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arXiv · 1507.06143

Semidefinite approximations of projections and polynomial images of semialgebraic sets

Victor Magron·Didier Henrion·Jean-Bernard Lasserre
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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.

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Victor Magron, Didier Henrion, Jean-Bernard Lasserre. 2015-07-22. Semidefinite approximations of projections and polynomial images of semialgebraic sets. https://arxiv.org/abs/1507.06143

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