Elliptic polytopes and invariant norms of linear operators

TL;DR

This paper studies the construction of elliptic polytopes formed by convex hulls of ellipses, analyzing the complexity of related optimization problems and proposing geometric methods for solutions in higher dimensions.

Contribution

It introduces new geometric methods for constructing elliptic polytopes and analyzes the complexity of determining ellipse inclusion in the convex hull.

Findings

Number of local extrema can grow exponentially with dimension

Explicit solutions for 2D and 3D cases

Efficient approximate methods for higher dimensions

Abstract

We address the problem of constructing elliptic polytopes in R^d, which are convex hulls of finitely many two-dimensional ellipses with a common center. Such sets arise in the study of spectral properties of matrices, asymptotics of long matrix products, in the Lyapunov stability, etc.. The main issue in the construction is to decide whether a given ellipse is in the convex hull of others. The computational complexity of this problem is analysed by considering an equivalent optimisation problem. We show that the number of local extrema of that problem may grow exponentially in d. For d=2,3, it admits an explicit solution for an arbitrary number of ellipses; for higher dimensions, several geometric methods for approximate solutions are derived. Those methods are analysed numerically and their efficiency is demonstrated in applications.