Given a compact semialgebraic set S of R^n and a polynomial map f from R^n toR^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 aball), we provide two methods to compute certified outer approximations of F.Method 1 exploits the fact that F can be defined with an existentialquantifier, while Method 2 computes approximations of the support of imagemeasures.The two methods output a sequence of superlevel sets defined with asingle polynomial that yield explicit outer approximations of F. Finding thecoefficients of this polynomial boils down to computing an optimal solution ofa convex semidefinite program. We provide guarantees of strong convergence to Fin L^1 norm on B, when the degree of the polynomial approximation tends toinfinity. Several examples of applications are provided, together withnumerical experiments.
展开▼
