Computational Comparison of Exact Solution Methods for 0-1 Quadratic Programs: Recommendations for Practitioners

Richard J Forrester,Noah Hunt-Isaak

doi:10.1155/2020/5974820

Abstract

This paper is concerned with binary quadratic programs (BQPs), which are among the most well-studied classes of nonlinear integer optimization problems because of their wide variety of applications. While a number of different solution approaches have been proposed for tackling BQPs, practitioners need techniques that are both efficient and easy to implement. We revisit two of the most widely used linearization strategies for BQPs and examine the effectiveness of enhancements to these formulations that have been suggested in the literature. We perform a detailed large-scale computational study over five different classes of BQPs to compare these two linearizations with a more recent linear reformulation and direct submission of the nonlinear integer program to an optimization solver. The goal is to provide practitioners with guidance on how to best approach solving BQPs in an effective and easily implemented manner.

Highlights

Binary quadratic programs (BQPs) are one of the most wellstudied classes of nonlinear integer optimization problems
We begin with our first question raised: When implementing the standard linearization, should you utilize the full formulation or use the formulation that is based on the sign of the quadratic objective coefficients? That is, how does STD compare with STD′ when submitted to an mixed-integer linear programming (MILP) solver?
Note that since STD and STD′ vary depending on the sign of the objective coefficients, we provide two performance profiles for both quadratic multidimensional knapsack problem (QMKP) and k quadratic knapsack problem (QKP), one for instances with positive quadratic coefficients and one for instances with quadratic coefficients of mixed sign

Summary

Introduction

Binary quadratic programs (BQPs) are one of the most wellstudied classes of nonlinear integer optimization problems. These problems appear in a wide variety of applications (see [1, 2] for examples) and are known to be NP-hard. Given the difficulty of implementing and maintaining custom algorithms, the most commonly used exact solution method for BQPs involves linearizing the nonlinear problem and subsequently submitting the equivalent linear form to a standard mixed-integer linear programming (MILP) solver. Note that our work is similar to that of [6, 7] who perform computational studies of different linearization strategies for BQPs. our focus is on comparing different linearizations and on how enhancements to the standard linearization and Glover’s method affect algorithmic performance

Objectives

Results

Conclusion