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Optimization Letters

, Volume 12, Issue 3, pp 435–442 | Cite as

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

  • Milan Korda
  • Didier Henrion
Original Paper
  • 118 Downloads

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. 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.

Keywords

Moment relaxations Polynomial sums of squares Convergence rate Semidefinite programming Approximation theory 

Notes

Acknowledgements

The authors would like to thank the three anonymous reviewers for their constructive comments that helped improve the manuscript as well as to J. Nie for pointing out that the complexity can be reduced if the semialgebraic set is defined by a single constraint. The research of M. Korda was supported by the Swiss National Science Foundation under Grant P2ELP2_165166.

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of CaliforniaSanta BarbaraUSA
  2. 2.LAASCNRSToulouseFrance
  3. 3.LAASUniversité de ToulouseToulouseFrance
  4. 4.Faculty of Electrical EngineeringCzech Technical University in PraguePragueCzech Republic

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