Publication: Piecewise linear secant approximation via algorithmic piecewise differentiation
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Piecewise linear secant approximation via algorithmic piecewise differentiation

- Article in a journal -
 

Author(s)
Andreas Griewank , Tom Streubel , Lutz Lehmann , Manuel Radons , Richard Hasenfelder

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
It is shown how piecewise differentiable functions F:ℝn↦ℝm that are defined by evaluation programmes can be approximated locally by a piecewise linear model based on a pair of sample points . We show that the discrepancy between function and model at any point x is of the bilinear order . As an application of the piecewise linearization procedure we devise a generalized Newton's method based on successive piecewise linearization and prove for it sufficient conditions for convergence and convergence rates equalling those of semismooth Newton. We conclude with the derivation of formulas for the numerically stable implementation of the aforedeveloped piecewise linearization methods.

Cross-References
Christianson2018Sio

AD Theory and Techniques
Piecewise Linear

BibTeX
@ARTICLE{
         Griewank2018Pls,
       crossref = "Christianson2018Sio",
       author = "Andreas Griewank and Tom Streubel and Lutz Lehmann and Manuel Radons and Richard
         Hasenfelder",
       title = "Piecewise linear secant approximation via algorithmic piecewise differentiation",
       journal = "Optimization Methods \& Software",
       volume = "33",
       number = "4--6",
       pages = "1108--1126",
       year = "2018",
       publisher = "Taylor \& Francis",
       doi = "10.1080/10556788.2017.1387256",
       url = "https://doi.org/10.1080/10556788.2017.1387256",
       eprint = "https://doi.org/10.1080/10556788.2017.1387256",
       abstract = "It is shown how piecewise differentiable functions
         F:ℝn↦ℝm that are defined by evaluation programmes can be approximated
         locally by a piecewise linear model based on a pair of sample points . We show that the discrepancy
         between function and model at any point x is of the bilinear order . As an application of the
         piecewise linearization procedure we devise a generalized Newton's method based on successive
         piecewise linearization and prove for it sufficient conditions for convergence and convergence rates
         equalling those of semismooth Newton. We conclude with the derivation of formulas for the
         numerically stable implementation of the aforedeveloped piecewise linearization methods.",
       booktitle = "Special issue of Optimization Methods \& Software: Advances in
         Algorithmic Differentiation",
       editor = "Bruce Christianson and Shaun A. Forth and Andreas Griewank",
       ad_theotech = "Piecewise Linear"
}


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