6164
2016
eng
reportzib
0
--
2016-12-28
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Piecewise linear secant approximation via Algorithmic Piecewise Differentiation
It is shown how piecewise differentiable functions \(F: R^n → R^m\) that are defined by evaluation programs can be approximated locally by a piecewise linear model based on a pair of sample points x̌ and x̂. We show that the discrepancy between function and model at any point x is of the bilinear order O(||x − x̌|| ||x − x̂||). This is a little surprising since x ∈ R^n may vary over the whole Euclidean space, and we utilize only two function samples F̌ = F(x̌) and F̂ = F(x̂), as well as the intermediates computed during their evaluation. 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 equaling those of semismooth Newton. We conclude with the derivation of formulas for the numerically stable implementation of the aforedeveloped piecewise linearization methods.
1438-0064
urn:nbn:de:0297-zib-61642
10.1080/10556788.2017.1387256
published at Optimization Methods and Software
Andreas Griewank
Tom Streubel
Tom Streubel
Lutz Lehmann
Richard Hasenfelder
Manuel Radons
ZIB-Report
16-54
eng
uncontrolled
Automatic differentiation
eng
uncontrolled
Computational graph
eng
uncontrolled
Lipschitz continuity
eng
uncontrolled
Generalized Hermite interpolation
eng
uncontrolled
ADOL-C
Mathematics of Computing
NUMERICAL ANALYSIS
COMPUTER SCIENCE (For papers involving machine computations and programs in a specific mathematical area, see Section -04 in that area)
Mathematical Optimization
MODAL-GasLab
Streubel, Tom
MODAL-Gesamt
Energy Network Optimization
https://opus4.kobv.de/opus4-zib/files/6164/newton_secant_approx_paper.pdf