Elliptic polytopes and invariant norms of linear operators

Abstract

Elliptic polytopes are convex hulls of several concentric plane ellipses in ${{\mathbb {R}}}^d$. They arise in applications as natural generalizations of usual polytopes. In particular, they define invariant convex bodies of linear operators, optimal Lyapunov norms for linear dynamical systems, etc. To construct elliptic polytopes one needs to decide whether a given ellipse is contained in the convex hull of other ellipses. We analyse the computational complexity of this problem and show that for $d=2, 3$, it admits an explicit solution. For larger d, two geometric methods for approximate solution are presented. Both use the convex optimization tools. The efficiency of the methods is demonstrated in two applications: the construction of extremal norms of linear operators and the computation of the joint spectral radius/Lyapunov exponent of a family of matrices.

Appendix

Theorem

(Theorem 5.4) For each $N\ge 2$ there exists a polyhedron in ${{\mathbb {R}}}^N$ with less than 2N facets and a positive semidefinite quadratic form of rank two on that polyhedron which has at least $2^{N-2}$ points of local maxima with different values of the function.

We begin with the following technical fact. Let us have a vector ${\varvec{a}}\in {{\mathbb {R}}}^2$ and a line $\ell$ on ${{\mathbb {R}}}^2$ which is not parallel to ${\varvec{a}}$. An affine symmetry about $\ell$ along ${\varvec{a}}$ is an affine transform that, for each ${\varvec{x}}\in \ell$ and $t \in {{\mathbb {R}}}$, maps the point ${\varvec{x}}+ t{\varvec{a}}$ to ${\varvec{x}}- t{\varvec{a}}$. If ${\varvec{a}}\perp \ell$, then this is the usual (orthogonal) symmetry.

Lemma A.1

Let O be an arbitrary point on the side of a convex polygon different from its midpoint. Then there exists an affine symmetry about this side arbitrarily close to an orthogonal symmetry such that the distances from O to the images of the vertices of this polygon are all different.

Proof

Let O be the origin and one of the basis vectors along that side. Then, the matrix of an arbitrary affine symmetry is $S = \left( {\begin{matrix} 1 &{}\quad a\\ 0 &{}\quad -1 \end{matrix}} \right)$, with $a\in {\mathbb {R}}$ arbitrary. If the images $A{\varvec{x}}$ and $A{\varvec{y}}$ of two vertices ${\varvec{x}}\ne {\varvec{y}}$ are equidistant from O, then $A({\varvec{x}}+ {\varvec{y}})$ and $A({\varvec{x}}- {\varvec{y}})$ are orthogonal and, hence, $({\varvec{x}}- {\varvec{y}})A^TA ({\varvec{x}}+ {\varvec{y}}) = 0$. This quadratic equation in a has at most two solutions. Hence, there exists only a finite number of values of a for which some of the images of the vertices are equidistant from O. $\square$

Proposition A.2

For every $n\ge 2$ and $\varepsilon > 0$, there exists a polyhedron $Q_n$ in ${{\mathbb {R}}}^{n+2}$ with at most $2n+3$ facets whose orthogonal projection to some two-dimensional plane is a $2^n$-gon such that: (1) Its distance (in the Hausdorff metric) to a regular $2^n$-gon centred at the origin is less than $\varepsilon$. (2) The distances from its $2^n$ vertices to the origin are all different.

Proof

Applying the construction (16) for $r= 1$, we obtain a polyhedron that consists of points $(x_1, \ldots , x_{2n+2})^T \in {{\mathbb {R}}}^{2n+2}$ satisfying the system (16). That system contains n linear equations and $2n+3$ linear inequalities. Hence, it defines an $(n+2)$-dimensional polyhedron with at most $2n+3$ facets. Its projection to the plane $(x_{2n+1}, x_{2n+2})$ is a regular $2^n$-gon. Now, in each iteration $j = 1, \dots , n$ of the construction (16), we replace the symmetry about the line $\ell _{\alpha _{n-j+1}}$ by a close affine symmetry about the same line. Invoking Lemma A.1 we can choose this symmetry so that the resulting polygon has all its vertices on different distances from the origin. Hence, the polygon obtained after the last iteration also possesses this property. $\square$

Proof of Theorem 5.4

After applying Proposition A.2 for $n=N-2$, we obtain a polyhedron $Q_{N-2} \subset {{\mathbb {R}}}^{N}$ whose two-dimensional projection to the plane $(x_{2N-3},x_{2N-2})$ is a $2^{N-2}$-gon close to a regular $2^{N-2}$-gon. Then, for the quadratic form $x_{2N-3}^2 + x_{2N-2}^2$, each vertex of this polygon is a local maximum and all the values in those points are different. $\square$

Fig. 9

Runtime t in seconds of the the methods complex polytope, corner cutting and projection. On the x-axis the theoretical minimal accuracy on a logarithmic scale is printed, on the y-axis the time the algorithm needed, also on a logarithmic scale. The true, obtained, accuracy is especially for the complex polytope method much higher. The colour indicates the number of vertices of the elliptic polytope. The dimension of the problem is not plotted, since it turned out to have only a very minor influence on the runtime. All algorithms were assessed using the same data set, the corner cutting method and the projection method were tested with different approximation factors. The left column is for data arising in the Invariant polytope algorithm. The right column is for data of ellipses and elliptic polytopes with normal distributed real and imaginary part. One can see clearly, that the complex polytope method is the most efficient algorithm when one compares the time the algorithm needs with its accuracy. This is even more true under the viewpoint that the complex polytope method on average yields an accuracy of 0.7071. Comparing the corner cutting method and the projection method, one sees that the latter clearly outperforms the former consistently

The following example is for the curious reader, who asks herself whether there exists a set of matrices, for which the spectral maximizing product has length greater than one and a complex leading eigenvalue.

Example A.3

For $\alpha ,\beta \in (-\pi /2,\pi /2)$, $\alpha \ne \beta$, the set $\{T_0,T_1\}$,

$$\begin{aligned} T_0 = \begin{pmatrix} 0 &{}\quad 0 &{}\quad 0\ \\ -\sin \alpha &{}\quad \cos \alpha &{}\quad 0\ \\ \,\cos \alpha &{}\quad \sin \alpha &{}\quad 0\ \end{pmatrix}, \quad T_1 = \begin{pmatrix} \ 0 &{}\quad -\sin \beta &{}\quad \cos \beta \ \\ \ 0 &{}\quad \,\cos \beta &{}\quad \sin \, \beta \ \\ \ 0 &{}\quad 0 &{}\quad 0\ \end{pmatrix}, \end{aligned}$$

has $T_0T_1$ as spectrum maximizing product, i.e. up to permutations and powers the normalized spectral radius of all other products of matrices $T_0$ and $T_1$ is strictly less than $\rho (T_0T_1)^{1/2}=1$ (Fig. 9).