Partial Gröbner bases for multiobjective integer linear optimization

This paper presents a new methodology for solving multiobjective integer linear programs (MOILP) using tools from algebraic geometry. We introduce the concept of partial Gr¨obner basis for a family of multiobjective programs where the right-hand side varies. This new structure extends the notion of...

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Detalles Bibliográficos
Autores: Blanco Izquierdo, Víctor, Puerto Albandoz, Justo
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2012
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/36535
Acceso en línea:http://hdl.handle.net/11441/36535
https://doi.org/10.1137/070698051
Access Level:acceso abierto
Palabra clave:multiple objective optimization
integer programming
Gröbner bases
test sets
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spelling Partial Gröbner bases for multiobjective integer linear optimizationBlanco Izquierdo, VíctorPuerto Albandoz, Justomultiple objective optimizationinteger programmingGröbner basestest setsThis paper presents a new methodology for solving multiobjective integer linear programs (MOILP) using tools from algebraic geometry. We introduce the concept of partial Gr¨obner basis for a family of multiobjective programs where the right-hand side varies. This new structure extends the notion of Gr¨obner basis for the single objective case to the case of multiple objectives, i.e., when there is a partial ordering instead of a total ordering over the feasible vectors. The main property of these bases is that the partial reduction of the integer elements in the kernel of the constraint matrix by the different blocks of the basis is zero. This property allows us to prove that this new construction is a test family for a family of multiobjective programs. An algorithm “´a la Buchberger” is developed to compute partial Gr¨obner bases, and two different approaches are derived, using this methodology, for computing the entire set of Pareto-optimal solutions of any MOILP problem. Some examples illustrate the application of the algorithm, and computational experiments are reported on several families of problems.Ministerio de Educación y CienciaSociety for Industrial and Applied MathematicsEstadística e Investigación OperativaMinisterio de Educación y Ciencia (MEC). España2012info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfapplication/pdfhttp://hdl.handle.net/11441/36535https://doi.org/10.1137/070698051reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésSIAM Journal on Discrete Mathematics, 23(2), 571-595MTM2007-67433-C02-01info:eu-repo/semantics/openAccessoai:idus.us.es:11441/365352026-06-17T12:51:07Z
dc.title.none.fl_str_mv Partial Gröbner bases for multiobjective integer linear optimization
title Partial Gröbner bases for multiobjective integer linear optimization
spellingShingle Partial Gröbner bases for multiobjective integer linear optimization
Blanco Izquierdo, Víctor
multiple objective optimization
integer programming
Gröbner bases
test sets
title_short Partial Gröbner bases for multiobjective integer linear optimization
title_full Partial Gröbner bases for multiobjective integer linear optimization
title_fullStr Partial Gröbner bases for multiobjective integer linear optimization
title_full_unstemmed Partial Gröbner bases for multiobjective integer linear optimization
title_sort Partial Gröbner bases for multiobjective integer linear optimization
dc.creator.none.fl_str_mv Blanco Izquierdo, Víctor
Puerto Albandoz, Justo
author Blanco Izquierdo, Víctor
author_facet Blanco Izquierdo, Víctor
Puerto Albandoz, Justo
author_role author
author2 Puerto Albandoz, Justo
author2_role author
dc.contributor.none.fl_str_mv Estadística e Investigación Operativa
Ministerio de Educación y Ciencia (MEC). España
dc.subject.none.fl_str_mv multiple objective optimization
integer programming
Gröbner bases
test sets
topic multiple objective optimization
integer programming
Gröbner bases
test sets
description This paper presents a new methodology for solving multiobjective integer linear programs (MOILP) using tools from algebraic geometry. We introduce the concept of partial Gr¨obner basis for a family of multiobjective programs where the right-hand side varies. This new structure extends the notion of Gr¨obner basis for the single objective case to the case of multiple objectives, i.e., when there is a partial ordering instead of a total ordering over the feasible vectors. The main property of these bases is that the partial reduction of the integer elements in the kernel of the constraint matrix by the different blocks of the basis is zero. This property allows us to prove that this new construction is a test family for a family of multiobjective programs. An algorithm “´a la Buchberger” is developed to compute partial Gr¨obner bases, and two different approaches are derived, using this methodology, for computing the entire set of Pareto-optimal solutions of any MOILP problem. Some examples illustrate the application of the algorithm, and computational experiments are reported on several families of problems.
publishDate 2012
dc.date.none.fl_str_mv 2012
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11441/36535
https://doi.org/10.1137/070698051
url http://hdl.handle.net/11441/36535
https://doi.org/10.1137/070698051
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv SIAM Journal on Discrete Mathematics, 23(2), 571-595
MTM2007-67433-C02-01
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Society for Industrial and Applied Mathematics
publisher.none.fl_str_mv Society for Industrial and Applied Mathematics
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,300719

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