www.Autodiff.org - Publication: Branch-locking AD techniques for nonsmooth composite functions and nonsmooth implicit functions

Branch-locking AD techniques for nonsmooth composite functions and nonsmooth implicit functions

- Article in a journal -

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 recent nonsmooth vector forward mode of algorithmic differentiation (ad) computes Nesterov's L-derivatives for nonsmooth composite functions; these L-derivatives provide useful sensitivity information to methods for nonsmooth optimization and equation solving. The established reverse ad mode evaluates gradients efficiently for smooth functions, but it does not extend directly to nonsmooth functions. Thus, this article examines branch-locking strategies to harness the benefits of smooth ad techniques even in the nonsmooth case, in order to improve the computational performance of the nonsmooth vector forward ad mode. In these strategies, each nonsmooth elemental function in the original composition is ‘locked’ into an appropriate linear ‘branch’. The original composition is thereby replaced with a smooth variant, which may be subjected to efficient ad techniques for smooth functions such as the reverse ad mode. In order to choose the correct linear branches, we use inexpensive probing steps to ascertain the composite function's local behaviour. A simple implementation in is described, and the developed techniques are extended to nonsmooth local implicit functions and inverse functions.

Cross-References
Christianson2018Sio

AD Theory and Techniques
Generalized Jacobian, Reverse Mode

BibTeX
@ARTICLE{
         Khan2018BlA,
       crossref = "Christianson2018Sio",
       author = "Kamil A. Khan",
       title = "Branch-locking {AD} techniques for nonsmooth composite functions and nonsmooth
         implicit functions",
       journal = "Optimization Methods \& Software",
       volume = "33",
       number = "4--6",
       pages = "1127--1155",
       year = "2018",
       publisher = "Taylor \& Francis",
       doi = "10.1080/10556788.2017.1341506",
       url = "https://doi.org/10.1080/10556788.2017.1341506",
       eprint = "https://doi.org/10.1080/10556788.2017.1341506",
       abstract = "A recent nonsmooth vector forward mode of algorithmic differentiation (AD) computes
         Nesterov's L-derivatives for nonsmooth composite functions; these L-derivatives provide useful
         sensitivity information to methods for nonsmooth optimization and equation solving. The established
         reverse AD mode evaluates gradients efficiently for smooth functions, but it does not extend
         directly to nonsmooth functions. Thus, this article examines branch-locking strategies to harness
         the benefits of smooth AD techniques even in the nonsmooth case, in order to improve the
         computational performance of the nonsmooth vector forward AD mode. In these strategies, each
         nonsmooth elemental function in the original composition is ‘locked’ into an
         appropriate linear ‘branch’. The original composition is thereby replaced with a
         smooth variant, which may be subjected to efficient AD techniques for smooth functions such as the
         reverse AD mode. In order to choose the correct linear branches, we use inexpensive probing steps to
         ascertain the composite function's local behaviour. A simple implementation in is described,
         and the developed techniques are extended to nonsmooth local implicit functions and inverse
         functions.",
       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, Reverse Mode"
}

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