Semidefinite Relaxations for Lebesgue and Gaussian Measures of Unions of Basic Semialgebraic Sets

• 成果类型: Article
• 署名作者: Lasserre, Jean B.; Emin, Youssouf
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Institut Polytechnique de Paris; Ecole Polytechnique
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2018.0980
• 发表日期: 2019
• 页码: 1477-1493
• 关键词: VOLUME

摘要

Given a finite Bard measure mu on R-n and basic semialgebraic sets Omega(i) subset of R-n, i=1, ..., p, we provide a systematic numerical scheme to approximate as closely as desired mu(U-i Omega(i)), when all moments of mu are available (and finite). More precisely, we provide a hierarchy of semidefinite programs whose associated sequence of optimal values is monotone and converges to the desired value from above. The same methodology applied to the complement R-n \ (U-i Omega(i))provides a monotone sequence that converges to the desired value from below. When p is the Lebesgue measure, we assume that Omega:= U-i Omega(i) is compact and contained in a known box B:=[-a,a](n), and in this case the complement is taken to be B \ Omega. In fact, not only mu (Omega) but also every finite vector of moments of mu(Omega) (the restriction of mu on Omega) can be approximated as closely as desired and so permits to approximate the integral on Omega of any given polynomial.