Abstract
Jeroslow and Lowe gave an exact geometric characterization of subsets of \(\mathbb R^n\) that are projections of mixed-integer linear sets, a.k.a MILP-representable sets. We give an alternate algebraic characterization by showing that a set is MILP-representable if and only if the set can be described as the intersection of finitely many affine Chvátal inequalities. These inequalities are a modification of a concept introduced by Blair and Jeroslow. This gives a sequential variable elimination scheme that, when applied to the MILP representation of a set, explicitly gives the affine Chvátal inequalities characterizing the set. This is related to the elimination scheme of Wiliams and Williams-Hooker, who describe projections of integer sets using disjunctions of affine Chvátal systems. Our scheme extends their work in two ways. First, we show that disjunctions are unnecessary, by showing how to find the affine Chvátal inequalities that cannot be discovered by the Williams-Hooker scheme. Second, disjunctions of Chvátal systems can give sets that are not projections of mixed-integer linear sets; so the Williams-Hooker approach does not give an exact characterization of MILP representability.
A. Basu—Supported by the NSF grant CMMI1452820.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
- 2.
This is precisely where we need to allow the arguments of the \(f_i\)’s to be non rational because the vector \(d' - A'x\) that arise from all possible x is sometimes non rational.
References
Balas, E.: Projecting systems of linear inequalities with binary variables. Ann. Oper. Res. 188(1), 19–31 (2011)
Blair, C., Jeroslow, R.: The value function of an integer program. Math. Program. 23, 237–273 (1982)
Conforti, M., Cornuéjols, G., Zambelli, G.: Integer Programming. Springer, Heidelberg (2014)
Jeroslow, R., Lowe, J.: Modelling with integer variables. In: Korte, B., Ritter, K. (eds.) Mathematical Programming at Oberwolfach II, vol. 22, pp. 167–184. Springer, Heidelberg (1984)
Ryan, J.: Integral monoid duality models. Technical report, Cornell University Operations Research and Industrial Engineering (1986)
Ryan, J.: Decomposing finitely generated integral monoids by elimination. Linear Algebra Appl. 153, 209–217 (1991)
Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)
Vielma, J.P.: Mixed integer linear programming formulation techniques. SIAM Rev. 57(1), 3–57 (2015)
Williams, H.P.: Fourier-Motzkin elimination extension to integer programming problems. J. Comb. Theor. A 21, 118–123 (1976)
Williams, H.P.: The elimination of integer variables. J. Oper. Res. Soc. 43, 387–393 (1992)
Williams, H., Hooker, J.: Integer programming as projection. Discrete Optim. 22, 291–311 (2016)
Wolsey, L.: The b-hull of an integer program. Discrete Appl. Math. 3(3), 193–201 (1981)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Basu, A., Martin, K., Ryan, C.T., Wang, G. (2017). Mixed-Integer Linear Representability, Disjunctions, and Variable Elimination. In: Eisenbrand, F., Koenemann, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2017. Lecture Notes in Computer Science(), vol 10328. Springer, Cham. https://doi.org/10.1007/978-3-319-59250-3_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-59250-3_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-59249-7
Online ISBN: 978-3-319-59250-3
eBook Packages: Computer ScienceComputer Science (R0)