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Lagrangian decomposition and mixed-integer quadratic programming reformulations for probabilistically constrained quadratic programs

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  • Zheng, Xiaojin
  • Sun, Xiaoling
  • Li, Duan
  • Cui, Xueting

Abstract

Probabilistically constrained quadratic programming (PCQP) problems arise naturally from many real-world applications and have posed a great challenge in front of the optimization society for years due to the nonconvex and discrete nature of its feasible set. We consider in this paper a special case of PCQP where the random vector has a finite discrete distribution. We first derive second-order cone programming (SOCP) relaxation and semidefinite programming (SDP) relaxation for the problem via a new Lagrangian decomposition scheme. We then give a mixed integer quadratic programming (MIQP) reformulation of the PCQP and show that the continuous relaxation of the MIQP is exactly the SOCP relaxation. This new MIQP reformulation is more efficient than the standard MIQP reformulation in the sense that its continuous relaxation is tighter than or at least as tight as that of the standard MIQP. We report preliminary computational results to demonstrate the tightness of the new convex relaxations and the effectiveness of the new MIQP reformulation.

Suggested Citation

  • Zheng, Xiaojin & Sun, Xiaoling & Li, Duan & Cui, Xueting, 2012. "Lagrangian decomposition and mixed-integer quadratic programming reformulations for probabilistically constrained quadratic programs," European Journal of Operational Research, Elsevier, vol. 221(1), pages 38-48.
    • Handle: RePEc:eee:ejores:v:221:y:2012:i:1:p:38-48
    DOI: 10.1016/j.ejor.2012.03.006
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    References listed on IDEAS

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    Cited by:

    1. Wu, Baiyi & Li, Duan & Jiang, Rujun, 2019. "Quadratic convex reformulation for quadratic programming with linear on–off constraints," European Journal of Operational Research, Elsevier, vol. 274(3), pages 824-836.
    2. Ghaddar, Bissan & Naoum-Sawaya, Joe & Kishimoto, Akihiro & Taheri, Nicole & Eck, Bradley, 2015. "A Lagrangian decomposition approach for the pump scheduling problem in water networks," European Journal of Operational Research, Elsevier, vol. 241(2), pages 490-501.
    3. Zheng, Xiaojin & Wu, Baiyi & Cui, Xueting, 2017. "Cell-and-bound algorithm for chance constrained programs with discrete distributions," European Journal of Operational Research, Elsevier, vol. 260(2), pages 421-431.
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    5. Hsia, Yong & Wu, Baiyi & Li, Duan, 2014. "New reformulations for probabilistically constrained quadratic programs," European Journal of Operational Research, Elsevier, vol. 233(3), pages 550-556.
    6. Ran Ji & Miguel A. Lejeune, 2018. "Risk-budgeting multi-portfolio optimization with portfolio and marginal risk constraints," Annals of Operations Research, Springer, vol. 262(2), pages 547-578, March.

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