Publication: Newton step methods for AD of an objective defined using implicit functions
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Newton step methods for AD of an objective defined using implicit functions

- Article in a journal -
 

Author(s)
Bradley M. Bell , Kasper Kristensen

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
We consider the problem of computing derivatives of an objective that is defined using implicit functions; i.e., implicit variables are computed by solving equations that are often nonlinear and solved by an iterative process. If one were to apply Algorithmic Differentiation (ad) directly, one would differentiate the iterative process. In this paper we present the Newton step methods for computing derivatives of the objective. These methods make it easy to take advantage of sparsity, forward mode, reverse mode, and other ad techniques. We prove that the partial Newton step method works if the number of steps is equal to the order of the derivatives. The full Newton step method obtains two derivatives order for each step except for the first step. There are alternative methods that avoid differentiating the iterative process; e.g., the method implemented in ADOL-C. An optimal control example demonstrates the advantage of the Newton step methods when computing both gradients and Hessians. We also discuss the Laplace approximation method for nonlinear mixed effects models as an example application.

Cross-References
Christianson2018Sio

BibTeX
@ARTICLE{
         Bell2018Nsm,
       crossref = "Christianson2018Sio",
       author = "Bradley M. Bell and Kasper Kristensen",
       title = "{N}ewton step methods for {AD} of an objective defined using implicit functions",
       journal = "Optimization Methods \& Software",
       volume = "33",
       number = "4--6",
       pages = "907--923",
       year = "2018",
       publisher = "Taylor \& Francis",
       doi = "10.1080/10556788.2017.1406936",
       url = "https://doi.org/10.1080/10556788.2017.1406936",
       eprint = "https://doi.org/10.1080/10556788.2017.1406936",
       abstract = "We consider the problem of computing derivatives of an objective that is defined
         using implicit functions; i.e., implicit variables are computed by solving equations that are often
         nonlinear and solved by an iterative process. If one were to apply Algorithmic Differentiation (AD)
         directly, one would differentiate the iterative process. In this paper we present the Newton step
         methods for computing derivatives of the objective. These methods make it easy to take advantage of
         sparsity, forward mode, reverse mode, and other AD techniques. We prove that the partial Newton step
         method works if the number of steps is equal to the order of the derivatives. The full Newton step
         method obtains two derivatives order for each step except for the first step. There are alternative
         methods that avoid differentiating the iterative process; e.g., the method implemented in ADOL-C. An
         optimal control example demonstrates the advantage of the Newton step methods when computing both
         gradients and Hessians. We also discuss the Laplace approximation method for nonlinear mixed effects
         models as an example application.",
       booktitle = "Special issue of Optimization Methods \& Software: Advances in
         Algorithmic Differentiation",
       editor = "Bruce Christianson and Shaun A. Forth and Andreas Griewank"
}


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