PittsburghV1, Hal, Curation, bibRecord, 000177

Cut-generating functions and S-free sets

Identifieur interne : 000177 ( Hal/Curation ); précédent : 000176; suivant : 000178

Cut-generating functions and S-free sets

Auteurs : Michele Conforti [France] ; Gérard Cornuéjols [États-Unis] ; Aris Daniilidis [Espagne] ; Claude Lemaréchal [France] ; Jérôme Malick [France]

Source :

Mathematics of Operations Research [ 0364-765X ] ; 2014-08.

RBID : Hal:hal-01123860

English descriptors

mix :
- Convex analysis, Generalized gauges, Integer programming, S-free sets, Separation.

Abstract

We consider the separation problem for sets X that are pre-images of a given set S by a linear mapping. Classical examples occur in integer programming, as well as in other optimization problems such as complementarity. One would like to generate valid inequalities that cut off some point not lying in X, without reference to the linear mapping. To this aim, we introduce a concept: cut-generating functions (cgf) and we develop a formal theory for them, largely based on convex analysis. They are intimately related to S-free sets and we study this relation, disclosing several definitions for minimal cgf's and maximal S-free sets. Our work unifies and puts in perspective a number of existing works on S-free sets; in particular, we show how cgf's recover the celebrated Gomory cuts.

Url:

https://hal.archives-ouvertes.fr/hal-01123860

DOI: 10.1287/moor.2014.0670

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<idno type="DOI">10.1287/moor.2014.0670</idno>
<series><title level="j">Mathematics of Operations Research</title>
<idno type="ISSN">0364-765X</idno>
<imprint><date type="datePub">2014-08</date>
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<profileDesc><textClass><keywords scheme="mix" xml:lang="en"><term>Convex analysis</term>
<term>Generalized gauges</term>
<term>Integer programming</term>
<term>S-free sets</term>
<term>Separation</term>
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<front><div type="abstract" xml:lang="en">We consider the separation problem for sets X that are pre-images of a given set S by a linear mapping. Classical examples occur in integer programming, as well as in other optimization problems such as complementarity. One would like to generate valid inequalities that cut off some point not lying in X, without reference to the linear mapping. To this aim, we introduce a concept: cut-generating functions (cgf) and we develop a formal theory for them, largely based on convex analysis. They are intimately related to S-free sets and we study this relation, disclosing several definitions for minimal cgf's and maximal S-free sets. Our work unifies and puts in perspective a number of existing works on S-free sets; in particular, we show how cgf's recover the celebrated Gomory cuts.</div>
</front>
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<hal api="V3">          <titleStmt>            <title xml:lang="en">Cut-generating functions and S-free sets</title>
            <author role="aut">              <persName>                <forename type="first">Michele</forename>
                <surname>Conforti</surname>
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            <author role="aut">              <persName>                <forename type="first">Gérard</forename>
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            <author role="aut">              <persName>                <forename type="first">Aris</forename>
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            <author role="aut">              <persName>                <forename type="first">Claude</forename>
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              <date type="whenModified">2017-10-13 13:52:03</date>
              <date type="whenReleased">2015-03-05 16:05:24</date>
              <date type="whenProduced">2014-08</date>
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            <idno type="halRefHtml">Mathematics of Operations Research, INFORMS, 2014, 40 (2), pp.276 - 391. 〈10.1287/moor.2014.0670〉</idno>
            <idno type="halRef">Mathematics of Operations Research, INFORMS, 2014, 40 (2), pp.276 - 391. 〈10.1287/moor.2014.0670〉</idno>
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            <idno type="stamp" n="INSMI">CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions</idno>
            <idno type="stamp" n="LJK" p="UGA">Laboratoire Jean Kuntzmann</idno>
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          <sourceDesc>            <biblStruct>              <analytic>                <title xml:lang="en">Cut-generating functions and S-free sets</title>
                <author role="aut">                  <persName>                    <forename type="first">Michele</forename>
                    <surname>Conforti</surname>
                  </persName>
                  <idno type="halauthorid">830520</idno>
                  <affiliation ref="#struct-43887"></affiliation>
                </author>
                <author role="aut">                  <persName>                    <forename type="first">Gérard</forename>
                    <surname>Cornuéjols</surname>
                  </persName>
                  <idno type="halauthorid">98448</idno>
                  <affiliation ref="#struct-87848"></affiliation>
                </author>
                <author role="aut">                  <persName>                    <forename type="first">Aris</forename>
                    <surname>DANIILIDIS</surname>
                  </persName>
                  <idno type="halauthorid">67550</idno>
                  <affiliation ref="#struct-73761"></affiliation>
                </author>
                <author role="aut">                  <persName>                    <forename type="first">Claude</forename>
                    <surname>Lemaréchal</surname>
                  </persName>
                  <idno type="halauthorid">97586</idno>
                  <affiliation ref="#struct-44907"></affiliation>
                </author>
                <author role="aut">                  <persName>                    <forename type="first">Jérôme</forename>
                    <surname>Malick</surname>
                  </persName>
                  <idno type="halauthorid">1109580</idno>
                  <affiliation ref="#struct-44907"></affiliation>
                </author>
              </analytic>
              <monogr>                <idno type="halJournalId" status="VALID">16938</idno>
                <idno type="issn">0364-765X</idno>
                <idno type="eissn">1526-5471</idno>
                <title level="j">Mathematics of Operations Research</title>
                <imprint>                  <publisher>INFORMS</publisher>
                  <biblScope unit="volume">40</biblScope>
                  <biblScope unit="issue">2</biblScope>
                  <biblScope unit="pp">276 - 391</biblScope>
                  <date type="datePub">2014-08</date>
                  <date type="dateEpub">2014-08-20</date>
                </imprint>
              </monogr>
              <idno type="doi">10.1287/moor.2014.0670</idno>
            </biblStruct>
          </sourceDesc>
          <profileDesc>            <langUsage>              <language ident="en">English</language>
            </langUsage>
            <textClass>              <keywords scheme="author">                <term xml:lang="en">S-free sets</term>
                <term xml:lang="en">Generalized gauges</term>
                <term xml:lang="en">Separation</term>
                <term xml:lang="en">Convex analysis</term>
                <term xml:lang="en">Integer programming</term>
              </keywords>
              <classCode scheme="halDomain" n="math.math-oc">Mathematics [math]/Optimization and Control [math.OC]</classCode>
              <classCode scheme="halDomain" n="info.info-ro">Computer Science [cs]/Operations Research [cs.RO]</classCode>
              <classCode scheme="halDomain" n="info.info-dm">Computer Science [cs]/Discrete Mathematics [cs.DM]</classCode>
              <classCode scheme="halDomain" n="math.math-na">Mathematics [math]/Numerical Analysis [math.NA]</classCode>
              <classCode scheme="halTypology" n="ART">Journal articles</classCode>
            </textClass>
            <abstract xml:lang="en">We consider the separation problem for sets X that are pre-images of a given set S by a linear mapping. Classical examples occur in integer programming, as well as in other optimization problems such as complementarity. One would like to generate valid inequalities that cut off some point not lying in X, without reference to the linear mapping. To this aim, we introduce a concept: cut-generating functions (cgf) and we develop a formal theory for them, largely based on convex analysis. They are intimately related to S-free sets and we study this relation, disclosing several definitions for minimal cgf's and maximal S-free sets. Our work unifies and puts in perspective a number of existing works on S-free sets; in particular, we show how cgf's recover the celebrated Gomory cuts.</abstract>
          </profileDesc>
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Cut-generating functions and S-free sets

Cut-generating functions and S-free sets

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