Global Optimization of Polynomials Using Gradient Tentacles and Sums of Squares

Lade...
Vorschaubild
Dateien
tentacle.pdf
tentacle.pdfGröße: 1.57 MBDownloads: 419
Datum
2006
Herausgeber:innen
Kontakt
ISSN der Zeitschrift
Electronic ISSN
ISBN
Bibliografische Daten
Verlag
Schriftenreihe
Auflagebezeichnung
DOI (zitierfähiger Link)
ArXiv-ID
Internationale Patentnummer
Angaben zur Forschungsförderung
Projekt
Open Access-Veröffentlichung
Open Access Green
Core Facility der Universität Konstanz
Gesperrt bis
Titel in einer weiteren Sprache
Publikationstyp
Zeitschriftenartikel
Publikationsstatus
Published
Erschienen in
SIAM Journal on Optimization. 2006, 17(3), pp. 920-942. ISSN 1052-6234. Available under: doi: 10.1137/050647098
Zusammenfassung

In this work, the combine the theory of generalized critical values with the theory of iterated rings of bounded elements (real holomorphy rings). We consider the problem of computing the global infimum of a real polynomial in several variables. Every global minimizer lies on the gradient variety. If the polynomial attains minimum, it is therefore equivalent to look for the greatest lower bound on its gradient variety. Nie, Demmel and Sturmfels proved recently a theorem about the existence of sums of squares certificates for such lower bounds. Based on these certificates, they find arbitrarily tight relaxations of the original problem that can be formulated as semidefinite programs and thus be solved efficiently. We deal here with the more general case when the polynomial is bounded from below but does not necessarily attain a minimum. In this case, the method of Nie, Demmel and Sturmfels might yield completely wrong results. In order to overcome this problem, we replace the gradient variety by larger semialgebraic sets which we call gradient tentacles. It now gets substantially harder to prove the existence of the necessary sums of squares certificates.

Zusammenfassung in einer weiteren Sprache
Fachgebiet (DDC)
510 Mathematik
Schlagwörter
global optimization, polynomial, preorder, sum of squares, semidefinite programming
Konferenz
Rezension
undefined / . - undefined, undefined
Forschungsvorhaben
Organisationseinheiten
Zeitschriftenheft
Datensätze
Zitieren
ISO 690SCHWEIGHOFER, Markus, 2006. Global Optimization of Polynomials Using Gradient Tentacles and Sums of Squares. In: SIAM Journal on Optimization. 2006, 17(3), pp. 920-942. ISSN 1052-6234. Available under: doi: 10.1137/050647098
BibTex
@article{Schweighofer2006Globa-15644,
  year={2006},
  doi={10.1137/050647098},
  title={Global Optimization of Polynomials Using Gradient Tentacles and Sums of Squares},
  number={3},
  volume={17},
  issn={1052-6234},
  journal={SIAM Journal on Optimization},
  pages={920--942},
  author={Schweighofer, Markus}
}
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/123456789/15644">
    <dspace:isPartOfCollection rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
    <dcterms:available rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2011-10-31T09:57:00Z</dcterms:available>
    <dcterms:issued>2006</dcterms:issued>
    <dcterms:title>Global Optimization of Polynomials Using Gradient Tentacles and Sums of Squares</dcterms:title>
    <foaf:homepage rdf:resource="http://localhost:8080/"/>
    <dc:language>eng</dc:language>
    <dcterms:rights rdf:resource="https://rightsstatements.org/page/InC/1.0/"/>
    <dc:date rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2011-10-31T09:57:00Z</dc:date>
    <dspace:hasBitstream rdf:resource="https://kops.uni-konstanz.de/bitstream/123456789/15644/2/tentacle.pdf"/>
    <dc:rights>terms-of-use</dc:rights>
    <dc:contributor>Schweighofer, Markus</dc:contributor>
    <dcterms:bibliographicCitation>SIAM Journal of Optimization 17 (2006), 3. - S. 920-942</dcterms:bibliographicCitation>
    <void:sparqlEndpoint rdf:resource="http://localhost/fuseki/dspace/sparql"/>
    <dcterms:abstract xml:lang="eng">In this work, the combine the theory of generalized critical values with the theory of iterated rings of bounded elements (real holomorphy rings). We consider the problem of computing the global infimum of a real polynomial in several variables. Every global minimizer lies on the gradient variety. If the polynomial attains minimum, it is therefore equivalent to look for the greatest lower bound on its gradient variety. Nie, Demmel and Sturmfels proved recently a theorem about the existence of sums of squares certificates for such lower bounds. Based on these certificates, they find arbitrarily tight relaxations of the original problem that can be formulated as semidefinite programs and thus be solved efficiently. We deal here with the more general case when the polynomial is bounded from below but does not necessarily attain a minimum. In this case, the method of Nie, Demmel and Sturmfels might yield completely wrong results. In order to overcome this problem, we replace the gradient variety by larger semialgebraic sets which we call gradient tentacles. It now gets substantially harder to prove the existence of the necessary sums of squares certificates.</dcterms:abstract>
    <bibo:uri rdf:resource="http://kops.uni-konstanz.de/handle/123456789/15644"/>
    <dcterms:hasPart rdf:resource="https://kops.uni-konstanz.de/bitstream/123456789/15644/2/tentacle.pdf"/>
    <dcterms:isPartOf rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
    <dc:creator>Schweighofer, Markus</dc:creator>
  </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
Diese Publikation teilen