Publikation:

The tracial moment problem and trace-optimization of polynomials

Lade...
Vorschaubild

Dateien

287 Burgdorf.pdf
287 Burgdorf.pdfGröße: 795.32 KBDownloads: 419

Datum

2011

Autor:innen

Cafuta, Kristijan
Povh, Janez

Herausgeber:innen

Kontakt

ISSN der Zeitschrift

Electronic ISSN

ISBN

Bibliografische Daten

Verlag

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
Working Paper/Technical Report
Publikationsstatus
Published

Erschienen in

Zusammenfassung

The main topic addressed in this paper is trace-optimization of polynomials in noncommuting (nc) variables: given an nc polynomial f, what is the smallest trace f(A) can attain for a tuple of matrices A? A relaxation using semide nite programming (SDP) based on sums of hermitian squares and commutators is proposed. While this relaxation is not always exact, it gives e ectively computable bounds on the optima. To test for exactness, the solution of the dual SDP is investigated. If it satis es a certain condition called atness, then the relaxation is exact. In this case it is shown how to extract global trace-optimizers with a procedure based on two ingredients. The first is the solution to the truncated tracial moment problem, and the other crucial component is the numerical implementation of the Artin-Wedderburn theorem for matrix -algebras due to Murota, Kanno, Kojima, Kojima, and Maehara. Trace-optimization of nc polynomials is a nontrivial extension of polynomial optimization in commuting variables on one side and eigenvalue optimization of nc polynomials on the other side { two topics with many applications, the most prominent being to linear systems engineering and quantum physics. The optimization problems discussed here facilitate new possibilities for applications, e.g. in operator algebras and statistical physics.

Zusammenfassung in einer weiteren Sprache

Fachgebiet (DDC)
510 Mathematik

Schlagwörter

sum of squares, noncommutative polynomial, semide nite programming, tracial moment problem, at extension, free positivity, real algebraic geometry

Konferenz

Rezension
undefined / . - undefined, undefined

Forschungsvorhaben

Organisationseinheiten

Zeitschriftenheft

Zugehörige Datensätze in KOPS

Zitieren

ISO 690BURGDORF, Sabine, Kristijan CAFUTA, Igor KLEP, Janez POVH, 2011. The tracial moment problem and trace-optimization of polynomials
BibTex
@techreport{Burgdorf2011traci-15284,
  year={2011},
  series={Konstanzer Schriften in Mathematik},
  title={The tracial moment problem and trace-optimization of polynomials},
  number={287},
  author={Burgdorf, Sabine and Cafuta, Kristijan and Klep, Igor and Povh, Janez}
}
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/123456789/15284">
    <bibo:uri rdf:resource="http://kops.uni-konstanz.de/handle/123456789/15284"/>
    <dc:language>eng</dc:language>
    <dcterms:available rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2011-09-05T08:48:29Z</dcterms:available>
    <dcterms:isPartOf rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
    <dc:date rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2011-09-05T08:48:29Z</dc:date>
    <dc:contributor>Burgdorf, Sabine</dc:contributor>
    <dc:creator>Cafuta, Kristijan</dc:creator>
    <dspace:hasBitstream rdf:resource="https://kops.uni-konstanz.de/bitstream/123456789/15284/2/287%20Burgdorf.pdf"/>
    <dc:rights>terms-of-use</dc:rights>
    <dcterms:issued>2011</dcterms:issued>
    <dcterms:abstract xml:lang="eng">The main topic addressed in this paper is trace-optimization of polynomials in noncommuting (nc) variables: given an nc polynomial f, what is the smallest trace f(A) can attain for a tuple of matrices A? A relaxation using semide nite programming (SDP) based on sums of hermitian squares and commutators is proposed. While this relaxation is not always exact, it gives e ectively computable bounds on the optima. To test for exactness, the solution of the dual SDP is investigated. If it satis es a certain condition called atness, then the relaxation is exact. In this case it is shown how to extract global trace-optimizers with a procedure based on two ingredients. The first is the solution to the truncated tracial moment problem, and the other crucial component is the numerical implementation of the Artin-Wedderburn theorem for matrix  -algebras due to Murota, Kanno, Kojima, Kojima, and Maehara.  Trace-optimization of nc polynomials is a nontrivial extension of polynomial optimization in commuting variables on one side and eigenvalue optimization of nc polynomials on the other side { two topics with many applications, the most prominent being to linear systems engineering and quantum physics. The optimization problems discussed here facilitate new possibilities for applications, e.g. in operator algebras and statistical physics.</dcterms:abstract>
    <foaf:homepage rdf:resource="http://localhost:8080/"/>
    <dcterms:hasPart rdf:resource="https://kops.uni-konstanz.de/bitstream/123456789/15284/2/287%20Burgdorf.pdf"/>
    <dc:creator>Povh, Janez</dc:creator>
    <dc:contributor>Klep, Igor</dc:contributor>
    <dcterms:rights rdf:resource="https://rightsstatements.org/page/InC/1.0/"/>
    <dcterms:title>The tracial moment problem and trace-optimization of polynomials</dcterms:title>
    <dspace:isPartOfCollection rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
    <dc:creator>Burgdorf, Sabine</dc:creator>
    <dc:creator>Klep, Igor</dc:creator>
    <void:sparqlEndpoint rdf:resource="http://localhost/fuseki/dspace/sparql"/>
    <dc:contributor>Cafuta, Kristijan</dc:contributor>
    <dc:contributor>Povh, Janez</dc:contributor>
  </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