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Globally Solving Nonconvex Quadratic Programs via Linear Integer Programming Techniques

Author

Listed:
  • Wei Xia

    (Industrial and Systems Engineering Department, Lehigh University, Bethlehem, Pennsylvania 18015;)

  • Juan C. Vera

    (Tilburg School of Economics and Management, Econometrics and Operations Research, Tilburg University, 5037 AB Tilburg, Netherlands)

  • Luis F. Zuluaga

    (Industrial and Systems Engineering Department, Lehigh University, Bethlehem, Pennsylvania 18015;)

Abstract

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 .

Suggested Citation

  • Wei Xia & Juan C. Vera & Luis F. Zuluaga, 2020. "Globally Solving Nonconvex Quadratic Programs via Linear Integer Programming Techniques," INFORMS Journal on Computing, INFORMS, vol. 32(1), pages 40-56, January.
    • Handle: RePEc:inm:orijoc:v:32:y:2020:i:1:p:40-56
    DOI: 10.1287/ijoc.2018.0883
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    References listed on IDEAS

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    2. Marcia Fampa & Jon Lee, 2021. "Convexification of bilinear forms through non-symmetric lifting," Journal of Global Optimization, Springer, vol. 80(2), pages 287-305, June.
    3. Riley Badenbroek & Etienne Klerk, 2022. "Simulated Annealing for Convex Optimization: Rigorous Complexity Analysis and Practical Perspectives," Journal of Optimization Theory and Applications, Springer, vol. 194(2), pages 465-491, August.
    4. Jacek Gondzio & E. Alper Yıldırım, 2021. "Global solutions of nonconvex standard quadratic programs via mixed integer linear programming reformulations," Journal of Global Optimization, Springer, vol. 81(2), pages 293-321, October.
    5. Bomze, Immanuel M. & Gabl, Markus, 2023. "Optimization under uncertainty and risk: Quadratic and copositive approaches," European Journal of Operational Research, Elsevier, vol. 310(2), pages 449-476.
    6. Riley Badenbroek & Etienne de Klerk, 2022. "An Analytic Center Cutting Plane Method to Determine Complete Positivity of a Matrix," INFORMS Journal on Computing, INFORMS, vol. 34(2), pages 1115-1125, March.
    7. G. Liuzzi & M. Locatelli & V. Piccialli & S. Rass, 2021. "Computing mixed strategies equilibria in presence of switching costs by the solution of nonconvex QP problems," Computational Optimization and Applications, Springer, vol. 79(3), pages 561-599, July.
    8. de Klerk, Etienne & Badenbroek, Riley, 2022. "Simulated annealing with hit-and-run for convex optimization: complexity analysis and practical perspectives," Other publications TiSEM 323b4588-65e0-4889-a555-9, Tilburg University, School of Economics and Management.

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