Global Solutions of Nonconvex Standard Quadratic Programs via Mixed Integer Linear Programming Reformulations

TL;DR

This paper introduces two novel mixed integer linear programming reformulations for solving nonconvex standard quadratic programs, significantly improving solution efficiency and scalability over existing methods.

Contribution

It proposes two new MILP reformulations with valid inequalities for standard quadratic programs, enhancing global solution capabilities.

Findings

Reformulations outperform existing methods in computational tests.

Significant improvements in solving larger instances.

Reformulations achieve orders of magnitude faster solutions.

Abstract

A standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We propose two alternative mixed integer linear programming formulations. Our first formulation is based on casting a standard quadratic program as a linear program with complementarity constraints. We then employ binary variables to linearize the complementarity constraints. For the second formulation, we first derive an overestimating function of the objective function and establish its tightness at any global minimizer. We then linearize the overestimating function using binary variables and obtain our second formulation. For both formulations, we propose a set of valid inequalities. Our extensive computational results illustrate that the proposed mixed integer linear programming reformulations significantly outperform other global solution approaches. On larger instances, we usually observe improvements of orders of magnitude.