Optimal complexity reduction of piecewise affine models based on hyperplane arrangements

Abstract

This work presents an algorithm that, given a piecewise affine (PWA) model, derives an equivalent PWA model that is minimal in the number of regions. The algorithm is based on the cells of the hyperplane arrangement that are already given when the PWA model is the result of the mode enumeration algorithm. In particular, the algorithm executes a branch and bound search on the markings of the cells of the hyperplane arrangement assuring optimality. As we refrain from solving additional LPs, the algorithm is not only optimal but also computationally attractive. The applicability of the algorithm can be extended to derive minimal PWA representations of general PWA models by first computing the hyperplane arrangement. An example illustrates the algorithm and shows its computational effectiveness.