# Representations and Analysis of Piecewise Linear Functions in Abs-normal form

It follows from the well known min/max representation given by Scholtes in his recent Springer book, that all piecewise linear continuous functions $y = F(x) : \mathbb{R}^n \to \mathbb{R}^m$ can be written in a so-called abs-normal form. This means in particular, that all nonsmoothness is encapsulated in $s$ absolute value functions that are applied to intermediate switching variables $z_i$ for $i=1, \ldots ,s$. The relation between the vectors $x, z$, and $y$ is described by four matrices $Y, L, J$, and $Z$, such that $\left(\begin{matrix} z \\ y \end{matrix}\right) = \left(\begin{matrix} b \\ c \end{matrix}\right) + \left(\begin{matrix} Z & L \\ J & Y \end{matrix}\right) \left(\begin{matrix} x \\ |z| \end{matrix}\right)$ which can be generated by ADOL-C or other Automatic Differentation Tools. Here $L$ is a strictly lower triangular matrix, and therefor $z_i$ can be computed successively from previous results. We show that in the square case $n=m$ the system of equations $F(x) = 0$ can be rewritten in terms of the variable vector $z$ as a linear complementarity problem (LCP). The transformation itself and the properties of the LCP depend on the Schur complement $S = L - Z J^{-1} Y$. We discuss associated linear algebra computations and highlight various theoretical and numerical effects via examples. %through examples.