Representation 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 \(y = F(x) : \mathbb {R}^n \rightarrow \mathbb {R}^m\) 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 This form can be generated by ADOL-C or other automatic differentation tools. Here \(L\) is a strictly lower triangular matrix, and therefore \( z_i\) 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 \(S = L - Z J^{-1} Y\).