Generalized Ellipsoids

TL;DR

This paper introduces generalized ellipsoids (GEs), a new family of symmetric convex bodies that extend traditional ellipsoids, with efficient algorithms for their characterization and applications in optimization and control.

Contribution

The paper defines GEs of degree d, proves their properties, provides efficient algorithms for their verification and representation, and demonstrates their applicability in various optimization problems.

Findings

GEs can be checked in strongly polynomial time.

Every GE has a semidefinite representation with size linear in dimension and degree.

GEs can approximate any symmetric convex body arbitrarily well.

Abstract

We introduce a family of symmetric convex bodies called generalized ellipsoids of degree $d$ (GE-$d$s), with ellipsoids corresponding to the case of $d=0$. Generalized ellipsoids (GEs) retain many geometric, algebraic, and algorithmic properties of ellipsoids. We show that the conditions that the parameters of a GE must satisfy can be checked in strongly polynomial time, and that one can search for GEs of a given degree by solving a semidefinite program whose size grows only linearly with dimension. We give an example of a GE which does not have a second-order cone representation, but show that every GE has a semidefinite representation whose size depends linearly on both its dimension and degree. In terms of expressiveness, we prove that for any integer $m\geq 2$, every symmetric full-dimensional polytope with $2m$ facets and every intersection of $m$ co-centered ellipsoids can be represented exactly as a GE-$d$ with $d \leq 2m-3$. Using this result, we show that every symmetric convex body can be approximated arbitrarily well by a GE-$d$ and we quantify the quality of the approximation as a function of the degree $d$. Finally, we present applications of GEs to several areas, such as time-varying portfolio optimization, stability analysis of switched linear systems, robust-to-dynamics optimization, and robust polynomial regression.