Piecewise linear secant approximation via Algorithmic Piecewise Differentiation
Abstract
It is shown how piecewise differentiable functions $F: \mathbb R^n \mapsto \mathbb R^m $ that are defined by evaluation programs can be approximated locally by a piecewise linear model based on a pair of sample points $\check x$ and $\hat x$. We show that the discrepancy between function and model at any point $x$ is of the bilinear order $O(\x\check x\ \x\hat x\)$. This is a little surprising since $x \in \mathbb R^n$ may vary over the whole Euclidean space, and we utilize only two function samples $\check F=F(\check x)$ and $\hat F=F(\hat x)$, 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2017
 arXiv:
 arXiv:1701.04368
 Bibcode:
 2017arXiv170104368G
 Keywords:

 Mathematics  Numerical Analysis
 EPrint:
 Submitted