Representation and Analysis of Piecewise Linear Functions in Abs-Normal Form

Abstract

International audienceIt 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 $$ \left[ \begin{array}{c} z \\ y \end{array}\right] = \left[ \begin{array}{c} c \\ b \end{array}\right] + \left[ \begin{array}{cc} Z &{} L \\ J &{} Y \end{array}\right] \left[ \begin{array}{c} x \\ |z |\end{array}\right] $$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 $$S = L - Z J^{-1} Y$$

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