www.Autodiff.org - Publication: Computationally relevant generalized derivatives: theory, evaluation and applications

Computationally relevant generalized derivatives: theory, evaluation and applications

- Article in a journal -

Author(s)
Paul I. Barton , Kamil A. Khan , Peter Stechlinski , Harry A. J. Watson

Published in
Special issue of Optimization Methods & Software: Advances in Algorithmic Differentiation Optimization Methods & Software

Editor(s)
Bruce Christianson, Shaun A. Forth, Andreas Griewank

Year
2018

Publisher
Taylor & Francis

Abstract
A new method for evaluating generalized derivatives in nonsmooth problems is reviewed. Lexicographic directional (LD-)derivatives are a recently developed tool in nonsmooth analysis for evaluating generalized derivative elements in a tractable and robust way. Applicable to problems in both steady-state and dynamic settings, LD-derivatives exhibit a number of advantages over current theory and algorithms. As highlighted in this article, the LD-derivative approach now admits a suitable theory for inverse and implicit functions, nonsmooth dynamical systems and optimization problems, among others. Moreover, this technique includes an extension of the standard vector forward mode of automatic differentiation (ad) and acts as the natural extension of classical calculus results to the nonsmooth case in many ways. The theory of LD-derivatives is placed in the context of state-of-the-art methods in nonsmooth analysis, with an application in multistream heat exchanger modelling and design used to illustrate the usefulness of the approach.

Cross-References
Christianson2018Sio

AD Theory and Techniques
Generalized Jacobian

BibTeX
@ARTICLE{
         Barton2018Crg,
       crossref = "Christianson2018Sio",
       author = "Paul I. Barton and Kamil A. Khan and Peter Stechlinski and Watson, Harry A. J.",
       title = "Computationally relevant generalized derivatives: theory, evaluation and
         applications",
       journal = "Optimization Methods \& Software",
       volume = "33",
       number = "4--6",
       pages = "1030--1072",
       year = "2018",
       publisher = "Taylor \& Francis",
       doi = "10.1080/10556788.2017.1374385",
       url = "https://doi.org/10.1080/10556788.2017.1374385",
       eprint = "https://doi.org/10.1080/10556788.2017.1374385",
       abstract = "A new method for evaluating generalized derivatives in nonsmooth problems is
         reviewed. Lexicographic directional (LD-)derivatives are a recently developed tool in nonsmooth
         analysis for evaluating generalized derivative elements in a tractable and robust way. Applicable to
         problems in both steady-state and dynamic settings, LD-derivatives exhibit a number of advantages
         over current theory and algorithms. As highlighted in this article, the LD-derivative approach now
         admits a suitable theory for inverse and implicit functions, nonsmooth dynamical systems and
         optimization problems, among others. Moreover, this technique includes an extension of the standard
         vector forward mode of automatic differentiation (AD) and acts as the natural extension of classical
         calculus results to the nonsmooth case in many ways. The theory of LD-derivatives is placed in the
         context of state-of-the-art methods in nonsmooth analysis, with an application in multistream heat
         exchanger modelling and design used to illustrate the usefulness of the approach.",
       booktitle = "Special issue of Optimization Methods \& Software: Advances in
         Algorithmic Differentiation",
       editor = "Bruce Christianson and Shaun A. Forth and Andreas Griewank",
       ad_theotech = "Generalized Jacobian"
}

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