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arXiv · 0807.2505

Approximate volume and integration for basic semi-algebraic sets

Didier Henrion·Jean Bernard Lasserre·Carlo Savorgnan
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Abstract

Given a basic compact semi-algebraic set $\K\subset\R^n$, we introduce a methodology that generates 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 that is 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 issues associated with the algorithms involved are briefly discussed.

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Didier Henrion, Jean Bernard Lasserre, Carlo Savorgnan. 2008-07-16. Approximate volume and integration for basic semi-algebraic sets. https://doi.org/10.1137/080730287

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