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This paper contributes to the theory of cutting planes for mixed integerlinear programs (MILPs). Minimal valid inequalities are well understood for arelaxation of an MILP in tableau form where all the nonbasic variables arecontinuous; they are derived using the gauge function of maximal lattice-freeconvex sets. In this paper we study lifting functions for the nonbasic integervariables starting from such minimal valid inequalities. We characterizeprecisely when the lifted coefficient is equal to the coefficient of thecorresponding continuous variable in every minimal lifting. The answer is anonconvex region that can be obtained as a finite union of convex polyhedra. Wethen establish a necessary and sufficient condition for the uniqueness of thelifting function.

Unique lifting of integer variables in minimal inequalities