Path-following proximal approach for solving Ill-posed convex semi-infinite programming problems
Identifieur interne : 001193 ( Istex/Corpus ); précédent : 001192; suivant : 001194Path-following proximal approach for solving Ill-posed convex semi-infinite programming problems
Auteurs : A. Kaplan ; R. TichatschkeSource :
- Journal of Optimization Theory and Applications [ 0022-3239 ] ; 1996-07-01.
Abstract
Abstract: For a class of ill-posed, convex semi-infinite programming problems, a regularized path-following strategy is developed. This approach consists in a coordinated application of adaptive discretization and prox-regularization procedures combined with a penalty method. At each iteration, only an approximate minimum of a strongly convex differentiable function has to be calculated, and this can be done by any fast-convergent algorithm. The use of prox-regularization ensures the convergence of the iterates to some solution of the original problem. Due to regularization, an efficient deleting rule is applicable, which excludes an essential part of the constraints in the discretized problems.
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DOI: 10.1007/BF02192249
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Kaplan</name><affiliation><mods:affiliation>Institute of Applied Mathematics, Humboldt University at Berlin, Berlin, Germany</mods:affiliation></affiliation></author><author><name sortKey="Tichatschke, R" sort="Tichatschke, R" uniqKey="Tichatschke R" first="R." last="Tichatschke">R. 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This approach consists in a coordinated application of adaptive discretization and prox-regularization procedures combined with a penalty method. At each iteration, only an approximate minimum of a strongly convex differentiable function has to be calculated, and this can be done by any fast-convergent algorithm. The use of prox-regularization ensures the convergence of the iterates to some solution of the original problem. Due to regularization, an efficient deleting rule is applicable, which excludes an essential part of the constraints in the discretized problems.</div></front></TEI><istex><corpusName>springer</corpusName><author><json:item><name>A. Kaplan</name><affiliations><json:string>Institute of Applied Mathematics, Humboldt University at Berlin, Berlin, Germany</json:string></affiliations></json:item><json:item><name>R. 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Due to regularization, an efficient deleting rule is applicable, which excludes an essential part of the constraints in the discretized problems.</abstract><qualityIndicators><score>6.176</score><pdfVersion>1.3</pdfVersion><pdfPageSize>432 x 648 pts</pdfPageSize><refBibsNative>false</refBibsNative><keywordCount>0</keywordCount><abstractCharCount>704</abstractCharCount><pdfWordCount>6015</pdfWordCount><pdfCharCount>29239</pdfCharCount><pdfPageCount>25</pdfPageCount><abstractWordCount>98</abstractWordCount></qualityIndicators><title>Path-following proximal approach for solving Ill-posed convex semi-infinite programming problems</title><refBibs><json:item><author><json:item><name>Hettich </name></json:item><json:item><name>R </name></json:item><json:item><name>Kortanek </name></json:item><json:item><name>K,O </name></json:item></author><host><pages><last>429</last><first>380</first></pages><author></author><title>Semi-Infinite Programming: Theory</title><publicationDate>1993</publicationDate></host><publicationDate>1993</publicationDate></json:item><json:item><host><author><json:item><name>Kortanek </name></json:item><json:item><name>K,O Hoon</name></json:item><json:item><name>N </name></json:item></author><title>A Central Cutting Plane Algorithm for Convex Semi-Infinite Programming Problems, Working Paper 90-08</title><publicationDate>1992</publicationDate></host></json:item><json:item><author><json:item><name>Panier </name></json:item><json:item><name>E </name></json:item><json:item><name>Tits </name></json:item><json:item><name>A,L </name></json:item></author><host><volume>34</volume><pages><last>908</last><first>903</first></pages><author></author><title>IEEE Transactions on Automatic Control</title><publicationDate>1989</publicationDate></host><title>A Globally Convergent Algorithm with Adaptively Refined Discretization for Semi-Infinite Optimization Problems Arising in Engineering Design</title><publicationDate>1989</publicationDate></json:item><json:item><author><json:item><name>Polak </name></json:item><json:item><name>E </name></json:item><json:item><name>He </name></json:item><json:item><name>L </name></json:item></author><host><volume>30</volume><pages><last>572</last><first>548</first></pages><author></author><title>SIAM Journal on Control and Optimization</title><publicationDate>1992</publicationDate></host><title>Rate-Preserving Diseretization Strategies for Semi- Infinite Programming and Optimal Control</title><publicationDate>1992</publicationDate></json:item><json:item><host><author><json:item><name>Zhou </name></json:item><json:item><name>J,L </name></json:item><json:item><name>Tits </name></json:item><json:item><name>A,L </name></json:item></author><title>An SQP Algorithm for Finely Discretized SIP Problems and Other Problems with Many Constraints</title><publicationDate>1992</publicationDate></host></json:item><json:item><author><json:item><name>Gollmer </name></json:item><json:item><name>R Guddat</name></json:item><json:item><name>J Guerra</name></json:item><json:item><name>F Nowack</name></json:item><json:item><name>D </name></json:item><json:item><name>Ruckman </name></json:item><json:item><name>J,J </name></json:item></author><host><pages><last>214</last><first>163</first></pages><author></author><title>Series on Approximation and Optimization</title><publicationDate>1992</publicationDate></host><title>Path-Following Methods in Nonlinear Optimization, I." 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</name></json:item><json:item><name>A </name></json:item><json:item><name>Tichatschke~,R </name></json:item></author><title>A Regularized Penalty Method for Solving Convex Semi-Infiniw Programs, Optimization</title><publicationDate>1992</publicationDate></host></json:item><json:item><author><json:item><name>Ekeland </name></json:item><json:item><name>I </name></json:item><json:item><name>Temam </name></json:item><json:item><name>R </name></json:item></author><host><author></author><title>Convex Analysis and Variational Problems</title><publicationDate>1976</publicationDate></host><publicationDate>1976</publicationDate></json:item><json:item><host><author><json:item><name>Polyak </name></json:item><json:item><name>B,Z </name></json:item></author><title>Introduction to Optimization, Optimization Software, Publishing Division</title><publicationDate>1987</publicationDate></host></json:item><json:item><host><author><json:item><name>Hettich </name></json:item><json:item><name>R </name></json:item><json:item><name>Zencke </name></json:item><json:item><name>P </name></json:item></author><title>Numerische Methoden der Approximation und Semi-Infinite Optimierung</title><publicationDate>1982</publicationDate></host></json:item><json:item><host><author><json:item><name>Henr~on </name></json:item><json:item><name>R </name></json:item></author><title>Strukturuntersuchungen zum Parametrischen Teilproblem in Semi- Infiniten Optimierungsaufgaben</title><publicationDate>1989</publicationDate></host></json:item><json:item><author><json:item><name>Jongen </name></json:item><json:item><name>H,T Jonker</name></json:item><json:item><name>P Twllt</name></json:item><json:item><name>F </name></json:item></author><host><volume>34</volume><pages><last>353</last><first>333</first></pages><author></author><title>Mathematical Programming</title><publicationDate>1986</publicationDate></host><title>Critical Sets in Parametric 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This approach consists in a coordinated application of adaptive discretization and prox-regularization procedures combined with a penalty method. At each iteration, only an approximate minimum of a strongly convex differentiable function has to be calculated, and this can be done by any fast-convergent algorithm. The use of prox-regularization ensures the convergence of the iterates to some solution of the original problem. Due to regularization, an efficient deleting rule is applicable, which excludes an essential part of the constraints in the discretized problems.</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="1996-07-01">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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Papers</ArticleCategory><ArticleFirstPage>113</ArticleFirstPage><ArticleLastPage>137</ArticleLastPage><ArticleHistory><RegistrationDate><Year>2005</Year><Month>9</Month><Day>9</Day></RegistrationDate></ArticleHistory><ArticleCopyright><CopyrightHolderName>Plenum Publishing Corporation</CopyrightHolderName><CopyrightYear>1996</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>90</VolumeIDStart><VolumeIDEnd>90</VolumeIDEnd><IssueIDStart>1</IssueIDStart><IssueIDEnd>1</IssueIDEnd></ArticleContext></ArticleInfo><ArticleHeader><AuthorGroup><Author AffiliationIDS="Aff1"><AuthorName DisplayOrder="Western"><GivenName>A.</GivenName><FamilyName>Kaplan</FamilyName></AuthorName><Role>Professor</Role></Author><Author AffiliationIDS="Aff2"><AuthorName DisplayOrder="Western"><GivenName>R.</GivenName><FamilyName>Tichatschke</FamilyName></AuthorName><Role>Professor</Role></Author><Affiliation ID="Aff1"><OrgDivision>Institute of Applied Mathematics</OrgDivision><OrgName>Humboldt University at Berlin</OrgName><OrgAddress><City>Berlin</City><Country>Germany</Country></OrgAddress></Affiliation><Affiliation ID="Aff2"><OrgDivision>Department of Mathematics</OrgDivision><OrgName>University Trier</OrgName><OrgAddress><City>Trier</City><Country>Germany</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>For a class of ill-posed, convex semi-infinite programming problems, a regularized path-following strategy is developed. This approach consists in a coordinated application of adaptive discretization and prox-regularization procedures combined with a penalty method. At each iteration, only an approximate minimum of a strongly convex differentiable function has to be calculated, and this can be done by any fast-convergent algorithm. The use of prox-regularization ensures the convergence of the iterates to some solution of the original problem. Due to regularization, an efficient deleting rule is applicable, which excludes an essential part of the constraints in the discretized problems.</Para></Abstract><KeywordGroup Language="En"><Heading>Key Words</Heading><Keyword>Regularization</Keyword><Keyword>penalty methods</Keyword><Keyword>parametric semi-infinite problems</Keyword><Keyword>adaptive discretization</Keyword></KeywordGroup><ArticleNote Type="CommunicatedBy"><SimplePara>Communicated by F. Giannessi</SimplePara></ArticleNote><ArticleNote Type="Misc"><SimplePara>This research was supported by the German Research Society (DFG).</SimplePara></ArticleNote><ArticleNote Type="Misc"><SimplePara>The authors are grateful to the anonymous referees for their valuable comments.</SimplePara></ArticleNote></ArticleHeader><NoBody></NoBody></Article></Issue></Volume></Journal></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Path-following proximal approach for solving Ill-posed convex semi-infinite programming problems</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>Path-following proximal approach for solving Ill-posed convex semi-infinite programming problems</title></titleInfo><name type="personal"><namePart type="given">A.</namePart><namePart type="family">Kaplan</namePart><affiliation>Institute of Applied Mathematics, Humboldt University at Berlin, Berlin, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role><description>Professor</description></name><name type="personal"><namePart type="given">R.</namePart><namePart type="family">Tichatschke</namePart><affiliation>Department of Mathematics, University Trier, Trier, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role><description>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">1996-07-01</dateIssued><copyrightDate encoding="w3cdtf">1996</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: For a class of ill-posed, convex semi-infinite programming problems, a regularized path-following strategy is developed. This approach consists in a coordinated application of adaptive discretization and prox-regularization procedures combined with a penalty method. At each iteration, only an approximate minimum of a strongly convex differentiable function has to be calculated, and this can be done by any fast-convergent algorithm. The use of prox-regularization ensures the convergence of the iterates to some solution of the original problem. Due to regularization, an efficient deleting rule is applicable, which excludes an essential part of the constraints in the discretized problems.</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">1996-07-01</dateIssued><copyrightDate encoding="w3cdtf">1996</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">18</identifier><identifier type="VolumeIssueCount">3</identifier><part><date>1996</date><detail type="volume"><number>90</number><caption>vol.</caption></detail><detail type="issue"><number>1</number><caption>no.</caption></detail><extent unit="pages"><start>113</start><end>137</end></extent></part><recordInfo><recordOrigin>Plenum Publishing Corporation, 1996</recordOrigin></recordInfo></relatedItem><identifier type="istex">FDBD23B97D8F08A8E8907E4DF25D8E0D58B51C3B</identifier><identifier type="DOI">10.1007/BF02192249</identifier><identifier type="ArticleID">BF02192249</identifier><identifier type="ArticleID">Art7</identifier><accessCondition type="use and reproduction" contentType="copyright">Plenum Publishing Corporation, 1996</accessCondition><recordInfo><recordContentSource>SPRINGER</recordContentSource><recordOrigin>Plenum Publishing Corporation, 1996</recordOrigin></recordInfo></mods></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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