Minimum volume semialgebraic sets for robust estimation

TL;DR

This paper introduces a tractable heuristic method for computing minimum volume polynomial sublevel sets that contain a given semialgebraic set, with applications in robust estimation and uncertainty propagation.

Contribution

It proposes an L^1-norm or trace heuristic using LMI relaxations for semialgebraic sets and simplifies to LP for discrete samples, advancing computational approaches in this area.

Findings

The heuristic provides a practical approach for minimum volume set estimation.

LMI relaxations enable handling complex semialgebraic sets.

Linear programming simplifies the problem for sample-based sets.

Abstract

Motivated by problems of uncertainty propagation and robust estimation we are interested in computing a polynomial sublevel set of fixed degree and minimum volume that contains a given semialgebraic set K. At this level of generality this problem is not tractable, even though it becomes convex e.g. when restricted to nonnegative homogeneous polynomials. Our contribution is to describe and justify a tractable L^1-norm or trace heuristic for this problem, relying upon hierarchies of linear matrix inequality (LMI) relaxations when K is semialgebraic, and simplifying to linear programming (LP) when K is a collection of samples, a discrete union of points.