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Article Dans Une Revue SIAM Journal on Optimization Année : 2015

Semidefinite approximations of projections and polynomial images of semialgebraic sets

Résumé

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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Dates et versions

hal-01075343 , version 1 (11-12-2014)
hal-01075343 , version 2 (22-07-2015)

Identifiants

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Victor Magron, Didier Henrion, Jean-Bernard Lasserre. Semidefinite approximations of projections and polynomial images of semialgebraic sets. SIAM Journal on Optimization, 2015, 25 (4), pp. 2143-2164. ⟨hal-01075343v2⟩
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