Constructive solution of the common invariant cone problem

TL;DR

This paper presents an efficient algorithm to determine the existence of a common invariant cone for a set of matrices, overcoming the undecidability in theory with practical solutions and applications.

Contribution

The authors introduce a practical algorithm that can find or disprove the existence of a common invariant cone for finite matrix sets, with analysis of cone properties.

Findings

Algorithm successfully finds common invariant cones in most tested cases.

The method can prove non-existence of a common invariant cone.

Applications demonstrated in dynamical systems and combinatorics.

Abstract

Sets of $d\times d$ matrices sharing a common invariant cone enjoy special properties, which are widely used in applications. However, finding this cone or even proving its existence/non-existence is hard. This problem is known to be algorithmically undecidable for general sets of matrices. We show that it can nevertheless be efficiently solved in practice. An algorithm that for a given finite set of matrices, either finds a common invariant cone or proves its non-existence is presented. Numerical results demonstrate that it works for a vast majority of matrix sets. The structure and properties of the minimal and maximal invariant cones are analyzed. Applications to dynamical systems and combinatorics are considered.