Numerical solution of a nonlinear parabolic control problem by a reduced SQP method
Identifieur interne : 001728 ( Istex/Corpus ); précédent : 001727; suivant : 001729Numerical solution of a nonlinear parabolic control problem by a reduced SQP method
Auteurs : F. S. Kupfer ; E. W. SachsSource :
- Computational Optimization and Applications [ 0926-6003 ] ; 1992-10-01.
Abstract
Abstract: We consider a control problem for a nonlinear diffusion equation with boundary input that occurs when heating ceramic products in a kiln. We interpret this control problem as a constrained optimization problem, and we develop a reduced SQP method that presents for this problem a new and efficient approach of its numerical solution. As opposed to Newton's method for the unconstrained problem, where at each iteration the state must be computed from a set of nonlinear equations,in the proposed algorithm only the linearized state equations need to be solved. Furthermore, by use of a secant update formula, the calculation of exact second derivatives is avoided. In this way the algorithm achieves a substantial decrease in the total cost compared to the implementation of Newton's method in [2]. Our method is practicable with regard to storage requirements, and by choosing an appropriate representation for the null space of the Jacobian of the constraints we are able to exploit the sparsity pattern of the Jacobian in the course of the iteration. We conclude with a presentation of numerical examples that demonstrate the fast two-step superlinear convergence behavior of the method.
Url:
DOI: 10.1007/BF00247656
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Sachs</name><affiliation><mods:affiliation>Universität Trier FB IV-Mathematik, Postfach 3825, W-5500, Trier, Germany</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:B5A47F1C1CCC4D3E03CD396094C8CF5BB2E44527</idno><date when="1992" year="1992">1992</date><idno type="doi">10.1007/BF00247656</idno><idno type="url">https://api.istex.fr/document/B5A47F1C1CCC4D3E03CD396094C8CF5BB2E44527/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">001728</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">001728</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Numerical solution of a nonlinear parabolic control problem by a reduced SQP method</title><author><name sortKey="Kupfer, F S" sort="Kupfer, F S" uniqKey="Kupfer F" first="F. S." last="Kupfer">F. S. Kupfer</name><affiliation><mods:affiliation>Universität Trier FB IV-Mathematik, Postfach 3825, W-5500, Trier, Germany</mods:affiliation></affiliation></author><author><name sortKey="Sachs, E W" sort="Sachs, E W" uniqKey="Sachs E" first="E. W." last="Sachs">E. W. 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We interpret this control problem as a constrained optimization problem, and we develop a reduced SQP method that presents for this problem a new and efficient approach of its numerical solution. As opposed to Newton's method for the unconstrained problem, where at each iteration the state must be computed from a set of nonlinear equations,in the proposed algorithm only the linearized state equations need to be solved. Furthermore, by use of a secant update formula, the calculation of exact second derivatives is avoided. In this way the algorithm achieves a substantial decrease in the total cost compared to the implementation of Newton's method in [2]. Our method is practicable with regard to storage requirements, and by choosing an appropriate representation for the null space of the Jacobian of the constraints we are able to exploit the sparsity pattern of the Jacobian in the course of the iteration. We conclude with a presentation of numerical examples that demonstrate the fast two-step superlinear convergence behavior of the method.</div></front></TEI><istex><corpusName>springer</corpusName><author><json:item><name>F. S. Kupfer</name><affiliations><json:string>Universität Trier FB IV-Mathematik, Postfach 3825, W-5500, Trier, Germany</json:string></affiliations></json:item><json:item><name>E. W. Sachs</name><affiliations><json:string>Universität Trier FB IV-Mathematik, Postfach 3825, W-5500, Trier, Germany</json:string></affiliations></json:item></author><articleId><json:string>BF00247656</json:string><json:string>Art5</json:string></articleId><language><json:string>eng</json:string></language><originalGenre><json:string>OriginalPaper</json:string></originalGenre><abstract>Abstract: We consider a control problem for a nonlinear diffusion equation with boundary input that occurs when heating ceramic products in a kiln. We interpret this control problem as a constrained optimization problem, and we develop a reduced SQP method that presents for this problem a new and efficient approach of its numerical solution. As opposed to Newton's method for the unconstrained problem, where at each iteration the state must be computed from a set of nonlinear equations,in the proposed algorithm only the linearized state equations need to be solved. Furthermore, by use of a secant update formula, the calculation of exact second derivatives is avoided. In this way the algorithm achieves a substantial decrease in the total cost compared to the implementation of Newton's method in [2]. Our method is practicable with regard to storage requirements, and by choosing an appropriate representation for the null space of the Jacobian of the constraints we are able to exploit the sparsity pattern of the Jacobian in the course of the iteration. We conclude with a presentation of numerical examples that demonstrate the fast two-step superlinear convergence behavior of the method.</abstract><qualityIndicators><score>7.268</score><pdfVersion>1.3</pdfVersion><pdfPageSize>453.56 x 684.28 pts</pdfPageSize><refBibsNative>false</refBibsNative><keywordCount>0</keywordCount><abstractCharCount>1200</abstractCharCount><pdfWordCount>5953</pdfWordCount><pdfCharCount>35118</pdfCharCount><pdfPageCount>23</pdfPageCount><abstractWordCount>189</abstractWordCount></qualityIndicators><title>Numerical solution of a nonlinear parabolic control problem by a reduced SQP method</title><refBibs><json:item><author><json:item><name>J Burger</name></json:item><json:item><name>C Machbub</name></json:item></author><host><volume>7</volume><pages><last>14</last><first>1</first></pages><author></author><title>Commun. Appl. Numer. Methods</title><publicationDate>1991</publicationDate></host><title>Comparison of numerical solutions of a one-dimensional nonlinear heat equation</title><publicationDate>1991</publicationDate></json:item><json:item><author><json:item><name>J Burger</name></json:item><json:item><name>M Pogu</name></json:item></author><host><volume>68</volume><pages><last>73</last><first>49</first></pages><author></author><title>J. Optim. Theory Appl</title><publicationDate>1991</publicationDate></host><title>Functional and numerical solution of a control problem originating from heat transfer</title><publicationDate>1991</publicationDate></json:item><json:item><author><json:item><name>R,H Byrd</name></json:item></author><host><volume>32</volume><pages><last>237</last><first>232</first></pages><author></author><title>Math. Programming</title><publicationDate>1985</publicationDate></host><title>An example of irregular convergence in some constrained optimization methods that use the projected Hessian</title><publicationDate>1985</publicationDate></json:item><json:item><author><json:item><name>R,H Byrd</name></json:item></author><host><volume>27</volume><pages><last>153</last><first>141</first></pages><author></author><title>SIAM J. Numer Anal</title><publicationDate>1990</publicationDate></host><title>On the convergence of constmined optimization methods with accurate Hessian information on a subspace</title><publicationDate>1990</publicationDate></json:item><json:item><author><json:item><name>R,H Byrd</name></json:item><json:item><name>J Nocedal</name></json:item></author><host><volume>49</volume><pages><last>323</last><first>285</first></pages><author></author><title>Math. Programming</title><publicationDate>1991</publicationDate></host><title>An analysis of reduced Hessian methods for constrained optimization</title><publicationDate>1991</publicationDate></json:item><json:item><author><json:item><name>Tf Coleman</name></json:item><json:item><name>A,R Conn</name></json:item></author><host><volume>21</volume><pages><last>769</last><first>755</first></pages><author></author><title>SIAM J. Numer. Anal</title><publicationDate>1984</publicationDate></host><title>On the local convergence of a quasi-Newton method for the nonlinear programming problem</title><publicationDate>1984</publicationDate></json:item><json:item><host><author><json:item><name>J,E Dennis</name></json:item><json:item><name>R,B </name></json:item></author><title>Numerical Methods for Unconstrained Optimization and Nonlinear Equations</title><publicationDate>1983</publicationDate></host></json:item><json:item><author><json:item><name>D Gabay</name></json:item></author><host><volume>16</volume><pages><last>44</last><first>18</first></pages><author></author><title>Math. Programming Study</title><publicationDate>1982</publicationDate></host><title>Reduced quasi-Newton methods with feasibility improvement for nonlinearly constrained optimization</title><publicationDate>1982</publicationDate></json:item><json:item><author><json:item><name>J,C Gilbert</name></json:item></author><host><pages><last>53</last><first>40</first></pages><author></author><title>Proc. of the Seventh Int. Conf on Analysis and Optimization of Systems</title><publicationDate>1986</publicationDate></host><title>Une methode de quasi-Newton reduite en optimisation sous contraintes avec priorite a la restaumtion</title><publicationDate>1986</publicationDate></json:item><json:item><author><json:item><name>J,C Gilbert</name></json:item></author><host><volume>20</volume><pages><last>450</last><first>421</first></pages><author></author><title>Optimization</title><publicationDate>1989</publicationDate></host><title>On the local and global convergence of a reduced quasi-Newton method</title><publicationDate>1989</publicationDate></json:item><json:item><host><author><json:item><name>C,B Gunvitz</name></json:item></author><title>Local convergence of a two-piece update of a projected Hessian matrix</title><publicationDate>1991</publicationDate></host></json:item><json:item><author><json:item><name>D,H Jacobson</name></json:item><json:item><name>D,Q Mayne</name></json:item></author><host><author></author><title>Differential Dynamic Programming</title><publicationDate>1970</publicationDate></host><publicationDate>1970</publicationDate></json:item><json:item><author><json:item><name>K Kunisch</name></json:item><json:item><name>E,W Sachs</name></json:item></author><host><author></author><title>SIAM J. Numer. Anal</title></host><title>Reduced SQP methods for parameter identification problems</title></json:item><json:item><author><json:item><name>F-S Kupfer</name></json:item></author><host><author></author><title>Fachbereich IV -Mathematick, tech. rep</title><publicationDate>1990</publicationDate></host><title>An infinite-dimensional convergence theory for reduced SQP methods in Hilbert space</title><publicationDate>1990</publicationDate></json:item><json:item><host><author><json:item><name>F-S Kupfer</name></json:item></author><title>Reduced successive quadratic programming in Hilbert space with applications to optimal control, doctoral thesis</title><publicationDate>1992</publicationDate></host></json:item><json:item><author><json:item><name>F-S Kupfer</name></json:item><json:item><name>E,W Sachs</name></json:item></author><host><author></author><title>Optimal Control of Partial Differential Equations</title><publicationDate>1990</publicationDate></host><title>A prospective look at SQP methods for semilinear pambolic control problems</title><publicationDate>1990</publicationDate></json:item><json:item><author><json:item><name>Eds Krabs</name></json:item></author><host><volume>149</volume><pages><last>157</last><first>143</first></pages><author></author><title>Springer Lect. Notes in Control and Information Sciences</title><publicationDate>1991</publicationDate></host><publicationDate>1991</publicationDate></json:item><json:item><author><json:item><name>J Nocedal</name></json:item><json:item><name>M,L Overton</name></json:item></author><host><volume>22</volume><pages><last>850</last><first>821</first></pages><author></author><title>SIAM J. Numer. Anal</title><publicationDate>1985</publicationDate></host><title>Projected Hessian updating algorithms for nonlinearly constrained optimization</title><publicationDate>1985</publicationDate></json:item><json:item><author><json:item><name>Y Yuan</name></json:item></author><host><volume>32</volume><pages><last>231</last><first>224</first></pages><author></author><title>Math. Programming</title><publicationDate>1985</publicationDate></host><title>An only 2-step q-superlinear convergence example for some algorithms that use reduced Hessian approximations</title><publicationDate>1985</publicationDate></json:item><json:item><author><json:item><name>J Zhang</name></json:item><json:item><name>D Zhu</name></json:item></author><host><volume>21</volume><pages><last>557</last><first>543</first></pages><author></author><title>Optimization</title><publicationDate>1990</publicationDate></host><title>A trust region type dogleg method for nonlinear optimization</title><publicationDate>1990</publicationDate></json:item></refBibs><genre><json:string>research-article</json:string></genre><host><volume>1</volume><pages><last>135</last><first>113</first></pages><issn><json:string>0926-6003</json:string></issn><issue>1</issue><subject><json:item><value>Convex and Discrete Geometry</value></json:item><json:item><value>Optimization</value></json:item><json:item><value>Operations 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récupérées via 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</publisher><pubPlace>Boston</pubPlace><availability><p>Kluwer Academic Publishers, 1992</p></availability><date>1992-01-28</date></publicationStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a" type="main" xml:lang="en">Numerical solution of a nonlinear parabolic control problem by a reduced SQP method</title><author xml:id="author-1"><persName><forename type="first">F.</forename><surname>Kupfer</surname></persName><affiliation>Universität Trier FB IV-Mathematik, Postfach 3825, W-5500, Trier, Germany</affiliation></author><author xml:id="author-2"><persName><forename type="first">E.</forename><surname>Sachs</surname></persName><affiliation>Universität Trier FB IV-Mathematik, Postfach 3825, W-5500, Trier, Germany</affiliation></author></analytic><monogr><title level="j">Computational Optimization and Applications</title><title level="j" type="abbrev">Comput Optim Applic</title><idno type="journal-ID">10589</idno><idno type="pISSN">0926-6003</idno><idno type="eISSN">1573-2894</idno><idno type="issue-article-count">5</idno><idno type="volume-issue-count">4</idno><imprint><publisher>Kluwer Academic Publishers</publisher><pubPlace>Boston</pubPlace><date type="published" when="1992-10-01"></date><biblScope unit="volume">1</biblScope><biblScope unit="issue">1</biblScope><biblScope unit="page" from="113">113</biblScope><biblScope unit="page" to="135">135</biblScope></imprint></monogr><idno type="istex">B5A47F1C1CCC4D3E03CD396094C8CF5BB2E44527</idno><idno type="DOI">10.1007/BF00247656</idno><idno type="ArticleID">BF00247656</idno><idno type="ArticleID">Art5</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>1992-01-28</date></creation><langUsage><language ident="en">en</language></langUsage><abstract xml:lang="en"><p>Abstract: We consider a control problem for a nonlinear diffusion equation with boundary input that occurs when heating ceramic products in a kiln. We interpret this control problem as a constrained optimization problem, and we develop a reduced SQP method that presents for this problem a new and efficient approach of its numerical solution. As opposed to Newton's method for the unconstrained problem, where at each iteration the state must be computed from a set of nonlinear equations,in the proposed algorithm only the linearized state equations need to be solved. Furthermore, by use of a secant update formula, the calculation of exact second derivatives is avoided. In this way the algorithm achieves a substantial decrease in the total cost compared to the implementation of Newton's method in [2]. Our method is practicable with regard to storage requirements, and by choosing an appropriate representation for the null space of the Jacobian of the constraints we are able to exploit the sparsity pattern of the Jacobian in the course of the iteration. We conclude with a presentation of numerical examples that demonstrate the fast two-step superlinear convergence behavior of the method.</p></abstract><textClass><keywords scheme="Journal Subject"><list><head>Mathematics</head><item><term>Convex and Discrete Geometry</term></item><item><term>Optimization</term></item><item><term>Operations Research, Mathematical Programming</term></item><item><term>Statistics, general</term></item><item><term>Operation Research/Decision Theory</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="1992-01-28">Created</change><change when="1992-10-01">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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Optimization and Applications</JournalTitle><JournalAbbreviatedTitle>Comput Optim Applic</JournalAbbreviatedTitle><JournalSubjectGroup><JournalSubject Type="Primary">Mathematics</JournalSubject><JournalSubject Type="Secondary">Convex and Discrete Geometry</JournalSubject><JournalSubject Type="Secondary">Optimization</JournalSubject><JournalSubject Type="Secondary">Operations Research, Mathematical Programming</JournalSubject><JournalSubject Type="Secondary">Statistics, general</JournalSubject><JournalSubject Type="Secondary">Operation Research/Decision Theory</JournalSubject></JournalSubjectGroup></JournalInfo><Volume><VolumeInfo VolumeType="Regular" TocLevels="0"><VolumeIDStart>1</VolumeIDStart><VolumeIDEnd>1</VolumeIDEnd><VolumeIssueCount>4</VolumeIssueCount></VolumeInfo><Issue IssueType="Regular"><IssueInfo TocLevels="0"><IssueIDStart>1</IssueIDStart><IssueIDEnd>1</IssueIDEnd><IssueArticleCount>5</IssueArticleCount><IssueHistory><CoverDate><Year>1992</Year><Month>10</Month></CoverDate></IssueHistory><IssueCopyright><CopyrightHolderName>Kluwer Academic Publishers</CopyrightHolderName><CopyrightYear>1992</CopyrightYear></IssueCopyright></IssueInfo><Article ID="Art5"><ArticleInfo Language="En" ArticleType="OriginalPaper" NumberingStyle="Unnumbered" TocLevels="0" ContainsESM="No"><ArticleID>BF00247656</ArticleID><ArticleDOI>10.1007/BF00247656</ArticleDOI><ArticleSequenceNumber>5</ArticleSequenceNumber><ArticleTitle Language="En">Numerical solution of a nonlinear parabolic control problem by a reduced SQP method</ArticleTitle><ArticleFirstPage>113</ArticleFirstPage><ArticleLastPage>135</ArticleLastPage><ArticleHistory><RegistrationDate><Year>2004</Year><Month>8</Month><Day>3</Day></RegistrationDate><Received><Year>1992</Year><Month>1</Month><Day>28</Day></Received><Revised><Year>1992</Year><Month>5</Month><Day>26</Day></Revised></ArticleHistory><ArticleCopyright><CopyrightHolderName>Kluwer Academic Publishers</CopyrightHolderName><CopyrightYear>1992</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>10589</JournalID><VolumeIDStart>1</VolumeIDStart><VolumeIDEnd>1</VolumeIDEnd><IssueIDStart>1</IssueIDStart><IssueIDEnd>1</IssueIDEnd></ArticleContext></ArticleInfo><ArticleHeader><AuthorGroup><Author AffiliationIDS="Aff1"><AuthorName DisplayOrder="Western"><GivenName>F.</GivenName><GivenName>S.</GivenName><FamilyName>Kupfer</FamilyName></AuthorName></Author><Author AffiliationIDS="Aff1"><AuthorName DisplayOrder="Western"><GivenName>E.</GivenName><GivenName>W.</GivenName><FamilyName>Sachs</FamilyName></AuthorName></Author><Affiliation ID="Aff1"><OrgName>Universität Trier FB IV-Mathematik</OrgName><OrgAddress><Postbox>Postfach 3825</Postbox><Postcode>W-5500</Postcode><City>Trier</City><Country>Germany</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>We consider a control problem for a nonlinear diffusion equation with boundary input that occurs when heating ceramic products in a kiln. We interpret this control problem as a constrained optimization problem, and we develop a reduced SQP method that presents for this problem a new and efficient approach of its numerical solution. As opposed to Newton's method for the unconstrained problem, where at each iteration the state must be computed from a set of nonlinear equations,in the proposed algorithm only the linearized state equations need to be solved. Furthermore, by use of a secant update formula, the calculation of exact second derivatives is avoided. In this way the algorithm achieves a substantial decrease in the total cost compared to the implementation of Newton's method in [2]. Our method is practicable with regard to storage requirements, and by choosing an appropriate representation for the null space of the Jacobian of the constraints we are able to exploit the sparsity pattern of the Jacobian in the course of the iteration. We conclude with a presentation of numerical examples that demonstrate the fast two-step superlinear convergence behavior of the method.</Para></Abstract><KeywordGroup Language="En"><Heading>Keywords</Heading><Keyword>optimal boundary control</Keyword><Keyword>nonlinear heat equation</Keyword><Keyword>reduced successive quadratic programming (SQP)</Keyword><Keyword>constrained optimization</Keyword><Keyword>BFGS-update</Keyword><Keyword>null space parametrization</Keyword><Keyword>two-step superlinear convergence</Keyword></KeywordGroup></ArticleHeader><NoBody></NoBody></Article></Issue></Volume></Journal></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Numerical solution of a nonlinear parabolic control problem by a reduced SQP method</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>Numerical solution of a nonlinear parabolic control problem by a reduced SQP method</title></titleInfo><name type="personal"><namePart type="given">F.</namePart><namePart type="given">S.</namePart><namePart type="family">Kupfer</namePart><affiliation>Universität Trier FB IV-Mathematik, Postfach 3825, W-5500, Trier, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">E.</namePart><namePart type="given">W.</namePart><namePart type="family">Sachs</namePart><affiliation>Universität Trier FB IV-Mathematik, Postfach 3825, W-5500, Trier, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="OriginalPaper"></genre><originInfo><publisher>Kluwer Academic Publishers</publisher><place><placeTerm type="text">Boston</placeTerm></place><dateCreated encoding="w3cdtf">1992-01-28</dateCreated><dateIssued encoding="w3cdtf">1992-10-01</dateIssued><copyrightDate encoding="w3cdtf">1992</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: We consider a control problem for a nonlinear diffusion equation with boundary input that occurs when heating ceramic products in a kiln. We interpret this control problem as a constrained optimization problem, and we develop a reduced SQP method that presents for this problem a new and efficient approach of its numerical solution. As opposed to Newton's method for the unconstrained problem, where at each iteration the state must be computed from a set of nonlinear equations,in the proposed algorithm only the linearized state equations need to be solved. Furthermore, by use of a secant update formula, the calculation of exact second derivatives is avoided. In this way the algorithm achieves a substantial decrease in the total cost compared to the implementation of Newton's method in [2]. Our method is practicable with regard to storage requirements, and by choosing an appropriate representation for the null space of the Jacobian of the constraints we are able to exploit the sparsity pattern of the Jacobian in the course of the iteration. We conclude with a presentation of numerical examples that demonstrate the fast two-step superlinear convergence behavior of the method.</abstract><relatedItem type="host"><titleInfo><title>Computational Optimization and Applications</title></titleInfo><titleInfo type="abbreviated"><title>Comput Optim Applic</title></titleInfo><genre type="journal" displayLabel="Archive Journal"></genre><originInfo><dateIssued encoding="w3cdtf">1992-10-01</dateIssued><copyrightDate encoding="w3cdtf">1992</copyrightDate></originInfo><subject><genre>Mathematics</genre><topic>Convex and Discrete Geometry</topic><topic>Optimization</topic><topic>Operations Research, Mathematical Programming</topic><topic>Statistics, general</topic><topic>Operation Research/Decision Theory</topic></subject><identifier type="ISSN">0926-6003</identifier><identifier type="eISSN">1573-2894</identifier><identifier type="JournalID">10589</identifier><identifier type="IssueArticleCount">5</identifier><identifier type="VolumeIssueCount">4</identifier><part><date>1992</date><detail type="volume"><number>1</number><caption>vol.</caption></detail><detail type="issue"><number>1</number><caption>no.</caption></detail><extent unit="pages"><start>113</start><end>135</end></extent></part><recordInfo><recordOrigin>Kluwer Academic Publishers, 1992</recordOrigin></recordInfo></relatedItem><identifier type="istex">B5A47F1C1CCC4D3E03CD396094C8CF5BB2E44527</identifier><identifier type="DOI">10.1007/BF00247656</identifier><identifier type="ArticleID">BF00247656</identifier><identifier type="ArticleID">Art5</identifier><accessCondition type="use and reproduction" contentType="copyright">Kluwer Academic Publishers, 1992</accessCondition><recordInfo><recordContentSource>SPRINGER</recordContentSource><recordOrigin>Kluwer Academic Publishers, 1992</recordOrigin></recordInfo></mods></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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