On the maxmin-$\omega$ eigenspaces and their over-approximation by zones

TL;DR

This paper introduces a novel over-approximation method using zones for maxmin-$7$ eigenspaces in maxmin-$7$ dynamical systems, improving eigenproblem analysis within tropical linear algebra and nonlinear Perron-Frobenius theory.

Contribution

It develops a new zone-based over-approximation approach for maxmin-$7$ eigenspaces, including a heuristic refinement procedure to identify eigenvalues and eigenvectors.

Findings

The zone over-approximation converges to the eigenspace in many cases.

The heuristic refinement often successfully identifies eigenvalues.

A column of the difference bound matrix can yield an eigenvector in successful cases.

Abstract

Maxmin-$\omega$ dynamical systems were previously introduced as a generalization of dynamical systems expressed by tropical linear algebra. To describe steady states of such systems one has to study an eigenproblem of the form $A\otimes_{\omega} x=\lambda+x$ where $\otimes_{\omega}$ is the maxmin-$\omega$ matrix-vector multiplication. This eigenproblem can be viewed in more general framework of nonlinear Perron-Frobenius theory. However, instead of studying such eigenspaces directly we develop a different approach: over-approximation by zones. These are traditionally convex sets of special kind which proved to be highly useful in computer science and also relevant in tropical convexity. We first construct a sequence of zones over-approximating a maxmin-$\omega$ eigenspace. Next, the limit of this sequence is refined in a heuristic procedure, which yields a refined zone and also the eigenvalue $\lambda$ with a high success rate. Based on the numerical experiments, in successful cases there is a column of the difference bound matrix (DBM) representation of the refined zone which yields an eigenvector.