Representation and Analysis of Piecewise Linear Functions in Abs-Normal Form
Résumé
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
$$ \left[ \begin{array}{c} z \\ y \end{array}\right] = \left[ \begin{array}{c} c \\ b \end{array}\right] + \left[ \begin{array}{cc} Z &{} L \\ J &{} Y \end{array}\right] \left[ \begin{array}{c} x \\ |z |\end{array}\right] $$
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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