?id=2491493
DOI10.1145/2491491.2491493