# Partial hyperplane activation for generalized intersection cuts

## TL;DR

This paper introduces partial hyperplane activation (PHA), a novel method to improve the computational viability of generalized intersection cuts in mixed integer programming, supported by theoretical and empirical analysis.

## Contribution

The paper develops and validates PHA algorithms as practical replacements for hyperplane activation in GIC, enhancing their computational feasibility.

## Key findings

- PHA shows promising computational performance on MIPLIB instances.
- Theoretical results support the validity of PHA algorithms.
- Insights into properties of strong cuts for future research.

## Abstract

The generalized intersection cut (GIC) paradigm is a recent framework for generating cutting planes in mixed integer programming with attractive theoretical properties. We investigate this computationally unexplored paradigm and observe that a key hyperplane activation procedure embedded in it is not computationally viable. To overcome this issue, we develop a novel replacement to this procedure called partial hyperplane activation (PHA), introduce a variant of PHA based on a notion of hyperplane tilting, and prove the validity of both algorithms. We propose several implementation strategies and parameter choices for our PHA algorithms and provide supporting theoretical results. We computationally evaluate these ideas in the COIN-OR framework on MIPLIB instances. Our findings shed light on the the strengths of the PHA approach as well as suggest properties related to strong cuts that can be targeted in the future.