# Unique lifting of integer variables in minimal inequalities

## TL;DR

This paper advances the understanding of lifting functions in mixed integer linear programming by characterizing when minimal valid inequalities can be uniquely lifted for nonbasic integer variables.

## Contribution

It provides a precise characterization of when the lifted coefficient equals the continuous variable's coefficient and establishes conditions for the uniqueness of the lifting function.

## Key findings

- The lifted coefficient equals the continuous variable's coefficient within a specific nonconvex region.
- A finite union of convex polyhedra describes the region where this equality holds.
- Necessary and sufficient conditions for the uniqueness of the lifting function are established.

## Abstract

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