English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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 $$S = L - Z J^{-1} Y$$

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