# The convex hull of a quadratic constraint over a polytope

## TL;DR

This paper proves that the convex hull of a quadratic constraint intersected with a bounded polyhedron can be represented using second-order cones, aiding in solving non-convex QCQPs more effectively.

## Contribution

It establishes that the convex hull of a quadratic constraint over a polytope is second-order cone representable, providing a constructive proof for this key convexification result.

## Key findings

- Convex hull is second-order cone representable.
- Constructive proof of the convex hull characterization.
- Facilitates convex relaxation of non-convex QCQPs.

## Abstract

A quadratically constrained quadratic program (QCQP) is an optimization problem in which the objective function is a quadratic function and the feasible region is defined by quadratic constraints. Solving non-convex QCQP to global optimality is a well-known NP-hard problem and a traditional approach is to use convex relaxations and branch-and-bound algorithms. This paper makes a contribution in this direction by showing that the exact convex hull of a general quadratic equation intersected with any bounded polyhedron is second-order cone representable. We present a simple constructive proof of this result.