# Intersection disjunctions for reverse convex sets

## TL;DR

This paper introduces a new framework for generating valid inequalities for reverse convex sets, extending intersection cuts to intersection disjunctions and multi-term disjunctions, applicable in optimization models like bilevel and polynomial optimization.

## Contribution

It develops a generalized approach to derive valid inequalities from solutions outside the convex set, expanding the classical intersection cut methodology with multi-term disjunctions.

## Key findings

- Introduces intersection disjunctions for reverse convex sets.
- Generalizes to multi-term disjunctions using recession directions.
- Provides a cut-generating linear program for valid inequalities.

## Abstract

We present a framework to obtain valid inequalities for a reverse convex set: the set of points in a polyhedron that lie outside a given open convex set. Reverse convex sets arise in many models, including bilevel optimization and polynomial optimization. An intersection cut is a well-known valid inequality for a reverse convex set that is generated from a basic solution that lies within the convex set. We introduce a framework for deriving valid inequalities for the reverse convex set from basic solutions that lie outside the convex set. We first propose an extension to intersection cuts that defines a two-term disjunction for a reverse convex set, which we refer to as an intersection disjunction. Next, we generalize this analysis to a multi-term disjunction by considering the convex set's recession directions. These disjunctions can be used in a cut-generating linear program to obtain valid inequalities for the reverse convex set.