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Successive Lagrangian relaxation algorithm for nonconvex quadratic optimization

Author

Listed:
  • Shinji Yamada

    (The University of Tokyo
    Tokio Marine & Nichido Fire Insurance Co., Ltd.)

  • Akiko Takeda

    (The Institute of Statistical Mathematics
    RIKEN)

Abstract

Optimization problems whose objective function and constraints are quadratic polynomials are called quadratically constrained quadratic programs (QCQPs). QCQPs are NP-hard in general and are important in optimization theory and practice. There have been many studies on solving QCQPs approximately. Among them, a semidefinite program (SDP) relaxation is a well-known convex relaxation method. In recent years, many researchers have tried to find better relaxed solutions by adding linear constraints as valid inequalities. On the other hand, the SDP relaxation requires a long computation time, and it has high space complexity for large-scale problems in practice; therefore, the SDP relaxation may not be useful for such problems. In this paper, we propose a new convex relaxation method that is weaker but faster than SDP relaxation methods. The proposed method transforms a QCQP into a Lagrangian dual optimization problem and successively solves subproblems while updating the Lagrange multipliers. The subproblem in our method is a QCQP with only one constraint for which we propose an efficient algorithm. Numerical experiments confirm that our method can quickly find a relaxed solution with an appropriate termination condition.

Suggested Citation

  • Shinji Yamada & Akiko Takeda, 2018. "Successive Lagrangian relaxation algorithm for nonconvex quadratic optimization," Journal of Global Optimization, Springer, vol. 71(2), pages 313-339, June.
    • Handle: RePEc:spr:jglopt:v:71:y:2018:i:2:d:10.1007_s10898-018-0617-2
    DOI: 10.1007/s10898-018-0617-2
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    References listed on IDEAS

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    1. Samuel Burer & Sunyoung Kim & Masakazu Kojima, 2014. "Faster, but weaker, relaxations for quadratically constrained quadratic programs," Computational Optimization and Applications, Springer, vol. 59(1), pages 27-45, October.
    2. Jos F. Sturm & Shuzhong Zhang, 2003. "On Cones of Nonnegative Quadratic Functions," Mathematics of Operations Research, INFORMS, vol. 28(2), pages 246-267, May.
    3. X. Zheng & X. Sun & D. Li, 2011. "Nonconvex quadratically constrained quadratic programming: best D.C. decompositions and their SDP representations," Journal of Global Optimization, Springer, vol. 50(4), pages 695-712, August.
    4. Hu, Yaohua & Yang, Xiaoqi & Sim, Chee-Khian, 2015. "Inexact subgradient methods for quasi-convex optimization problems," European Journal of Operational Research, Elsevier, vol. 240(2), pages 315-327.
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