Publication: Evaluating an Element of the Clarke Generalized Jacobian of a Piecewise Differentiable Function
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Evaluating an Element of the Clarke Generalized Jacobian of a Piecewise Differentiable Function

- incollection -
 

Author(s)
Kamil A. Khan , Paul I. Barton

Published in
Recent Advances in Algorithmic Differentiation

Editor(s)
Shaun Forth, Paul Hovland, Eric Phipps, Jean Utke, Andrea Walther

Year
2012

Publisher
Springer

Abstract
The (Clarke) generalized Jacobian of a locally Lipschitz continuous function is a derivative-like set-valued mapping that contains slope information. Several methods for optimization and equation solving require evaluation of generalized Jacobian elements. However, since the generalized Jacobian does not satisfy calculus rules sharply, this evaluation can be difficult. In this work, a method is presented for evaluating generalized Jacobian elements of a nonsmooth function that is expressed as a finite composition of absolute value functions and continuously differentiable functions. The method makes use of the principles of automatic differentiation and the theory of piecewise differentiable functions, and is guaranteed to be computationally tractable relative to the cost of a function evaluation.

Cross-References
Forth2012RAi

AD Theory and Techniques
Generalized Jacobian

BibTeX
@INCOLLECTION{
         Khan2012EaE,
       title = "Evaluating an Element of the {C}larke Generalized {J}acobian of a Piecewise
         Differentiable Function",
       doi = "10.1007/978-3-642-30023-3_11",
       author = "Kamil A. Khan and Paul I. Barton",
       abstract = "The (Clarke) generalized Jacobian of a locally Lipschitz continuous function is a
         derivative-like set-valued mapping that contains slope information. Several methods for optimization
         and equation solving require evaluation of generalized Jacobian elements. However, since the
         generalized Jacobian does not satisfy calculus rules sharply, this evaluation can be difficult. In
         this work, a method is presented for evaluating generalized Jacobian elements of a nonsmooth
         function that is expressed as a finite composition of absolute value functions and continuously
         differentiable functions. The method makes use of the principles of automatic differentiation and
         the theory of piecewise differentiable functions, and is guaranteed to be computationally tractable
         relative to the cost of a function evaluation.",
       pages = "115--125",
       crossref = "Forth2012RAi",
       booktitle = "Recent Advances in Algorithmic Differentiation",
       series = "Lecture Notes in Computational Science and Engineering",
       publisher = "Springer",
       address = "Berlin",
       volume = "87",
       editor = "Shaun Forth and Paul Hovland and Eric Phipps and Jean Utke and Andrea Walther",
       isbn = "978-3-540-68935-5",
       issn = "1439-7358",
       year = "2012",
       ad_theotech = "Generalized Jacobian"
}


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