# A geometric approach to cut-generating functions

## TL;DR

This paper surveys recent advances in the geometric approach to generating cutting planes for integer programming, highlighting new theoretical tools and breakthroughs that connect convex analysis, geometry, and number theory.

## Contribution

It provides a comprehensive overview of recent developments in geometric cut-generating functions, emphasizing innovative methods and key breakthroughs in the field.

## Key findings

- Renewed interest in corner polyhedron and intersection cuts
- Integration of convex analysis, geometry, and number theory tools
- Recent breakthroughs in cut-generating function theory

## Abstract

The cutting-plane approach to integer programming was initiated more that 40 years ago: Gomory introduced the corner polyhedron as a relaxation of a mixed integer set in tableau form and Balas introduced intersection cuts for the corner polyhedron. This line of research was left dormant for several decades until relatively recently, when a paper of Andersen, Louveaux, Weismantel and Wolsey generated renewed interest in the corner polyhedron and intersection cuts. Recent developments rely on tools drawn from convex analysis, geometry and number theory, and constitute an elegant bridge between these areas and integer programming. We survey these results and highlight recent breakthroughs in this area.