Journal
SIAM REVIEW
Volume 51, Issue 4, Pages 722-743Publisher
SIAM PUBLICATIONS
DOI: 10.1137/080730287
Keywords
computational geometry; volume; integration; K-moment problem; semidefinite programming
Categories
Funding
- (French) ANR [NT05-3-41612]
- Czech Ministry of Education [MSM6840770038]
- Czech Republic [102/08/0186]
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Given a basic compact semialgebraic set K subset of 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, which permits the approximation of the integral on K of any given polynomial; the 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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