Piecewise Liner Differentiation and Algebra

A minor modification of algorithmic differentiation tools allows the computation of piecewise linear approximations for non-smooth Lipschitz-continuous functions  that can be represented by a straight-line code, which consists of smooth elementary operations and absolute value expressions. The piecewise linear approximations provide essential information about the original function and reflect the non-smoothness more accurately than a simple linearization. In contrast to the smooth case, where the derivatives can be represented by simple vectors and matrices, the piecewise linearization requires a more sophisticated representation. In this talk we will focus on one possible algebraic representation of the piecewise linearization, namely, the abs-normal form (ANF). Although the ANF is very convenient for small applications and allows for the computation of elements from the generalized derivative, which then can be used within some optimization methods, it lacks efficiency for large scale applications. Therefore, we discuss the numerical challenges of this structure and present a more suitable extension of the ANF, which can be regarded as a cornerstone of an efficient piecewise linear algebra system.

Torsten Bosse is the current Wilkinson Postdoctoral Fellow at Argonne National Laboratory, US. His research interest are in non-linear optimization, One-shot-methods for design optimization problems, and piecewise linear algorithmic differentiation for non-smooth functions.  He received his PhD in Mathematics at the Humboldt-Universität zu Berlin, Germany, in 2014.