On the MIR Closure of Polyhedra

Authors:
Sanjeeb Dash;Oktay Günlük;Andrea Lodi
Affiliations:
IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598,;IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598,;DEIS, University of Bologna, viale Risorgimento 2 - 40136 Bologna, Italy
Venue:
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
Year:
2007

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Abstract

We study the mixed-integer rounding (MIR) closure of polyhedra. The MIR closure of a polyhedron is equal to its split closure and the associated separation problem is NP-hard. We describe a mixed-integer programming (MIP) model with linear constraints and a non-linear objective for separating an arbitrary point from the MIR closure of a given mixed-integer set. We linearize the objective using additional variables to produce a linear MIP model that solves the separation problem approximately, with an accuracy that depends on the number of additional variables used. Our analysis yields a short proof of the result of Cook, Kannan and Schrijver (1990) that the split closure of a polyhedron is again a polyhedron. We also present some computational results with our approximate separation model.