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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| Published in: | Optimization letters Vol. 12; no. 3; pp. 435 - 442 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.05.2018
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| ISSN: | 1862-4472, 1862-4480 |
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| 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. |
|---|---|
| AbstractList | 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. |
| Author | Korda, Milan Henrion, Didier |
| Author_xml | – sequence: 1 givenname: Milan surname: Korda fullname: Korda, Milan email: milan.korda@engineering.ucsb.edu organization: Department of Mechanical Engineering, University of California – sequence: 2 givenname: Didier surname: Henrion fullname: Henrion, Didier organization: LAAS, CNRS, LAAS, Université de Toulouse, Faculty of Electrical Engineering, Czech Technical University in Prague |
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| Cites_doi | 10.2307/2371513 10.1512/iumj.1993.42.42045 10.3844/jmssp.2010.240.245 10.1137/080730287 10.1016/j.jco.2006.07.002 10.1016/j.jco.2004.01.005 10.1016/j.sysconle.2016.11.010 |
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| Keywords | Convergence rate Polynomial sums of squares Moment relaxations Semidefinite programming Approximation theory |
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| References | Nie, Schweighofer (CR6) 2007; 23 Laurent, Putinar, Sullivan (CR5) 2009 CR3 Schweighofer (CR8) 2004; 20 Putinar (CR7) 1993; 42 Weyl (CR9) 1939; 61 Henrion, Lasserre, Savorgnan (CR2) 2009; 51 Bhaya (CR1) 2010; 6 Lasserre (CR4) 2010 ES Bhaya (1186_CR1) 2010; 6 D Henrion (1186_CR2) 2009; 51 M Laurent (1186_CR5) 2009 J Nie (1186_CR6) 2007; 23 M Putinar (1186_CR7) 1993; 42 H Weyl (1186_CR9) 1939; 61 JB Lasserre (1186_CR4) 2010 1186_CR3 M Schweighofer (1186_CR8) 2004; 20 |
| References_xml | – volume: 61 start-page: 461 year: 1939 end-page: 472 ident: CR9 article-title: On the volume of tubes publication-title: Am. J. Math. doi: 10.2307/2371513 – year: 2009 ident: CR5 article-title: Sums of squares, moment matrices and polynomial optimization publication-title: Emerging Applications of Algebraic Geometry – year: 2010 ident: CR4 publication-title: Moments, Positive Polynomials and Their Applications – volume: 42 start-page: 969 year: 1993 end-page: 984 ident: CR7 article-title: Positive polynomials on compact semi-algebraic sets publication-title: Indiana Univ. Math. J. doi: 10.1512/iumj.1993.42.42045 – volume: 6 start-page: 240 year: 2010 end-page: 245 ident: CR1 article-title: Interpolating operators for multiapproximation publication-title: J. Math. Stat. doi: 10.3844/jmssp.2010.240.245 – ident: CR3 – volume: 51 start-page: 722 issue: 4 year: 2009 end-page: 743 ident: CR2 article-title: Approximate volume and integration for basic semialgebraic sets publication-title: SIAM Rev. doi: 10.1137/080730287 – volume: 23 start-page: 135 year: 2007 end-page: 150 ident: CR6 article-title: On the complexity of Putinar’s Positivstellensatz publication-title: J. Complex. doi: 10.1016/j.jco.2006.07.002 – volume: 20 start-page: 529 year: 2004 end-page: 543 ident: CR8 article-title: On the complexity of Schmüdgen’s Positivstellensatz publication-title: J. Complex. doi: 10.1016/j.jco.2004.01.005 – volume: 42 start-page: 969 year: 1993 ident: 1186_CR7 publication-title: Indiana Univ. Math. J. doi: 10.1512/iumj.1993.42.42045 – volume: 51 start-page: 722 issue: 4 year: 2009 ident: 1186_CR2 publication-title: SIAM Rev. doi: 10.1137/080730287 – volume-title: Moments, Positive Polynomials and Their Applications year: 2010 ident: 1186_CR4 – ident: 1186_CR3 doi: 10.1016/j.sysconle.2016.11.010 – volume-title: Emerging Applications of Algebraic Geometry year: 2009 ident: 1186_CR5 – volume: 61 start-page: 461 year: 1939 ident: 1186_CR9 publication-title: Am. J. Math. doi: 10.2307/2371513 – volume: 6 start-page: 240 year: 2010 ident: 1186_CR1 publication-title: J. Math. Stat. doi: 10.3844/jmssp.2010.240.245 – volume: 23 start-page: 135 year: 2007 ident: 1186_CR6 publication-title: J. Complex. doi: 10.1016/j.jco.2006.07.002 – volume: 20 start-page: 529 year: 2004 ident: 1186_CR8 publication-title: J. Complex. doi: 10.1016/j.jco.2004.01.005 |
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| Snippet | 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... |
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| Title | Convergence rates of moment-sum-of-squares hierarchies for volume approximation of semialgebraic sets |
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