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Global solutions of nonconvex standard quadratic programs via mixed integer linear programming reformulations

Author

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  • Jacek Gondzio

    (The University of Edinburgh)

  • E. Alper Yıldırım

    (The University of Edinburgh)

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 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 several orders of magnitude.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:81:y:2021:i:2:d:10.1007_s10898-021-01017-y
    DOI: 10.1007/s10898-021-01017-y
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    References listed on IDEAS

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    1. Immanuel M. Bomze & Werner Schachinger & Reinhard Ullrich, 2018. "The Complexity of Simple Models—A Study of Worst and Typical Hard Cases for the Standard Quadratic Optimization Problem," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 651-674, May.
    2. Luana E. Gibbons & Donald W. Hearn & Panos M. Pardalos & Motakuri V. Ramana, 1997. "Continuous Characterizations of the Maximum Clique Problem," Mathematics of Operations Research, INFORMS, vol. 22(3), pages 754-768, August.
    3. 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.
    4. Peter Dickinson & Luuk Gijben, 2014. "On the computational complexity of membership problems for the completely positive cone and its dual," Computational Optimization and Applications, Springer, vol. 57(2), pages 403-415, March.
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    Cited by:

    1. Riccardo Bisori & Matteo Lapucci & Marco Sciandrone, 2022. "A study on sequential minimal optimization methods for standard quadratic problems," 4OR, Springer, vol. 20(4), pages 685-712, December.
    2. Bomze, Immanuel M. & Gabl, Markus & Maggioni, Francesca & Pflug, Georg Ch., 2022. "Two-stage stochastic standard quadratic optimization," European Journal of Operational Research, Elsevier, vol. 299(1), pages 21-34.
    3. 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.

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