## Journal article: Global Optimization of Polynomials Using Gradient Tentacles and Sums of Squares

No Thumbnail Available
2006
Journal article
##### Abstract
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.
510 Mathematics
##### Keywords
global optimization , polynomial , preorder , sum of squares , semidefinite programming
##### Published in
SIAM Journal on Optimization ; 17 (2006), 3. - pp. 920-942. - ISSN 1052-6234
##### Cite This
ISO 690SCHWEIGHOFER, Markus, 2006. Global Optimization of Polynomials Using Gradient Tentacles and Sums of Squares. In: SIAM Journal on Optimization. 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#" >
<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>

Yes