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Representation and analysis of piecewise linear functions in abs-normal form

Abstract:

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) : Rn → Rm 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 zi 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 (Formula presented). This form can be generated by ADOL-C or other automatic differentation tools. Here L is a strictly lower triangular matrix, and therefore zi 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–1Y.