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TitlePartial Gr\"obner bases for multiobjective integer linear optimization.
Author(s) Víctor Blanco, Justo Puerto
TypeArticle in Journal
AbstractThis paper presents a new methodology for solving multiobjective integer linear programs (MOILP) using tools from algebraic geometry. We introduce the concept of partial Gröbner basis for a family of multiobjective programs where the right-hand side varies. This new structure extends the notion of Gröbner 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 “á la Buchberger” is developed to compute partial Gröbner 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.

ISSN0895-4801; 1095-7146/e
URL http://epubs.siam.org/doi/abs/10.1137/070698051
LanguageEnglish
JournalSIAM J. Discrete Math.
Volume23
Number2
Pages571--595
PublisherSociety for Industrial and Applied Mathematics (SIAM), Philadelphia, PA
Year2009
Edition0
Translation No
Refereed No
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