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New reformulations for 0-1 quadratic programming problem using quadratic nonconvex reformulation techniques and valid inequalities

Cheng Lu, Yu Fei, Jing Zhou, Zhibin Deng, Guangtai Qu

TL;DR

Several new reformulations for 0-1 quadratic programming problems using the QNR technique provide more flexibility in adding nonconvex quadratic constraints into the problem formulation, so that some valid inequalities can be incorporated into the formulation to tighten the lower bound of the problem.

Abstract

It is well-known that the quadratic convex reformulation (QCR) technique can speed up some general-purpose solvers such as CPLEX and Gurobi. Recently, the method of quadratic nonconvex reformulation (QNR) was proposed, which provides an alternative way for accelerating a solver via reformulation technique. This paper proposes several new reformulations for 0-1 quadratic programming problems using the QNR technique. Such a technique provides more flexibility in adding nonconvex quadratic constraints into the problem formulation, so that some valid inequalities, such as the triangle inequalities, can be incorporated into the formulation to tighten the lower bound of the problem. We analyze the effects of the proposed reformulations on the lower bounds implemented in the solver, and propose some methods to maximize the McCormick relaxation bounds of the reformulations. Our numerical experiments compare the proposed reformulations with the existing quadratic convex reformulations, showing the effectiveness of the proposed reformulations on 0-1 quadratic programming problems.

New reformulations for 0-1 quadratic programming problem using quadratic nonconvex reformulation techniques and valid inequalities

TL;DR

Several new reformulations for 0-1 quadratic programming problems using the QNR technique provide more flexibility in adding nonconvex quadratic constraints into the problem formulation, so that some valid inequalities can be incorporated into the formulation to tighten the lower bound of the problem.

Abstract

It is well-known that the quadratic convex reformulation (QCR) technique can speed up some general-purpose solvers such as CPLEX and Gurobi. Recently, the method of quadratic nonconvex reformulation (QNR) was proposed, which provides an alternative way for accelerating a solver via reformulation technique. This paper proposes several new reformulations for 0-1 quadratic programming problems using the QNR technique. Such a technique provides more flexibility in adding nonconvex quadratic constraints into the problem formulation, so that some valid inequalities, such as the triangle inequalities, can be incorporated into the formulation to tighten the lower bound of the problem. We analyze the effects of the proposed reformulations on the lower bounds implemented in the solver, and propose some methods to maximize the McCormick relaxation bounds of the reformulations. Our numerical experiments compare the proposed reformulations with the existing quadratic convex reformulations, showing the effectiveness of the proposed reformulations on 0-1 quadratic programming problems.
Paper Structure (11 sections, 1 theorem, 35 equations, 4 tables)

This paper contains 11 sections, 1 theorem, 35 equations, 4 tables.

Table of Contents

  1. Introduction
  2. A review of quadratic convex reformulation
  3. Quadratic nonconvex reformulations
  4. The lower bound methods implemented in Gurobi
  5. A simple quadratic nonconvex reformulation for \ref{['BQP']}
  6. Enhancing quadratic nonconvex reformulations using valid inequalities
  7. A specific reformulation by incorporating triangle inequalities
  8. Numerical experiments
  9. Numerical results on public test instances from Biq Mac Library
  10. Numerical results on randomly generated test instances
  11. Conclusions

Key Result

Theorem 1

The parameter $(\lambda^\ast,Z^\ast,\gamma^\ast)$ that maximizes the McCormick relaxation bound of QNRE can be obtained by setting $\lambda^\ast=\lambda^\dag$, $\gamma^\ast=\gamma^\dag$, and where $(\lambda^\dag,\gamma^\dag,\tau^\dag,M^\dag,N^\dag,R^\dag,S^\dag)$ denotes the optimal solution of QNR-dual1.

Theorems & Definitions (7)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • proof
  • Remark 4
  • Remark 5