We study the mixed-integer rounding (MIR) closures of polyhedral sets. The MIR closure of **a** polyhedral set is equal to its split closure **a**nd the **a**ssociated separation problem is NP-hard. We describe **a** mixed-integer programming (MIP) model with linear constraints **a**nd **a** non-linear objective for separating **a**n **a**rbitrary point from the MIR closure of **a** given mixed-integer set. We linearize the objective using **a**dditional variables to produce **a** linear MIP model that solves the separation problem **a**pproximately, with **a**n **a**ccuracy that depends on the number of **a**dditional variables used. Our **a**nalysis yields **a** short proof of the result of Cook, Kannan **a**nd Schrijver (1990) that the split closure of **a** polyhedral set is **a**gain **a** polyhedron. We **a**lso present some computational results with our **a**pproximate separation model.

By:* Sanjeeb Dash, Oktay Günlük, Andrea Lodi*

Published in: Mathematical Programming , volume 121, (no 1), pages 33-60 in 2010

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