New LP bound in multivariate Lipschitz optimization: Theory and applications
Identifieur interne : 001390 ( Istex/Corpus ); précédent : 001389; suivant : 001391New LP bound in multivariate Lipschitz optimization: Theory and applications
Auteurs : R. Horst ; M. Nast ; N. V. ThoaiSource :
- Journal of Optimization Theory and Applications [ 0022-3239 ] ; 1995-08-01.
Abstract
Abstract: Most numerically promising methods for solving multivariate unconstrained Lipschitz optimization problems of dimension greater than two use rectangular or simplicial branch-and-bound techniques with computationally cheap but rather crude lower bounds. Generalizations to constrained problems, however, require additional devices to detect sufficiently many infeasible partition sets. In this article, a new lower bounding procedure is proposed for simplicial methods yielding considerably better bounds at the expense of two linear programs in each iteration. Moreover, the resulting approach can solve easily linearly constrained problems, since in this case infeasible partition sets are automatically detected by the lower bounding procedure. Finally, it is shown that the lower bounds can be further improved when the method is applied to solve systems of inequalities. Implementation issues, numerical experiments, and comparisons are discussed in some detail.
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DOI: 10.1007/BF02192085
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Generalizations to constrained problems, however, require additional devices to detect sufficiently many infeasible partition sets. In this article, a new lower bounding procedure is proposed for simplicial methods yielding considerably better bounds at the expense of two linear programs in each iteration. Moreover, the resulting approach can solve easily linearly constrained problems, since in this case infeasible partition sets are automatically detected by the lower bounding procedure. Finally, it is shown that the lower bounds can be further improved when the method is applied to solve systems of inequalities. Implementation issues, numerical experiments, and comparisons are discussed in some detail.</div></front></TEI><istex><corpusName>springer</corpusName><author><json:item><name>R. Horst</name><affiliations><json:string>Fachbereich IV-Department of Mathematics, University of Trier, Trier, Germany</json:string></affiliations></json:item><json:item><name>M. 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GROBID</resp><name resp="ISTEX-API">ISTEX-API (INIST-CNRS)</name></respStmt><respStmt><resp>Références bibliographiques récupérées via GROBID</resp><name resp="ISTEX-API">ISTEX-API (INIST-CNRS)</name></respStmt></titleStmt><publicationStmt><authority>ISTEX</authority><publisher>Kluwer Academic Publishers-Plenum Publishers</publisher><pubPlace>New York</pubPlace><availability><p>Plenum Publishing Corporation, 1995</p></availability><date>1995</date></publicationStmt><notesStmt><note>Contributed Papers</note></notesStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a" type="main" xml:lang="en">New LP bound in multivariate Lipschitz optimization: Theory and applications</title><author xml:id="author-1"><persName><forename type="first">R.</forename><surname>Horst</surname></persName><note type="biography">Professor</note><affiliation>Professor</affiliation><affiliation>Fachbereich IV-Department of Mathematics, University of Trier, Trier, Germany</affiliation></author><author xml:id="author-2"><persName><forename type="first">M.</forename><surname>Nast</surname></persName><note type="biography">Assistant</note><affiliation>Assistant</affiliation><affiliation>Fachbereich IV-Department of Mathematics, University of Trier, Trier, Germany</affiliation></author><author xml:id="author-3"><persName><forename type="first">N.</forename><surname>Thoai</surname></persName><note type="biography">Associate Professor</note><affiliation>Associate Professor</affiliation><affiliation>Institute of Mathematics, Bo Ho, Hanoi, Vietnam</affiliation></author></analytic><monogr><title level="j">Journal of Optimization Theory and Applications</title><title level="j" type="abbrev">J Optim Theory Appl</title><idno type="journal-ID">10957</idno><idno type="pISSN">0022-3239</idno><idno type="eISSN">1573-2878</idno><idno type="issue-article-count">16</idno><idno type="volume-issue-count">3</idno><imprint><publisher>Kluwer Academic Publishers-Plenum Publishers</publisher><pubPlace>New York</pubPlace><date type="published" when="1995-08-01"></date><biblScope unit="volume">86</biblScope><biblScope unit="issue">2</biblScope><biblScope unit="page" from="369">369</biblScope><biblScope unit="page" to="388">388</biblScope></imprint></monogr><idno type="istex">1A01A941FB2BB53038BCE09025E7E7CA3A7FAC81</idno><idno type="DOI">10.1007/BF02192085</idno><idno type="ArticleID">BF02192085</idno><idno type="ArticleID">Art5</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>1995</date></creation><langUsage><language ident="en">en</language></langUsage><abstract xml:lang="en"><p>Abstract: Most numerically promising methods for solving multivariate unconstrained Lipschitz optimization problems of dimension greater than two use rectangular or simplicial branch-and-bound techniques with computationally cheap but rather crude lower bounds. Generalizations to constrained problems, however, require additional devices to detect sufficiently many infeasible partition sets. In this article, a new lower bounding procedure is proposed for simplicial methods yielding considerably better bounds at the expense of two linear programs in each iteration. Moreover, the resulting approach can solve easily linearly constrained problems, since in this case infeasible partition sets are automatically detected by the lower bounding procedure. Finally, it is shown that the lower bounds can be further improved when the method is applied to solve systems of inequalities. Implementation issues, numerical experiments, and comparisons are discussed in some detail.</p></abstract><textClass><keywords scheme="Journal Subject"><list><head>Mathematics</head><item><term>Theory of Computation</term></item><item><term>Applications of Mathematics</term></item><item><term>Optimization</term></item><item><term>Calculus of Variations and Optimal Control</term></item><item><term>Optimization</term></item><item><term>Engineering, general</term></item><item><term>Operations Research/Decision Theory</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="1995-08-01">Published</change><change xml:id="refBibs-istex" who="#ISTEX-API" when="2016-11-22">References added</change><change xml:id="refBibs-istex" who="#ISTEX-API" when="2017-01-20">References 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NumberingStyle="Unnumbered"><JournalID>10957</JournalID><JournalPrintISSN>0022-3239</JournalPrintISSN><JournalElectronicISSN>1573-2878</JournalElectronicISSN><JournalTitle>Journal of Optimization Theory and Applications</JournalTitle><JournalAbbreviatedTitle>J Optim Theory Appl</JournalAbbreviatedTitle><JournalSubjectGroup><JournalSubject Type="Primary">Mathematics</JournalSubject><JournalSubject Type="Secondary">Theory of Computation</JournalSubject><JournalSubject Type="Secondary">Applications of Mathematics</JournalSubject><JournalSubject Type="Secondary">Optimization</JournalSubject><JournalSubject Type="Secondary">Calculus of Variations and Optimal Control</JournalSubject><JournalSubject Type="Secondary">Optimization</JournalSubject><JournalSubject Type="Secondary">Engineering, general</JournalSubject><JournalSubject Type="Secondary">Operations Research/Decision Theory</JournalSubject></JournalSubjectGroup></JournalInfo><Volume><VolumeInfo VolumeType="Regular" TocLevels="0"><VolumeIDStart>86</VolumeIDStart><VolumeIDEnd>86</VolumeIDEnd><VolumeIssueCount>3</VolumeIssueCount></VolumeInfo><Issue IssueType="Regular"><IssueInfo TocLevels="0"><IssueIDStart>2</IssueIDStart><IssueIDEnd>2</IssueIDEnd><IssueArticleCount>16</IssueArticleCount><IssueHistory><CoverDate><Year>1995</Year><Month>8</Month></CoverDate></IssueHistory><IssueCopyright><CopyrightHolderName>Plenum Publishing Corporation</CopyrightHolderName><CopyrightYear>1995</CopyrightYear></IssueCopyright></IssueInfo><Article ID="Art5"><ArticleInfo Language="En" ArticleType="OriginalPaper" NumberingStyle="Unnumbered" TocLevels="0" ContainsESM="No"><ArticleID>BF02192085</ArticleID><ArticleDOI>10.1007/BF02192085</ArticleDOI><ArticleSequenceNumber>5</ArticleSequenceNumber><ArticleTitle Language="En">New LP bound in multivariate Lipschitz optimization: Theory and applications</ArticleTitle><ArticleCategory>Contributed Papers</ArticleCategory><ArticleFirstPage>369</ArticleFirstPage><ArticleLastPage>388</ArticleLastPage><ArticleHistory><RegistrationDate><Year>2005</Year><Month>9</Month><Day>9</Day></RegistrationDate></ArticleHistory><ArticleCopyright><CopyrightHolderName>Plenum Publishing Corporation</CopyrightHolderName><CopyrightYear>1995</CopyrightYear></ArticleCopyright><ArticleGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ArticleGrants><ArticleContext><JournalID>10957</JournalID><VolumeIDStart>86</VolumeIDStart><VolumeIDEnd>86</VolumeIDEnd><IssueIDStart>2</IssueIDStart><IssueIDEnd>2</IssueIDEnd></ArticleContext></ArticleInfo><ArticleHeader><AuthorGroup><Author AffiliationIDS="Aff1"><AuthorName DisplayOrder="Western"><GivenName>R.</GivenName><FamilyName>Horst</FamilyName></AuthorName><Role>Professor</Role></Author><Author AffiliationIDS="Aff1"><AuthorName DisplayOrder="Western"><GivenName>M.</GivenName><FamilyName>Nast</FamilyName></AuthorName><Role>Assistant</Role></Author><Author AffiliationIDS="Aff2" PresentAffiliationID="Aff1"><AuthorName DisplayOrder="Western"><GivenName>N.</GivenName><GivenName>V.</GivenName><FamilyName>Thoai</FamilyName></AuthorName><Role>Associate Professor</Role></Author><Affiliation ID="Aff1"><OrgDivision>Fachbereich IV-Department of Mathematics</OrgDivision><OrgName>University of Trier</OrgName><OrgAddress><City>Trier</City><Country>Germany</Country></OrgAddress></Affiliation><Affiliation ID="Aff2"><OrgName>Institute of Mathematics</OrgName><OrgAddress><Street>Bo Ho</Street><City>Hanoi</City><Country>Vietnam</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>Most numerically promising methods for solving multivariate unconstrained Lipschitz optimization problems of dimension greater than two use rectangular or simplicial branch-and-bound techniques with computationally cheap but rather crude lower bounds.</Para><Para>Generalizations to constrained problems, however, require additional devices to detect sufficiently many infeasible partition sets. In this article, a new lower bounding procedure is proposed for simplicial methods yielding considerably better bounds at the expense of two linear programs in each iteration. Moreover, the resulting approach can solve easily linearly constrained problems, since in this case infeasible partition sets are automatically detected by the lower bounding procedure.</Para><Para>Finally, it is shown that the lower bounds can be further improved when the method is applied to solve systems of inequalities. Implementation issues, numerical experiments, and comparisons are discussed in some detail.</Para></Abstract><KeywordGroup Language="En"><Heading>Key Words</Heading><Keyword>Global optimization</Keyword><Keyword>Lipschitz optimization</Keyword><Keyword>systems of inequalities</Keyword><Keyword>branch-and-bounds methods</Keyword></KeywordGroup><ArticleNote Type="CommunicatedBy"><SimplePara>Communicated by G. Leitmann</SimplePara></ArticleNote><ArticleNote Type="Misc"><SimplePara>The authors are indebted to the Editor-in-Chief of this journal for his valuable suggestions which have considerably improved the final version of this article.</SimplePara></ArticleNote></ArticleHeader><NoBody></NoBody></Article></Issue></Volume></Journal></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>New LP bound in multivariate Lipschitz optimization: Theory and applications</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>New LP bound in multivariate Lipschitz optimization: Theory and applications</title></titleInfo><name type="personal"><namePart type="given">R.</namePart><namePart type="family">Horst</namePart><affiliation>Fachbereich IV-Department of Mathematics, University of Trier, Trier, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role><description>Professor</description></name><name type="personal"><namePart type="given">M.</namePart><namePart type="family">Nast</namePart><affiliation>Fachbereich IV-Department of Mathematics, University of Trier, Trier, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role><description>Assistant</description></name><name type="personal"><namePart type="given">N.</namePart><namePart type="given">V.</namePart><namePart type="family">Thoai</namePart><affiliation>Institute of Mathematics, Bo Ho, Hanoi, Vietnam</affiliation><role><roleTerm type="text">author</roleTerm></role><description>Associate Professor</description></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="OriginalPaper"></genre><originInfo><publisher>Kluwer Academic Publishers-Plenum Publishers</publisher><place><placeTerm type="text">New York</placeTerm></place><dateIssued encoding="w3cdtf">1995-08-01</dateIssued><copyrightDate encoding="w3cdtf">1995</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType></physicalDescription><abstract lang="en">Abstract: Most numerically promising methods for solving multivariate unconstrained Lipschitz optimization problems of dimension greater than two use rectangular or simplicial branch-and-bound techniques with computationally cheap but rather crude lower bounds. Generalizations to constrained problems, however, require additional devices to detect sufficiently many infeasible partition sets. In this article, a new lower bounding procedure is proposed for simplicial methods yielding considerably better bounds at the expense of two linear programs in each iteration. Moreover, the resulting approach can solve easily linearly constrained problems, since in this case infeasible partition sets are automatically detected by the lower bounding procedure. Finally, it is shown that the lower bounds can be further improved when the method is applied to solve systems of inequalities. Implementation issues, numerical experiments, and comparisons are discussed in some detail.</abstract><note>Contributed Papers</note><relatedItem type="host"><titleInfo><title>Journal of Optimization Theory and Applications</title></titleInfo><titleInfo type="abbreviated"><title>J Optim Theory Appl</title></titleInfo><genre type="journal" displayLabel="Archive Journal"></genre><originInfo><dateIssued encoding="w3cdtf">1995-08-01</dateIssued><copyrightDate encoding="w3cdtf">1995</copyrightDate></originInfo><subject><genre>Mathematics</genre><topic>Theory of Computation</topic><topic>Applications of Mathematics</topic><topic>Optimization</topic><topic>Calculus of Variations and Optimal Control</topic><topic>Optimization</topic><topic>Engineering, general</topic><topic>Operations Research/Decision Theory</topic></subject><identifier type="ISSN">0022-3239</identifier><identifier type="eISSN">1573-2878</identifier><identifier type="JournalID">10957</identifier><identifier type="IssueArticleCount">16</identifier><identifier type="VolumeIssueCount">3</identifier><part><date>1995</date><detail type="volume"><number>86</number><caption>vol.</caption></detail><detail type="issue"><number>2</number><caption>no.</caption></detail><extent unit="pages"><start>369</start><end>388</end></extent></part><recordInfo><recordOrigin>Plenum Publishing Corporation, 1995</recordOrigin></recordInfo></relatedItem><identifier type="istex">1A01A941FB2BB53038BCE09025E7E7CA3A7FAC81</identifier><identifier type="DOI">10.1007/BF02192085</identifier><identifier type="ArticleID">BF02192085</identifier><identifier type="ArticleID">Art5</identifier><accessCondition type="use and reproduction" contentType="copyright">Plenum Publishing Corporation, 1995</accessCondition><recordInfo><recordContentSource>SPRINGER</recordContentSource><recordOrigin>Plenum Publishing Corporation, 1995</recordOrigin></recordInfo></mods></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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