Convergence rates of moment-sum-of-squares hierarchies for volume approximation of semialgebraic sets - Korda, Milan; Henrion, Didier (Optimization letters, 2018)
Moment-sum-of-squares hierarchies of semidefinite programs can be used to approximate the volume of a given compact basic semialgebraic set K . The idea consists of approximating from above the indicator function of K with a sequence of polynomials of increasing degree d , so that the integrals of these polynomials generate a convergence sequence of upper bounds on the volume of K . Under certain assumptions, we show that the asymptotic rate of this convergence is at least O ( 1 / log log d ) in general and O ( 1 / log d ) provided that the semialgebraic set is defined by a single inequality.