Approximate volume and integration for basic semi-algebraic sets
Résumé
Given a basic compact semi-algebraic set $K$ in $R^n$ we introduce a methodology that provides a sequence converging to the volume of $K$. This sequence is obtained from optimal values of a hierarchy of either semidefinite or linear programs. Not only the volume but also every finite vector of moments of the probability measure uniformly distributed on $K$ can be approximated as closely as desired, and so permits to approximate the integral on $K$ of any given polynomial; extension to integration against some weight functions is also provided. Finally, some numerical aspects are discussed.
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