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# On the exactness of Lasserre relaxations for compact convex basic closed semialgebraic sets

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Publikationstyp: | Working Paper/Technical Report |

Publikationsstatus: | Submitted |

URI (zitierfähiger Link): | http://nbn-resolving.de/urn:nbn:de:bsz:352-0-406718 |

Autor/innen: | Kriel, Tom-Lukas; Schweighofer, Markus |

Erscheinungsjahr: | 2017 |

ArXiv-ID: | arXiv:1704.07231 |

Zusammenfassung: |
Consider a finite system of non-strict real polynomial inequalities and suppose its solution set S⊆R
^{n} is convex, has nonempty interior and is compact. Suppose that the system satisfies the Archimedean condition, which is slightly stronger than the compactness of S. Suppose that each defining polynomial satisfies a second order strict quasiconcavity condition where it vanishes on S (which is very natural because of the convexity of S) or its Hessian has a certain matrix sums of squares certificate for negative-semidefiniteness on S (fulfilled trivially by linear polynomials). Then we show that the system possesses an exact Lasserre relaxation.In their seminal work of 2009, Helton and Nie showed under the same conditions that S is the projection of a spectrahedron, i.e., it has a semidefinite representation. The semidefinite representation used by Helton and Nie arises from glueing together Lasserre relaxations of many small pieces obtained in a non-constructive way. By refining and varying their approach, we show that we can simply take a Lasserre relaxation of the original system itself. Such a result was provided by Helton and Nie with much more machinery only under very technical conditions and after changing the description of S. |

MSC-Klassifikation: | Primary 14P10, 52A20; Secondary 13J30, 52A41, 90C22, 90C26 |

Fachgebiet (DDC): | 510 Mathematik |

Schlagwörter: | moment relaxation, Lasserre relaxation, basic closed semialgebraic set, sum of squares, polynomial optimization, semidefinite programming, linear matrix inequality, spectrahedron, semidefinitely representable set |

Link zur Lizenz: | Nutzungsbedingungen |

Universitätsbibliographie: | Ja |

Prüfsumme:
MD5:acb4c4e75bfec3f8ba087621d53908a4

KRIEL, Tom-Lukas, Markus SCHWEIGHOFER, 2017. On the exactness of Lasserre relaxations for compact convex basic closed semialgebraic sets

@techreport{Kriel2017-04-24T13:50:50Zexact-38753, title={On the exactness of Lasserre relaxations for compact convex basic closed semialgebraic sets}, year={2017}, author={Kriel, Tom-Lukas and Schweighofer, Markus} }

Kriel_0-406718.pdf | 63 |