Semidefinite Approximations of Projections and Polynomial Images of SemiAlgebraic Sets

Abstract

Given a compact semialgebraic set $\mathbf{S} \subset \mathbb{R}^n$ and a polynomial map $\mathbf{f} : \mathbb{R}^n \to \mathbb{R}^m$, we consider the problem of approximating the image set $\mathbf{F}=\mathbf{f}(\mathbf{S}) \subset \mathbb{R}^m$. This includes in particular the projection of $\mathbf{S}$ on $\mathbb{R}^m$ for $n \geq m$. Assuming that $\mathbf{F} \subset \mathbf{B}$, with $\mathbf{B} \subset \mathbb{R}^m$ being a “simple” set (e.g., a box or a ball), we provide two methods to compute certified outer approximations of $\mathbf{F}$. Method 1 exploits the fact that $\mathbf{F}$ 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 $\mathbf{F}$. 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 $\mathbf{F}$ in $L_1$ norm on $\mathbf{B}$, when the degree of the polynomial approximation tends to infinity. Several examples of applications are provided, together with numerical experiments.