A global optimization approach for solving the convex multiplicative programming problem
Identifieur interne : 001144 ( Istex/Corpus ); précédent : 001143; suivant : 001145A global optimization approach for solving the convex multiplicative programming problem
Auteurs : Nguyen Van ThoaiSource :
- Journal of Global Optimization [ 0925-5001 ] ; 1991-12-01.
Abstract
Abstract: We consider a convex multiplicative programming problem of the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGG7bGaam% OzamaaBaaaleaacaaIXaaabeaakiaacIcacaWG4bGaaiykaiabgwSi% xlaadAgadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamiEaiaacMcaca% GG6aGaamiEaiabgIGiolaadIfacaGG9baaaa!4A08!\[\{ f_1 (x) \cdot f_2 (x):x \in X\} \] where X is a compact convex set of ℝ n and f 1, f 2 are convex functions which have nonnegative values over X. Using two additional variables we transform this problem into a problem with a special structure in which the objective function depends only on two of the (n+2) variables. Following a decomposition concept in global optimization we then reduce this problem to a master problem of minimizing a quasi-concave function over a convex set in ℝ2 2. This master problem can be solved by an outer approximation method which requires performing a sequence of simplex tableau pivoting operations. The proposed algorithm is finite when the functions f i, (i=1, 2) are affine-linear and X is a polytope and it is convergent for the general convex case.
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DOI: 10.1007/BF00130830
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Using two additional variables we transform this problem into a problem with a special structure in which the objective function depends only on two of the (n+2) variables. Following a decomposition concept in global optimization we then reduce this problem to a master problem of minimizing a quasi-concave function over a convex set in ℝ2 2. This master problem can be solved by an outer approximation method which requires performing a sequence of simplex tableau pivoting operations. The proposed algorithm is finite when the functions f i, (i=1, 2) are affine-linear and X is a polytope and it is convergent for the general convex case.</div></front></TEI><istex><corpusName>springer</corpusName><author><json:item><name>Nguyen Van Thoai</name><affiliations><json:string>Fachbereich IV-Mathematik, Universität Trier, 5500, Trier, Germany</json:string></affiliations></json:item></author><articleId><json:string>BF00130830</json:string><json:string>Art4</json:string></articleId><language><json:string>eng</json:string></language><originalGenre><json:string>OriginalPaper</json:string></originalGenre><abstract>Abstract: We consider a convex multiplicative programming problem of the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGG7bGaam% OzamaaBaaaleaacaaIXaaabeaakiaacIcacaWG4bGaaiykaiabgwSi% xlaadAgadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamiEaiaacMcaca% GG6aGaamiEaiabgIGiolaadIfacaGG9baaaa!4A08!\[\{ f_1 (x) \cdot f_2 (x):x \in X\} \] where X is a compact convex set of ℝ n and f 1, f 2 are convex functions which have nonnegative values over X. Using two additional variables we transform this problem into a problem with a special structure in which the objective function depends only on two of the (n+2) variables. Following a decomposition concept in global optimization we then reduce this problem to a master problem of minimizing a quasi-concave function over a convex set in ℝ2 2. This master problem can be solved by an outer approximation method which requires performing a sequence of simplex tableau pivoting operations. The proposed algorithm is finite when the functions f i, (i=1, 2) are affine-linear and X is a polytope and it is convergent for the general convex case.</abstract><qualityIndicators><score>6.058</score><pdfVersion>1.3</pdfVersion><pdfPageSize>449.96 x 677 pts</pdfPageSize><refBibsNative>false</refBibsNative><keywordCount>0</keywordCount><abstractCharCount>1328</abstractCharCount><pdfWordCount>4162</pdfWordCount><pdfCharCount>25127</pdfCharCount><pdfPageCount>17</pdfPageCount><abstractWordCount>158</abstractWordCount></qualityIndicators><title>A global optimization approach for solving the convex multiplicative programming problem</title><refBibs><json:item><author><json:item><name> While Solving Problem</name></json:item></author><host><pages><last>70</last><first>62</first></pages><author></author><title>On a Class of Quadratic Programming</title><publicationDate>1984</publicationDate></host><title>method for the calculation of the sets V(Sk) in Rq is required As mentioned in Section 2, the methods discussed in Horst et al. 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Nonconvex Quadratic Problems</title><publicationDate>1988</publicationDate></host></json:item><json:item><author><json:item><name>R,T Rockafellar</name></json:item><json:item><name>N,J Schaible</name></json:item><json:item><name>S </name></json:item></author><host><pages><last>352</last><first>347</first></pages><author></author><title>Convex Analysis Minimization of Ratios</title><publicationDate>1970</publicationDate></host><publicationDate>1970</publicationDate></json:item><json:item><host><pages><last>136</last><first>133</first></pages><author><json:item><name>K Swarup</name></json:item></author><title>Programming with Indefinite Quadratic Function with Linear Constraints</title><publicationDate>1966</publicationDate></host></json:item><json:item><author><json:item><name>H Tuy</name></json:item></author><host><volume>16</volume><pages><last>352</last><first>335</first></pages><author></author><title>Optimization</title><publicationDate>1985</publicationDate></host><title>Concave Minimization under Linear Constraints with Special Structure</title><publicationDate>1985</publicationDate></json:item><json:item><author><json:item><name>H Tuy</name></json:item><json:item><name> Hanoi</name></json:item><json:item><name>H Tuy</name></json:item><json:item><name>B,T Tam</name></json:item></author><host><author></author><title>The Complementary Convex Structure in Global Optimization, Preprint, Institute of Mathematics</title><publicationDate>1990</publicationDate></host><title>An Efficient Solution Method for Rank Two Quasiconcave Minimization Problems</title><publicationDate>1990</publicationDate></json:item></refBibs><genre><json:string>research-article</json:string></genre><host><volume>1</volume><pages><last>357</last><first>341</first></pages><issn><json:string>0925-5001</json:string></issn><issue>4</issue><subject><json:item><value>Computer Science, general</value></json:item><json:item><value>Real 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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>Dordrecht</pubPlace><availability><p>Kluwer Academic Publishers, 1991</p></availability><date>1991</date></publicationStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a" type="main" xml:lang="en">A global optimization approach for solving the convex multiplicative programming problem</title><author xml:id="author-1"><persName><forename type="first">Nguyen</forename><surname>Van Thoai</surname></persName><affiliation>Fachbereich IV-Mathematik, Universität Trier, 5500, Trier, Germany</affiliation></author></analytic><monogr><title level="j">Journal of Global Optimization</title><title level="j" type="sub">An International Journal Dealing with Theoretical and Computational Aspects of Seeking Global Optima and Their Applications in Science, Management, and Engineer</title><title level="j" type="abbrev">J Glob Optim</title><idno type="journal-ID">10898</idno><idno type="pISSN">0925-5001</idno><idno type="eISSN">1573-2916</idno><idno type="issue-article-count">8</idno><idno type="volume-issue-count">4</idno><imprint><publisher>Kluwer Academic Publishers</publisher><pubPlace>Dordrecht</pubPlace><date type="published" when="1991-12-01"></date><biblScope unit="volume">1</biblScope><biblScope unit="issue">4</biblScope><biblScope unit="page" from="341">341</biblScope><biblScope unit="page" to="357">357</biblScope></imprint></monogr><idno type="istex">B37A76B1360E3D61D60A028FE4F5DFC78F766243</idno><idno type="DOI">10.1007/BF00130830</idno><idno type="ArticleID">BF00130830</idno><idno type="ArticleID">Art4</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>1991</date></creation><langUsage><language ident="en">en</language></langUsage><abstract xml:lang="en"><p>Abstract: We consider a convex multiplicative programming problem of the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGG7bGaam% OzamaaBaaaleaacaaIXaaabeaakiaacIcacaWG4bGaaiykaiabgwSi% xlaadAgadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamiEaiaacMcaca% GG6aGaamiEaiabgIGiolaadIfacaGG9baaaa!4A08!\[\{ f_1 (x) \cdot f_2 (x):x \in X\} \] where X is a compact convex set of ℝ n and f 1, f 2 are convex functions which have nonnegative values over X. Using two additional variables we transform this problem into a problem with a special structure in which the objective function depends only on two of the (n+2) variables. Following a decomposition concept in global optimization we then reduce this problem to a master problem of minimizing a quasi-concave function over a convex set in ℝ2 2. This master problem can be solved by an outer approximation method which requires performing a sequence of simplex tableau pivoting operations. The proposed algorithm is finite when the functions f i, (i=1, 2) are affine-linear and X is a polytope and it is convergent for the general convex case.</p></abstract><textClass><keywords scheme="Journal Subject"><list><head>Economics / Management Science</head><item><term>Computer Science, general</term></item><item><term>Real Functions</term></item><item><term>Optimization</term></item><item><term>Operation Research/Decision Theory</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="1991-12-01">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="2017-01-21">References 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Global Optimization</JournalTitle><JournalSubTitle>An International Journal Dealing with Theoretical and Computational Aspects of Seeking Global Optima and Their Applications in Science, Management, and Engineer</JournalSubTitle><JournalAbbreviatedTitle>J Glob Optim</JournalAbbreviatedTitle><JournalSubjectGroup><JournalSubject Type="Primary">Economics / Management Science</JournalSubject><JournalSubject Type="Secondary">Computer Science, general</JournalSubject><JournalSubject Type="Secondary">Real Functions</JournalSubject><JournalSubject Type="Secondary">Optimization</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>4</IssueIDStart><IssueIDEnd>4</IssueIDEnd><IssueArticleCount>8</IssueArticleCount><IssueHistory><CoverDate><DateString>1991</DateString><Year>1991</Year><Month>12</Month></CoverDate></IssueHistory><IssueCopyright><CopyrightHolderName>Kluwer Academic Publishers</CopyrightHolderName><CopyrightYear>1991</CopyrightYear></IssueCopyright></IssueInfo><Article ID="Art4"><ArticleInfo Language="En" ArticleType="OriginalPaper" NumberingStyle="Unnumbered" TocLevels="0" ContainsESM="No"><ArticleID>BF00130830</ArticleID><ArticleDOI>10.1007/BF00130830</ArticleDOI><ArticleSequenceNumber>4</ArticleSequenceNumber><ArticleTitle Language="En">A global optimization approach for solving the convex multiplicative programming problem</ArticleTitle><ArticleFirstPage>341</ArticleFirstPage><ArticleLastPage>357</ArticleLastPage><ArticleHistory><RegistrationDate><Year>2004</Year><Month>5</Month><Day>31</Day></RegistrationDate><Accepted><Year>1991</Year><Month>10</Month><Day>16</Day></Accepted></ArticleHistory><ArticleCopyright><CopyrightHolderName>Kluwer Academic Publishers</CopyrightHolderName><CopyrightYear>1991</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>10898</JournalID><VolumeIDStart>1</VolumeIDStart><VolumeIDEnd>1</VolumeIDEnd><IssueIDStart>4</IssueIDStart><IssueIDEnd>4</IssueIDEnd></ArticleContext></ArticleInfo><ArticleHeader><AuthorGroup><Author AffiliationIDS="Aff1" PresentAffiliationID="Aff2"><AuthorName DisplayOrder="Western"><GivenName>Nguyen</GivenName><Particle>Van</Particle><FamilyName>Thoai</FamilyName></AuthorName></Author><Affiliation ID="Aff1"><OrgDivision>Fachbereich IV-Mathematik</OrgDivision><OrgName>Universität Trier</OrgName><OrgAddress><Postcode>5500</Postcode><City>Trier</City><Country>Germany</Country></OrgAddress></Affiliation><Affiliation ID="Aff2"><OrgName>Institute of Mathematics Hanoi</OrgName><OrgAddress><Country>Vietnam</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>We consider a <Emphasis Type="Italic">convex multiplicative programming problem</Emphasis> of the form</Para><Para>% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGG7bGaam% OzamaaBaaaleaacaaIXaaabeaakiaacIcacaWG4bGaaiykaiabgwSi% xlaadAgadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamiEaiaacMcaca% GG6aGaamiEaiabgIGiolaadIfacaGG9baaaa!4A08!\[\{ f_1 (x) \cdot f_2 (x):x \in X\} \]</Para><Para>where <Emphasis Type="Italic">X</Emphasis> is a compact convex set of ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> and <Emphasis Type="Italic">f</Emphasis><Subscript>1</Subscript>, <Emphasis Type="Italic">f</Emphasis><Subscript>2</Subscript> are convex functions which have nonnegative values over <Emphasis Type="Italic">X</Emphasis>.</Para><Para>Using two additional variables we transform this problem into a problem with a special structure in which the objective function depends only on two of the (<Emphasis Type="Italic">n</Emphasis>+2) variables. Following a decomposition concept in global optimization we then reduce this problem to a <Emphasis Type="Italic">master problem</Emphasis> of minimizing a quasi-concave function over a convex set in ℝ<Superscript>2</Superscript><Subscript>2</Subscript>. This master problem can be solved by an outer approximation method which requires performing a sequence of simplex tableau pivoting operations. The proposed algorithm is finite when the functions <Emphasis Type="Italic">f</Emphasis><Subscript>i</Subscript>, (<Emphasis Type="Italic">i</Emphasis>=1, 2) are affine-linear and <Emphasis Type="Italic">X</Emphasis> is a polytope and it is convergent for the general convex case.</Para></Abstract><KeywordGroup Language="En"><Heading>Key words</Heading><Keyword>Multiplicative programming</Keyword><Keyword>global optimization</Keyword><Keyword>decomposition</Keyword><Keyword>outer approximation</Keyword></KeywordGroup><ArticleNote Type="Misc"><SimplePara>Partly supported by the Deutsche Forschungsgemeinschaft Project CONMIN.</SimplePara></ArticleNote></ArticleHeader><NoBody></NoBody></Article></Issue></Volume></Journal></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>A global optimization approach for solving the convex multiplicative programming problem</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>A global optimization approach for solving the convex multiplicative programming problem</title></titleInfo><name type="personal"><namePart type="given">Nguyen</namePart><namePart type="family">Van Thoai</namePart><affiliation>Fachbereich IV-Mathematik, Universität Trier, 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">Dordrecht</placeTerm></place><dateIssued encoding="w3cdtf">1991-12-01</dateIssued><copyrightDate encoding="w3cdtf">1991</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 convex multiplicative programming problem of the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGG7bGaam% OzamaaBaaaleaacaaIXaaabeaakiaacIcacaWG4bGaaiykaiabgwSi% xlaadAgadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamiEaiaacMcaca% GG6aGaamiEaiabgIGiolaadIfacaGG9baaaa!4A08!\[\{ f_1 (x) \cdot f_2 (x):x \in X\} \] where X is a compact convex set of ℝ n and f 1, f 2 are convex functions which have nonnegative values over X. Using two additional variables we transform this problem into a problem with a special structure in which the objective function depends only on two of the (n+2) variables. Following a decomposition concept in global optimization we then reduce this problem to a master problem of minimizing a quasi-concave function over a convex set in ℝ2 2. This master problem can be solved by an outer approximation method which requires performing a sequence of simplex tableau pivoting operations. The proposed algorithm is finite when the functions f i, (i=1, 2) are affine-linear and X is a polytope and it is convergent for the general convex case.</abstract><relatedItem type="host"><titleInfo><title>Journal of Global Optimization</title><subTitle>An International Journal Dealing with Theoretical and Computational Aspects of Seeking Global Optima and Their Applications in Science, Management, and Engineer</subTitle></titleInfo><titleInfo type="abbreviated"><title>J Glob Optim</title></titleInfo><genre type="journal" displayLabel="Archive Journal"></genre><originInfo><dateIssued encoding="w3cdtf">1991-12-01</dateIssued><copyrightDate encoding="w3cdtf">1991</copyrightDate></originInfo><subject><genre>Economics / Management Science</genre><topic>Computer Science, general</topic><topic>Real Functions</topic><topic>Optimization</topic><topic>Operation Research/Decision Theory</topic></subject><identifier type="ISSN">0925-5001</identifier><identifier type="eISSN">1573-2916</identifier><identifier type="JournalID">10898</identifier><identifier type="IssueArticleCount">8</identifier><identifier type="VolumeIssueCount">4</identifier><part><date>1991</date><detail type="volume"><number>1</number><caption>vol.</caption></detail><detail type="issue"><number>4</number><caption>no.</caption></detail><extent unit="pages"><start>341</start><end>357</end></extent></part><recordInfo><recordOrigin>Kluwer Academic Publishers, 1991</recordOrigin></recordInfo></relatedItem><identifier type="istex">B37A76B1360E3D61D60A028FE4F5DFC78F766243</identifier><identifier type="DOI">10.1007/BF00130830</identifier><identifier type="ArticleID">BF00130830</identifier><identifier type="ArticleID">Art4</identifier><accessCondition type="use and reproduction" contentType="copyright">Kluwer Academic Publishers, 1991</accessCondition><recordInfo><recordContentSource>SPRINGER</recordContentSource><recordOrigin>Kluwer Academic Publishers, 1991</recordOrigin></recordInfo></mods></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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