A. Hempel, P. J. Goulart and J. Lygeros
in European Control Conference, Zürich, Switzerland, pp. 2657-2662, July 2013.
@inproceedings{HGL:2013,
author = {A. Hempel and P. J. Goulart and J. Lygeros},
title = {Every Continuous Piecewise Affine Function Can Be Obtained by Solving a Parametric Linear Program},
booktitle = {European Control Conference},
year = {2013},
pages = {2657-2662},
url = {http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6669386}
}
It is well-known that solutions to parametric linear or quadratic programs are continuous piecewise affine functions of the parameter. In this paper we prove the converse, i.e. that every continuous piecewise affine function can be identified with the solution to a parametric linear program. In particular, we provide a constructive proof that every piecewise affine function can be expressed as the linear mapping of the solution to a parametric linear program with at most twice as many variables as the dimension of the image of the piecewise affine function. Our method is illustrated via two small numerical examples.