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

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 t...

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Bibliographic Details
Published in:Optimization letters Vol. 12; no. 3; pp. 435 - 442
Main Authors: Korda, Milan, Henrion, Didier
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2018
Subjects:
Computational Intelligence
Mathematics
Mathematics and Statistics
Numerical and Computational Physics
Operations Research/Decision Theory
Optimization
Original Paper
Simulation
Convergence rate
Polynomial sums of squares
Moment relaxations
Semidefinite programming
Approximation theory
ISSN:1862-4472, 1862-4480
Online Access:Get full text
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Summary: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.
ISSN:1862-4472
1862-4480
DOI:10.1007/s11590-017-1186-x