Cut-generating functions and S-free sets
Identifieur interne : 000177 ( Hal/Curation ); précédent : 000176; suivant : 000178Cut-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.
English descriptors
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.
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DOI: 10.1287/moor.2014.0670
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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>
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<term xml:lang="en">Convex analysis</term>
<term xml:lang="en">Integer programming</term>
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<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>
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<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>
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