Piecewise linear secant approximation via Algorithmic Piecewise Differentiation
This provides a new theoretical framework for piecewise linear approximation that could benefit optimization and numerical analysis, though it is an incremental theoretical contribution.
The authors present a method to approximate piecewise differentiable functions locally using only two function evaluations and their computational intermediates, achieving bilinear error bounds. They develop a generalized Newton's method with convergence guarantees matching semismooth Newton.
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.