# Exact Augmented Lagrangian Duality for Mixed Integer Quadratic Programming

## TL;DR

This paper establishes theoretical guarantees for the augmented Lagrangian dual approach in solving mixed integer quadratic programming problems, showing conditions under which it achieves zero duality gap.

## Contribution

It proves that the augmented Lagrangian dual can reach zero duality gap asymptotically and finitely for MIQP under certain conditions, with polynomial bounds on penalty weights.

## Key findings

- ALD reaches zero duality gap asymptotically as penalty weight increases.
- Finite penalty weights suffice for zero duality gap with any norm.
- Polynomial bounds on penalty weights ensure zero duality gap.

## Abstract

Mixed integer quadratic programming (MIQP) is the problem of minimizing a convex quadratic function over mixed integer points in a rational polyhedron. This paper focuses on the augmented Lagrangian dual (ALD) for MIQP. ALD augments the usual Lagrangian dual with a weighted nonlinear penalty on the dualized constraints. We first prove that ALD will reach a zero duality gap asymptotically as the weight on the penalty goes to infinity under some mild conditions on the penalty function. We next show that a finite penalty weight is enough for a zero gap when we use any norm as the penalty function. Finally, we prove a polynomially bound on the weight on the penalty term to obtain a zero gap.