Certifying convergence of Lasserre’s hierarchy via flat truncation
Consider the optimization problem of minimizing a polynomial function subject to polynomial constraints. A typical approach for solving it globally is applying Lasserre’s hierarchy of semidefinite relaxations, based on either Putinar’s or Schmüdgen’s Positivstellensatz. A practical question in appli...
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| Published in | Mathematical programming Vol. 142; no. 1-2; pp. 485 - 510 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.12.2013
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0025-5610 1436-4646 |
| DOI | 10.1007/s10107-012-0589-9 |
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| Abstract | Consider the optimization problem of minimizing a polynomial function subject to polynomial constraints. A typical approach for solving it globally is applying Lasserre’s hierarchy of semidefinite relaxations, based on either Putinar’s or Schmüdgen’s Positivstellensatz. A practical question in applications is: how to certify its convergence and get minimizers? In this paper, we propose
flat truncation
as a certificate for this purpose. Assume the set of global minimizers is nonempty and finite. Our main results are: (1) Putinar type Lasserre’s hierarchy has finite convergence if and only if flat truncation holds, under some generic assumptions; the same conclusion holds for the Schmüdgen type one under weaker assumptions. (2) Flat truncation is asymptotically satisfied for Putinar type Lasserre’s hierarchy if the archimedean condition holds; the same conclusion holds for the Schmüdgen type one if the feasible set is compact. (3) We show that flat truncation can be used as a certificate to check exactness of standard SOS relaxations and Jacobian SDP relaxations. |
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| AbstractList | Consider the optimization problem of minimizing a polynomial function subject to polynomial constraints. A typical approach for solving it globally is applying Lasserre’s hierarchy of semidefinite relaxations, based on either Putinar’s or Schmüdgen’s Positivstellensatz. A practical question in applications is: how to certify its convergence and get minimizers? In this paper, we propose
flat truncation
as a certificate for this purpose. Assume the set of global minimizers is nonempty and finite. Our main results are: (1) Putinar type Lasserre’s hierarchy has finite convergence if and only if flat truncation holds, under some generic assumptions; the same conclusion holds for the Schmüdgen type one under weaker assumptions. (2) Flat truncation is asymptotically satisfied for Putinar type Lasserre’s hierarchy if the archimedean condition holds; the same conclusion holds for the Schmüdgen type one if the feasible set is compact. (3) We show that flat truncation can be used as a certificate to check exactness of standard SOS relaxations and Jacobian SDP relaxations. Consider the optimization problem of minimizing a polynomial function subject to polynomial constraints. A typical approach for solving it globally is applying Lasserre's hierarchy of semidefinite relaxations, based on either Putinar's or Schmuedgen's Positivstellensatz. A practical question in applications is: how to certify its convergence and get minimizers? In this paper, we propose flat truncation as a certificate for this purpose. Assume the set of global minimizers is nonempty and finite. Our main results are: (1) Putinar type Lasserre's hierarchy has finite convergence if and only if flat truncation holds, under some generic assumptions; the same conclusion holds for the Schmuedgen type one under weaker assumptions. (2) Flat truncation is asymptotically satisfied for Putinar type Lasserre's hierarchy if the archimedean condition holds; the same conclusion holds for the Schmuedgen type one if the feasible set is compact. (3) We show that flat truncation can be used as a certificate to check exactness of standard SOS relaxations and Jacobian SDP relaxations. Consider the optimization problem of minimizing a polynomial function subject to polynomial constraints. A typical approach for solving it globally is applying Lasserre's hierarchy of semidefinite relaxations, based on either Putinar's or Schmüdgen's Positivstellensatz. A practical question in applications is: how to certify its convergence and get minimizers? In this paper, we propose flat truncation as a certificate for this purpose. Assume the set of global minimizers is nonempty and finite. Our main results are: (1) Putinar type Lasserre's hierarchy has finite convergence if and only if flat truncation holds, under some generic assumptions; the same conclusion holds for the Schmüdgen type one under weaker assumptions. (2) Flat truncation is asymptotically satisfied for Putinar type Lasserre's hierarchy if the archimedean condition holds; the same conclusion holds for the Schmüdgen type one if the feasible set is compact. (3) We show that flat truncation can be used as a certificate to check exactness of standard SOS relaxations and Jacobian SDP relaxations.[PUBLICATION ABSTRACT] |
| Author | Nie, Jiawang |
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| Keywords | Semidefinite program 65K05 Sum of squares 90C22 Preordering Quadratic module Flat truncation Lasserre’s hierarchy |
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| References_xml | – reference: NieJSchweighoferMOn the complexity of Putinar’s PositivstellensatzJ. Complex.200723135150229701910.1016/j.jco.2006.07.0021143.13028 – reference: BochnakJCosteMRoyM-FReal Algebraic Geometry1998BerlinSpringer0912.14023 – reference: ConwayJBA Course in Functional Analysis1990BerlinSpringer0706.46003 – reference: CurtoRFialkowLTruncated K-moment problems in several variablesJ. Oper. Theory20055418922621688671119.47304 – reference: ParriloPASturmfelsBBasuSGonzalez-VegaLMinimizing polynomial functionsAlgorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science, vol. 60 of DIMACS Series in Discrete Mathematics and Computer Science2003ProvidenceAMS8399 – reference: ScheidererCSums of squares of regular functions on real algebraic varietiesTrans. Am. Math. Soc.199935210391069167523010.1090/S0002-9947-99-02522-2 – reference: NieJRanestadKAlgebraic degree of polynomial optimizationSIAM J. 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