Details:
Title  Partial Gr\"obner bases for multiobjective integer linear optimization.  Author(s)  Víctor Blanco, Justo Puerto  Type  Article in Journal  Abstract  This 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 righthand 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 Paretooptimal solutions of any MOILP problem. Some examples illustrate the application of the algorithm, and computational experiments are reported on several families of problems.
 ISSN  08954801; 10957146/e 
URL 
http://epubs.siam.org/doi/abs/10.1137/070698051 
Language  English  Journal  SIAM J. Discrete Math.  Volume  23  Number  2  Pages  571595  Publisher  Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA  Year  2009  Edition  0  Translation 
No  Refereed 
No 
