Pointwise quasi-Newton method for unconstrained optimal control problems, II
Identifieur interne : 000A85 ( Istex/Corpus ); précédent : 000A84; suivant : 000A86Pointwise quasi-Newton method for unconstrained optimal control problems, II
Auteurs : C. T. Kelley ; E. W. Sachs ; B. WatsonSource :
- Journal of Optimization Theory and Applications [ 0022-3239 ] ; 1991-12-01.
Abstract
Abstract: In this paper, the necessary optimality conditions for an unconstrained optimal control problem are used to derive a quasi-Newton method where the update involves only second-order derivative terms. A pointwise update which was presented in a previous paper by the authors is changed to allow for more general second-order sufficiency conditions in the control problem. In particular, pointwise versions of the Broyden, PSB, and SR1 update are considered. A convergence rate theorem is given for the Broyden and PSB versions.
Url:
DOI: 10.1007/BF00941402
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Watson</name><affiliation><mods:affiliation>Fachbereich IV—Mathematik, Universität Trier, Trier, Germany</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:DD441FCEBE9ACC8A7B0E1814C140676D1600C88C</idno><date when="1991" year="1991">1991</date><idno type="doi">10.1007/BF00941402</idno><idno type="url">https://api.istex.fr/document/DD441FCEBE9ACC8A7B0E1814C140676D1600C88C/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">000A85</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">000A85</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Pointwise quasi-Newton method for unconstrained optimal control problems, II</title><author><name sortKey="Kelley, C T" sort="Kelley, C T" uniqKey="Kelley C" first="C. T." last="Kelley">C. T. Kelley</name><affiliation><mods:affiliation>Department of Mathematics, North Carolina State University, Raleigh, North Carolina</mods:affiliation></affiliation></author><author><name sortKey="Sachs, E W" sort="Sachs, E W" uniqKey="Sachs E" first="E. W." last="Sachs">E. W. Sachs</name><affiliation><mods:affiliation>Fachbereich IV—Mathematik, Universität Trier, Trier, Germany</mods:affiliation></affiliation></author><author><name sortKey="Watson, B" sort="Watson, B" uniqKey="Watson B" first="B." last="Watson">B. Watson</name><affiliation><mods:affiliation>Fachbereich IV—Mathematik, Universität Trier, Trier, Germany</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">Journal of Optimization Theory and Applications</title><title level="j" type="abbrev">J Optim Theory Appl</title><idno type="ISSN">0022-3239</idno><idno type="eISSN">1573-2878</idno><imprint><publisher>Kluwer Academic Publishers-Plenum Publishers</publisher><pubPlace>New York</pubPlace><date type="published" when="1991-12-01">1991-12-01</date><biblScope unit="volume">71</biblScope><biblScope unit="issue">3</biblScope><biblScope unit="page" from="535">535</biblScope><biblScope unit="page" to="547">547</biblScope></imprint><idno type="ISSN">0022-3239</idno></series><idno type="istex">DD441FCEBE9ACC8A7B0E1814C140676D1600C88C</idno><idno type="DOI">10.1007/BF00941402</idno><idno type="ArticleID">BF00941402</idno><idno type="ArticleID">Art6</idno></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0022-3239</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: In this paper, the necessary optimality conditions for an unconstrained optimal control problem are used to derive a quasi-Newton method where the update involves only second-order derivative terms. 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Watson</name><affiliations><json:string>Fachbereich IV—Mathematik, Universität Trier, Trier, Germany</json:string></affiliations></json:item></author><articleId><json:string>BF00941402</json:string><json:string>Art6</json:string></articleId><language><json:string>eng</json:string></language><originalGenre><json:string>OriginalPaper</json:string></originalGenre><abstract>Abstract: In this paper, the necessary optimality conditions for an unconstrained optimal control problem are used to derive a quasi-Newton method where the update involves only second-order derivative terms. A pointwise update which was presented in a previous paper by the authors is changed to allow for more general second-order sufficiency conditions in the control problem. In particular, pointwise versions of the Broyden, PSB, and SR1 update are considered. A convergence rate theorem is given for the Broyden and PSB versions.</abstract><qualityIndicators><score>4.38</score><pdfVersion>1.3</pdfVersion><pdfPageSize>432 x 648 pts</pdfPageSize><refBibsNative>false</refBibsNative><keywordCount>0</keywordCount><abstractCharCount>535</abstractCharCount><pdfWordCount>3408</pdfWordCount><pdfCharCount>16766</pdfCharCount><pdfPageCount>13</pdfPageCount><abstractWordCount>81</abstractWordCount></qualityIndicators><title>Pointwise quasi-Newton method for unconstrained optimal control problems, II</title><refBibs><json:item><author><json:item><name>Kelley </name></json:item><json:item><name>C,T </name></json:item><json:item><name>Sachs </name></json:item><json:item><name>E,W </name></json:item></author><host><volume>55</volume><pages><last>176</last><first>159</first></pages><author></author><title>Numerische Mathematik</title><publicationDate>1989</publicationDate></host><title>A Pointwise Quasi-Newton Method for Unconstrained Optimal Control Problems</title><publicationDate>1989</publicationDate></json:item><json:item><author><json:item><name>Kelley </name></json:item><json:item><name>C,T </name></json:item><json:item><name>Sachs </name></json:item><json:item><name>E,W </name></json:item></author><host><volume>25</volume><pages><last>1517</last><first>1503</first></pages><author></author><title>SIAM Journal on Control and Optimization</title><publicationDate>1987</publicationDate></host><title>Quasi-Newton Methods and Unconstrained Optimal Control Problems</title><publicationDate>1987</publicationDate></json:item><json:item><author><json:item><name>Toint </name></json:item><json:item><name>P,L </name></json:item></author><host><pages><last>435</last><first>416</first></pages><issue>8</issue><author></author><title>SIAM Journal on Scientific and Statistical Computing</title><publicationDate>1987</publicationDate></host><title>On Large-Scale Nonlinear Least-Squares Calculations</title><publicationDate>1987</publicationDate></json:item><json:item><author><json:item><name>Lee </name></json:item><json:item><name>E,B </name></json:item><json:item><name>Markus </name></json:item><json:item><name>L </name></json:item></author><host><author></author><title>Foundations of Optimal Control Theory</title><publicationDate>1967</publicationDate></host><publicationDate>1967</publicationDate></json:item><json:item><author><json:item><name>Maurer </name></json:item><json:item><name>H </name></json:item></author><host><volume>14</volume><pages><last>77</last><first>63</first></pages><author></author><title>Mathematical Programming Study</title><publicationDate>1981</publicationDate></host><title>First-Order and Second-Order Sufficient Optimality Conditions in Mathematical Programming and Optimal Control</title><publicationDate>1981</publicationDate></json:item><json:item><host><author><json:item><name>Dennis </name></json:item><json:item><name>J,E </name></json:item><json:item><name>Schnabel </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>Keller </name></json:item><json:item><name>H,B </name></json:item></author><host><author></author><title>CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM</title><publicationDate>1982</publicationDate></host><title>Numerical Solutions of Two-Point Boundary-Value Problems</title><publicationDate>1982</publicationDate></json:item><json:item><host><author><json:item><name>Watson </name></json:item><json:item><name>B Quasi-Newton Methoden</name></json:item><json:item><name>J~r Minimierungsprobteme Mit Strukturierter</name></json:item><json:item><name> Hesse-Matrix</name></json:item></author></host></json:item><json:item><host><author><json:item><name>Osborne </name></json:item><json:item><name>M,R Sun</name></json:item><json:item><name>L,P </name></json:item></author><title>A New Approach to the Symmetric Rank-One Updating Algorithm</title><publicationDate>1988</publicationDate></host></json:item><json:item><author><json:item><name>Conn </name></json:item><json:item><name>A,R Gould</name></json:item><json:item><name>N,I M </name></json:item><json:item><name>Toint </name></json:item><json:item><name>P,L </name></json:item></author><host><pages><last>195</last><first>177</first></pages><author></author><title>Convergence of Quasi- Newton Matrices Generated by the Symmetric Rank-One Update</title><publicationDate>1991</publicationDate></host><publicationDate>1991</publicationDate></json:item></refBibs><genre><json:string>research-article</json:string></genre><host><volume>71</volume><pages><last>547</last><first>535</first></pages><issn><json:string>0022-3239</json:string></issn><issue>3</issue><subject><json:item><value>Theory of 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Papers</ArticleCategory><ArticleFirstPage>535</ArticleFirstPage><ArticleLastPage>547</ArticleLastPage><ArticleHistory><RegistrationDate><Year>2004</Year><Month>12</Month><Day>21</Day></RegistrationDate></ArticleHistory><ArticleCopyright><CopyrightHolderName>Plenum Publishing Corporation</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>10957</JournalID><VolumeIDStart>71</VolumeIDStart><VolumeIDEnd>71</VolumeIDEnd><IssueIDStart>3</IssueIDStart><IssueIDEnd>3</IssueIDEnd></ArticleContext></ArticleInfo><ArticleHeader><AuthorGroup><Author AffiliationIDS="Aff1"><AuthorName DisplayOrder="Western"><GivenName>C.</GivenName><GivenName>T.</GivenName><FamilyName>Kelley</FamilyName></AuthorName><Role>Professor</Role></Author><Author AffiliationIDS="Aff2"><AuthorName DisplayOrder="Western"><GivenName>E.</GivenName><GivenName>W.</GivenName><FamilyName>Sachs</FamilyName></AuthorName><Role>Professor</Role></Author><Author AffiliationIDS="Aff2" PresentAffiliationID="Aff3"><AuthorName DisplayOrder="Western"><GivenName>B.</GivenName><FamilyName>Watson</FamilyName></AuthorName><Role>Graduate Student</Role></Author><Affiliation ID="Aff1"><OrgDivision>Department of Mathematics</OrgDivision><OrgName>North Carolina State University</OrgName><OrgAddress><City>Raleigh</City><State>North Carolina</State></OrgAddress></Affiliation><Affiliation ID="Aff2"><OrgDivision>Fachbereich IV—Mathematik</OrgDivision><OrgName>Universität Trier</OrgName><OrgAddress><City>Trier</City><Country>Germany</Country></OrgAddress></Affiliation><Affiliation ID="Aff3"><OrgName>IBM Luxembourg</OrgName><OrgAddress><City>Hesperange</City><Country>Luxembourg</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>In this paper, the necessary optimality conditions for an unconstrained optimal control problem are used to derive a quasi-Newton method where the update involves only second-order derivative terms. A pointwise update which was presented in a previous paper by the authors is changed to allow for more general second-order sufficiency conditions in the control problem. In particular, pointwise versions of the Broyden, PSB, and SR1 update are considered. A convergence rate theorem is given for the Broyden and PSB versions.</Para></Abstract><KeywordGroup Language="En"><Heading>Key Words</Heading><Keyword>Quasi-Newton methods</Keyword><Keyword>optimal control</Keyword></KeywordGroup><ArticleNote Type="CommunicatedBy"><SimplePara>Communicated by R. A. Tapia</SimplePara></ArticleNote><ArticleNote Type="Misc"><SimplePara>This research was supported by NSF Grant No. DMS-89-00410, by NSF Grant No. INT-88-00560, by AFOSR Grant No. AFOSR-89-0044, and by the Deutsche Forschungsgemeinschaft.</SimplePara></ArticleNote></ArticleHeader><NoBody></NoBody></Article></Issue></Volume></Journal></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Pointwise quasi-Newton method for unconstrained optimal control problems, II</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>Pointwise quasi-Newton method for unconstrained optimal control problems, II</title></titleInfo><name type="personal"><namePart type="given">C.</namePart><namePart type="given">T.</namePart><namePart type="family">Kelley</namePart><affiliation>Department of Mathematics, North Carolina State University, Raleigh, North Carolina</affiliation><role><roleTerm type="text">author</roleTerm></role><description>Professor</description></name><name type="personal"><namePart type="given">E.</namePart><namePart type="given">W.</namePart><namePart type="family">Sachs</namePart><affiliation>Fachbereich IV—Mathematik, Universität Trier, Trier, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role><description>Professor</description></name><name type="personal"><namePart type="given">B.</namePart><namePart type="family">Watson</namePart><affiliation>Fachbereich IV—Mathematik, Universität Trier, Trier, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role><description>Graduate Student</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">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: In this paper, the necessary optimality conditions for an unconstrained optimal control problem are used to derive a quasi-Newton method where the update involves only second-order derivative terms. A pointwise update which was presented in a previous paper by the authors is changed to allow for more general second-order sufficiency conditions in the control problem. In particular, pointwise versions of the Broyden, PSB, and SR1 update are considered. 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