### Abstract

Original language | English |
---|---|

Number of pages | 22 |

Journal | INFORMS Journal on Computing |

Early online date | Jul 2019 |

DOIs | |

Publication status | E-pub ahead of print - Jul 2019 |

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### Keywords

- non-convex quadratic programming
- global optimization
- mixed-integer linear optimization
- KKT conditions
- branch and bound
- Hoffman bound

### Cite this

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**Globally solving non-convex quadratic programs via linear integer programming techniques.** / Xia, Wei; Vera, J. C.; Zuluaga, Luis F.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Globally solving non-convex quadratic programs via linear integer programming techniques

AU - Xia, Wei

AU - Vera, J. C.

AU - Zuluaga, Luis F.

PY - 2019/7

Y1 - 2019/7

N2 - We reformulate a (indefinite) quadratic program (QP) as a mixed-integer linear programming (MILP) problem by first reformulating a QP as a linear complementary problem, and then using binary variables and big-M constraints to model its complementary constraints. To obtain such reformulation, we use fundamental results on the solution of perturbed linear systems to impose bounds on the QP’s dual variables without eliminating any of its (globally) optimal primal solutions. Reformulating a nonconvex QP as a MILP problem allows the use of current state-of-the-art MILP solvers to find its global optimal solution. To illustrate this, we compare the performance of this MILP-based solution approach, labeled quadprogIP, with quadprogBB, BARON, and CPLEX. In practice, quadprogIP is shown to typically outperform by orders of magnitude quadprogBB, BARON, and CPLEX on standard QPs. Also, unlike quadprogBB, quadprogIP is able to solve QP instances in which the dual feasible set is unbounded. The MATLAB code quadprogIP and the instances used to perform the reported numerical experiments are publicly available at https://github.com/xiawei918/quadprogIP.

AB - We reformulate a (indefinite) quadratic program (QP) as a mixed-integer linear programming (MILP) problem by first reformulating a QP as a linear complementary problem, and then using binary variables and big-M constraints to model its complementary constraints. To obtain such reformulation, we use fundamental results on the solution of perturbed linear systems to impose bounds on the QP’s dual variables without eliminating any of its (globally) optimal primal solutions. Reformulating a nonconvex QP as a MILP problem allows the use of current state-of-the-art MILP solvers to find its global optimal solution. To illustrate this, we compare the performance of this MILP-based solution approach, labeled quadprogIP, with quadprogBB, BARON, and CPLEX. In practice, quadprogIP is shown to typically outperform by orders of magnitude quadprogBB, BARON, and CPLEX on standard QPs. Also, unlike quadprogBB, quadprogIP is able to solve QP instances in which the dual feasible set is unbounded. The MATLAB code quadprogIP and the instances used to perform the reported numerical experiments are publicly available at https://github.com/xiawei918/quadprogIP.

KW - non-convex quadratic programming

KW - global optimization

KW - mixed-integer linear optimization

KW - KKT conditions

KW - branch and bound

KW - Hoffman bound

U2 - 10.1287/ijoc.2018.0883

DO - 10.1287/ijoc.2018.0883

M3 - Article

JO - INFORMS Journal on Computing

JF - INFORMS Journal on Computing

SN - 1091-9856

ER -