Partial Gröbner bases for multiobjective integer linear optimization
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by
Victor Blanco, Justo Puerto
2008
Abstract
In this paper we present a new methodology for solving multiobjective integer
linear programs 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.,
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. It 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
efficient solutions of any multiobjective integer linear problem (MOILP). Some
examples illustrate the application of the algorithms and computational
experiments are reported on several families of problems.
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