Semidefinite representation for convex hulls of real algebraic curves

Lade...
Vorschaubild
Dateien
Scheiderer_2-1n6fa7dg51tl00.pdf
Scheiderer_2-1n6fa7dg51tl00.pdfGröße: 534.63 KBDownloads: 259
Datum
2018
Herausgeber:innen
Kontakt
ISSN der Zeitschrift
Electronic ISSN
ISBN
Bibliografische Daten
Verlag
Schriftenreihe
Auflagebezeichnung
DOI (zitierfähiger Link)
Internationale Patentnummer
Angaben zur Forschungsförderung
Projekt
Open Access-Veröffentlichung
Open Access Green
Core Facility der Universität Konstanz
Gesperrt bis
Titel in einer weiteren Sprache
Forschungsvorhaben
Organisationseinheiten
Zeitschriftenheft
Publikationstyp
Zeitschriftenartikel
Publikationsstatus
Published
Erschienen in
SIAM Journal on Applied Algebra and Geometry. 2018, 2(1), pp. 1-25. eISSN 2470-6566. Available under: doi: 10.1137/17M1115113
Zusammenfassung

We show that the closed convex hull of any one-dimensional semialgebraic subset of $\mathbb{R}^n$ is a spectrahedral shadow, meaning that it can be written as a linear image of the solution set of some linear matrix inequality. This is proved by an application of the moment relaxation method. Given a nonsingular affine real algebraic curve $C$ and a compact semialgebraic subset $K$ of its $\mathbb{R}$-points, the preordering $\mathscr{P}(K)$ of all regular functions on $C$ that are nonnegative on $K$ is known to be finitely generated. Our main result, from which all others are derived, says that $\mathscr{P}(K)$ is stable, meaning that uniform degree bounds exist for weighted sum of squares representations of elements of $\mathscr{P}(K)$. We also extend this last result to the case where $K$ is only virtually compact. The main technical tool for the proof of stability is the archimedean local-global principle. As a consequence of our results we show that every convex semialgebraic subset of $\mathbb{R}^2$ is a spectrahedral shadow.

Zusammenfassung in einer weiteren Sprache
Fachgebiet (DDC)
510 Mathematik
Schlagwörter
spectrahedral shadows, convex algebraic geometry, real algebraic curves, convex hull, linear matrix inequalities, moment relaxation, semidefinite programming, Helton-Nie conjecture
Konferenz
Rezension
undefined / . - undefined, undefined
Zitieren
ISO 690SCHEIDERER, Claus, 2018. Semidefinite representation for convex hulls of real algebraic curves. In: SIAM Journal on Applied Algebra and Geometry. 2018, 2(1), pp. 1-25. eISSN 2470-6566. Available under: doi: 10.1137/17M1115113
BibTex
@article{Scheiderer2018Semid-23348.2,
  year={2018},
  doi={10.1137/17M1115113},
  title={Semidefinite representation for convex hulls of real algebraic curves},
  number={1},
  volume={2},
  journal={SIAM Journal on Applied Algebra and Geometry},
  pages={1--25},
  author={Scheiderer, Claus}
}
RDF
<rdf:RDF
    xmlns:dcterms="http://purl.org/dc/terms/"
    xmlns:dc="http://purl.org/dc/elements/1.1/"
    xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
    xmlns:bibo="http://purl.org/ontology/bibo/"
    xmlns:dspace="http://digital-repositories.org/ontologies/dspace/0.1.0#"
    xmlns:foaf="http://xmlns.com/foaf/0.1/"
    xmlns:void="http://rdfs.org/ns/void#"
    xmlns:xsd="http://www.w3.org/2001/XMLSchema#" > 
  <rdf:Description rdf:about="https://kops.uni-konstanz.de/server/rdf/resource/123456789/23348.2">
    <dcterms:isPartOf rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
    <foaf:homepage rdf:resource="http://localhost:8080/"/>
    <dcterms:available rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2018-09-10T11:54:49Z</dcterms:available>
    <dcterms:rights rdf:resource="https://rightsstatements.org/page/InC/1.0/"/>
    <dc:contributor>Scheiderer, Claus</dc:contributor>
    <dc:creator>Scheiderer, Claus</dc:creator>
    <dcterms:hasPart rdf:resource="https://kops.uni-konstanz.de/bitstream/123456789/23348.2/1/Scheiderer_2-1n6fa7dg51tl00.pdf"/>
    <dc:language>eng</dc:language>
    <dcterms:issued>2018</dcterms:issued>
    <dc:date rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2018-09-10T11:54:49Z</dc:date>
    <void:sparqlEndpoint rdf:resource="http://localhost/fuseki/dspace/sparql"/>
    <dspace:isPartOfCollection rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
    <dcterms:title>Semidefinite representation for convex hulls of real algebraic curves</dcterms:title>
    <bibo:uri rdf:resource="https://kops.uni-konstanz.de/handle/123456789/23348.2"/>
    <dspace:hasBitstream rdf:resource="https://kops.uni-konstanz.de/bitstream/123456789/23348.2/1/Scheiderer_2-1n6fa7dg51tl00.pdf"/>
    <dcterms:abstract xml:lang="eng">We show that the closed convex hull of any one-dimensional semialgebraic subset of $\mathbb{R}^n$ is a spectrahedral shadow, meaning that it can be written as a linear image of the solution set of some linear matrix inequality. This is proved by an application of the moment relaxation method. Given a nonsingular affine real algebraic curve $C$ and a compact semialgebraic subset $K$ of its $\mathbb{R}$-points, the preordering $\mathscr{P}(K)$ of all regular functions on $C$ that are nonnegative on $K$ is known to be finitely generated. Our main result, from which all others are derived, says that $\mathscr{P}(K)$ is stable, meaning that uniform degree bounds exist for weighted sum of squares representations of elements of $\mathscr{P}(K)$. We also extend this last result to the case where $K$ is only virtually compact. The main technical tool for the proof of stability is the archimedean local-global principle. As a consequence of our results we show that every convex semialgebraic subset of $\mathbb{R}^2$ is a spectrahedral shadow.</dcterms:abstract>
    <dc:rights>terms-of-use</dc:rights>
  </rdf:Description>
</rdf:RDF>
Interner Vermerk
xmlui.Submission.submit.DescribeStep.inputForms.label.kops_note_fromSubmitter
Kontakt
URL der Originalveröffentl.
Prüfdatum der URL
Prüfungsdatum der Dissertation
Finanzierungsart
Kommentar zur Publikation
Allianzlizenz
Corresponding Authors der Uni Konstanz vorhanden
Internationale Co-Autor:innen
Universitätsbibliographie
Ja
Begutachtet
Unbekannt
Diese Publikation teilen

Versionsgeschichte

Gerade angezeigt 1 - 2 von 2
VersionDatumZusammenfassung
2*
2018-08-27 09:32:04
2013-06-04 10:14:28
* Ausgewählte Version