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Convergence rates of moment-sum-of-squares hierarchies for volume approximation of semialgebraic sets

Milan Korda(milan.korda***at***engineering.ucsb.edu)
Didier Henrion(henrion***at***laas.fr)

Abstract: 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 $K$. We show that the asymptotic rate of this convergence is at least $O(1 / \log \log d)$.

Keywords: moment relaxations, polynomial sums of squares, convergence rate, semidefinite programming, approximation theory

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 2: Infinite Dimensional Optimization (Other )

Citation:

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Entry Submitted: 12/12/2016
Entry Accepted: 12/13/2016
Entry Last Modified: 12/12/2016

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