Cutting planes from extended LP formulations

Given a mixed-integer set defined by linear inequalities and integrality requirements on some of the variables, we consider extended formulations of its continuous (LP) relaxation and study the effect of adding cutting planes in the extended space. In terms of optimization, extended LP formulations do not lead to better bounds as their projection onto the original space is precisely the original LP relaxation. However, adding cutting planes in the extended space can lead to stronger bounds. In this paper we show that for every 0–1 mixed-integer set with n integer and k continuous variables, there is an extended LP formulation with $$2n+k-1$$2n+k-1 variables whose elementary split closure is integral. The proof is constructive but it requires an inner description of the LP relaxation. We then extend this idea to general mixed-integer sets and construct the best extended LP formulation for such sets with respect to lattice-free cuts. We also present computational results on the two-row continuous group relaxation showing the strength of cutting planes derived from extended LP formulations.

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