Linearly constrained global minimization of functions with concave minorants
Identifieur interne : 000B15 ( Istex/Corpus ); précédent : 000B14; suivant : 000B16Linearly constrained global minimization of functions with concave minorants
Auteurs : R. Horst ; M. NastSource :
- Journal of Optimization Theory and Applications [ 0022-3239 ] ; 1996-03-01.
Abstract
Abstract: In this note, we show how a recent approach for solving linearly constrained multivariate Lipschitz optimization problems and corresponding systems of inequalities can be generalized to solve optimization problems where the objective function is only assumed to possess a concave minorant at each point. This class of functions includes not only Lipschitz functions and some generalizations, such as certain ρ-convex functions and Hölder functions with exponent greater than one, but also all functions which can be expressed as differences of two convex functions (d.c. functions). Thus, in particular, a new approach is obtained for the important problem of minimizing a d.c. function over a polytope.
Url:
DOI: 10.1007/BF02192209
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Nast</name><affiliation><mods:affiliation>Fachbereich IV-Department of Mathematics, University of Trier, Trier, Germany</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:246DE1AED33EFDBBC73D8365008B2FD803392391</idno><date when="1996" year="1996">1996</date><idno type="doi">10.1007/BF02192209</idno><idno type="url">https://api.istex.fr/document/246DE1AED33EFDBBC73D8365008B2FD803392391/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">000B15</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">000B15</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Linearly constrained global minimization of functions with concave minorants</title><author><name sortKey="Horst, R" sort="Horst, R" uniqKey="Horst R" first="R." last="Horst">R. Horst</name><affiliation><mods:affiliation>Fachbereich IV-Department of Mathematics, University of Trier, Trier, Germany</mods:affiliation></affiliation></author><author><name sortKey="Nast, M" sort="Nast, M" uniqKey="Nast M" first="M." last="Nast">M. Nast</name><affiliation><mods:affiliation>Fachbereich IV-Department of Mathematics, University of 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="1996-03-01">1996-03-01</date><biblScope unit="volume">88</biblScope><biblScope unit="issue">3</biblScope><biblScope unit="page" from="751">751</biblScope><biblScope unit="page" to="763">763</biblScope></imprint><idno type="ISSN">0022-3239</idno></series><idno type="istex">246DE1AED33EFDBBC73D8365008B2FD803392391</idno><idno type="DOI">10.1007/BF02192209</idno><idno type="ArticleID">BF02192209</idno><idno type="ArticleID">Art13</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 note, we show how a recent approach for solving linearly constrained multivariate Lipschitz optimization problems and corresponding systems of inequalities can be generalized to solve optimization problems where the objective function is only assumed to possess a concave minorant at each point. This class of functions includes not only Lipschitz functions and some generalizations, such as certain ρ-convex functions and Hölder functions with exponent greater than one, but also all functions which can be expressed as differences of two convex functions (d.c. functions). Thus, in particular, a new approach is obtained for the important problem of minimizing a d.c. function over a polytope.</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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Optimization</title><publicationDate>1995</publicationDate></host></json:item><json:item><host><author><json:item><name>Horst </name></json:item><json:item><name>R Tuv</name></json:item><json:item><name>H </name></json:item></author><title>Global Optimization: Deterministic Approaches</title><publicationDate>1993</publicationDate></host></json:item><json:item><author><json:item><name>Horst </name></json:item><json:item><name>R </name></json:item><json:item><name>Pardalos </name></json:item><json:item><name>P,M </name></json:item></author><host><author></author><title>Handbook of Global Optimization</title><publicationDate>1995</publicationDate></host><publicationDate>1995</publicationDate></json:item><json:item><host><author><json:item><name>Khamisov </name></json:item><json:item><name>O </name></json:item></author><title>Functions with Concave Minorants : A General View</title><publicationDate>1994</publicationDate></host></json:item><json:item><author><json:item><name>Horst </name></json:item><json:item><name>R Nast</name></json:item><json:item><name>M </name></json:item><json:item><name>Thoai </name></json:item><json:item><name>N,V </name></json:item></author><host><volume>86</volume><pages><last>388</last><first>369</first></pages><author></author><title>Journal of Optimization Theory and Applications</title><publicationDate>1995</publicationDate></host><title>New LP-Bound in Multivariate Lipschitz Optimization: Theory and Applications</title><publicationDate>1995</publicationDate></json:item><json:item><host><author><json:item><name>Rockafellar </name></json:item><json:item><name>R,T </name></json:item></author><title>Convex Analysis</title><publicationDate>1970</publicationDate></host></json:item><json:item><author><json:item><name>Vxal </name></json:item><json:item><name>I,P </name></json:item></author><host><volume>8</volume><pages><last>259</last><first>231</first></pages><author></author><title>Mathematics of Operations Research</title><publicationDate>1983</publicationDate></host><title>Strong and Weak Convexity of Sets and Functions</title><publicationDate>1983</publicationDate></json:item><json:item><host><pages><last>24</last><first>13</first></pages><author><json:item><name>Piyavskii </name></json:item><json:item><name>S,A </name></json:item></author><title>An Algorithm for Finding the Absolute Minimum of a Function, Theory of Optimal Solutions</title><publicationDate>1967</publicationDate></host></json:item><json:item><author><json:item><name>Piyavskii </name></json:item><json:item><name>S,A </name></json:item></author><host><volume>12</volume><pages><last>67</last><first>57</first></pages><author></author><title>USSR Computational Mathematics and Mathematical Physics</title><publicationDate>1972</publicationDate></host><title>An Algorithm for Finding the Absolute Extremum of a Function</title><publicationDate>1972</publicationDate></json:item><json:item><author><json:item><name>Horst 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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><Affiliation ID="Aff1"><OrgDivision>Fachbereich IV-Department of Mathematics</OrgDivision><OrgName>University of Trier</OrgName><OrgAddress><City>Trier</City><Country>Germany</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>In this note, we show how a recent approach for solving linearly constrained multivariate Lipschitz optimization problems and corresponding systems of inequalities can be generalized to solve optimization problems where the objective function is only assumed to possess a concave minorant at each point. This class of functions includes not only Lipschitz functions and some generalizations, such as certain ρ-convex functions and Hölder functions with exponent greater than one, but also all functions which can be expressed as differences of two convex functions (d.c. functions). Thus, in particular, a new approach is obtained for the important problem of minimizing a d.c. function over a polytope.</Para></Abstract><KeywordGroup Language="En"><Heading>Key Words</Heading><Keyword>Global optimization</Keyword><Keyword>functions with concave minorants</Keyword><Keyword>d.c. optimization</Keyword><Keyword>branch-and-bounds methods</Keyword><Keyword>systems of inequalities</Keyword></KeywordGroup><ArticleNote Type="CommunicatedBy"><SimplePara>Communicated by G. 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This class of functions includes not only Lipschitz functions and some generalizations, such as certain ρ-convex functions and Hölder functions with exponent greater than one, but also all functions which can be expressed as differences of two convex functions (d.c. functions). 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