Semidefinite approximations of projections and polynomial images of semialgebraic sets

TL;DR

This paper introduces two semidefinite programming-based methods to compute certified outer approximations of polynomial images of semialgebraic sets, including projections, with guarantees of convergence.

Contribution

It proposes novel semidefinite approximation techniques for polynomial images and projections of semialgebraic sets with convergence guarantees.

Findings

Methods produce explicit polynomial superlevel set approximations.

Guarantees of strong convergence in L^1 norm as degree increases.

Numerical experiments demonstrate effectiveness of the approaches.

Abstract

Given a compact semialgebraic set S of R^n and a polynomial map f from R^n to R^m, we consider the problem of approximating the image set F = f(S) in R^m. This includes in particular the projection of S on R^m for n greater than m. Assuming that F is included in a set B which is "simple" (e.g. a box or a ball), we provide two methods to compute certified outer approximations of F. Method 1 exploits the fact that F can be defined with an existential quantifier, while Method 2 computes approximations of the support of image measures.The two methods output a sequence of superlevel sets defined with a single polynomial that yield explicit outer approximations of F. Finding the coefficients of this polynomial boils down to computing an optimal solution of a convex semidefinite program. We provide guarantees of strong convergence to F in L^1 norm on B, when the degree of the polynomial approximation tends to infinity. Several examples of applications are provided, together with numerical experiments.