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

Abstract

Given a finite Borel measure [Formula: see text] on [Formula: see text] and basic semialgebraic sets [Formula: see text], [Formula: see text], we provide a systematic numerical scheme to approximate as closely as desired [Formula: see text], when all moments of [Formula: see text] 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 [Formula: see text] provides a monotone sequence that converges to the desired value from below. When [Formula: see text] is the Lebesgue measure, we assume that [Formula: see text] is compact and contained in a known box [Formula: see text], and in this case the complement is taken to be [Formula: see text]. In fact, not only [Formula: see text] but also every finite vector of moments of [Formula: see text] (the restriction of [Formula: see text] on [Formula: see text]) can be approximated as closely as desired and so permits to approximate the integral on [Formula: see text] of any given polynomial.