# Characterization of the Split Closure via Geometric Lifting

## TL;DR

This paper explores the geometric properties of split cuts in mixed-integer programming, demonstrating their equivalence to k-cuts and revealing limitations in their strength through specific examples.

## Contribution

It establishes the equivalence between split cuts and k-cuts via geometric lifting and analyzes their limitations in finite and infinite-dimensional relaxations.

## Key findings

- k-cuts are equivalent to split cuts for 1-row relaxations
- Split cuts for finite relaxations are restrictions of those for infinite-dimensional relaxations
- Split closures can have arbitrarily bad integrality gaps in certain integer programs

## Abstract

We analyze split cuts from the perspective of cut generating functions via geometric lifting. We show that $\alpha$-cuts, a natural higher-dimensional generalization of the $k$-cuts of Cornu\'{e}jols et al., gives all the split cuts for the mixed-integer corner relaxation. As an immediate consequence we obtain that the $k$-cuts are equivalent to split cuts for the 1-row mixed-integer relaxation. Further, we show that split cuts for finite-dimensional corner relaxations are restrictions of split cuts for the infinite-dimensional relaxation. In a final application of this equivalence, we exhibit a family of pure-integer programs whose split closures have arbitrarily bad integrality gap. This complements the mixed-integer example provided by Basu et al [On the relative strength of split, triangle and quadrilateral cuts, Math. Program. 126(2):281--314, 2011].