Best approximation in max-plus semimodules

TL;DR

This paper introduces new theoretical results on projectors and distances in max-plus semimodules, enabling efficient algorithms for solving systems of max-plus linear inequalities.

Contribution

It provides explicit formulas for distances and minimizers in max-plus spaces, and develops a cyclic projection algorithm for max-plus linear inequalities.

Findings

Explicit formula for distance in Hilbert's projective metric

Characterization of minimizers in max-plus semimodules

A cyclic projection algorithm for max-plus inequalities

Abstract

We establish new results concerning projectors on max-plus spaces, as well as separating half-spaces, and derive an explicit formula for the distance in Hilbert's projective metric between a point and a half-space over the max-plus semiring, as well as explicit descriptions of the set of minimizers. As a consequence, we obtain a cyclic projection type algorithm to solve systems of max-plus linear inequalities.