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

摘要

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) : R~n → 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,...,s. The relation between the vectors x, z, and y is described by four matrices Y, L, J, and Z, such that [z y]=[c b]+[Z L J Y][x |z|] This form can be generated by ADOL-C or other automatic differenta-tion 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 - ZJ~(-1)Y.