2017/06/26 by Jean B. Lasserre, Lasserre, Jean, Youssouf Emin +1
Computer Science · Engineering · Mathematics · #Advanced Control Systems Optimization #Advanced Optimization Algorithms Research #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.1706.08253
openalex publication_date 2017/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a finite Borel measure μ on R n and basic semi-algebraic sets Ω_i ⊂ R n , i = 1,. .. , p, we provide a systematic numerical scheme to approximate as closely as desired μ(∪_i Ω_i), when all moments of μ 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 (∪_i Ω_i) provides a monotone sequence that converges to the desired value from below. When μ is the Lebesgue measure we assume that Ω := ∪_i Ω_i is compact and contained in a known box B and in this case the complement is taken to be B Ω. In fact, not only μ(Ω) but also every finite vector of moments of μ_Ω (the restriction of μ on Ω) can be approximated as closely as desired, and so permits to approximate the integral on Ω of any given polynomial.