Inner and Outer Approximations of Star-Convex Semialgebraic Sets

TL;DR

This paper introduces a new scale-invariant objective for approximating star-convex semialgebraic sets with polynomial sublevel-sets, providing tighter inner and outer approximations than existing heuristics.

Contribution

It proposes a novel objective that yields both inner and outer approximations of star-convex sets, minimizing their volume ratio, and offers methods to verify star-convexity via these approximations.

Findings

Approximations are often tighter than existing heuristics.

The proposed method is scale-invariant and easy to interpret.

Numerical examples demonstrate improved approximation quality.

Abstract

We consider the problem of approximating a semialgebraic set with a sublevel-set of a polynomial function. In this setting, it is standard to seek a minimum volume outer approximation and/or maximum volume inner approximation. As there is no known relationship between the coefficients of an arbitrary polynomial and the volume of its sublevel sets, previous works have proposed heuristics based on the determinant and trace objectives commonly used in ellipsoidal fitting. For the case of star-convex semialgebraic sets, we propose a novel objective which yields both an outer and an inner approximation while minimizing the ratio of their respective volumes. This objective is scale-invariant and easily interpreted. Numerical examples are given which show that the approximations obtained are often tighter than those returned by existing heuristics. We also provide methods for establishing the star-convexity of a semialgebraic set by finding inner and outer approximations of its kernel.