IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v71y2018i2d10.1007_s10898-018-0617-2.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-018-0617-2
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10898-018-0617-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Mohammad Keyanpour & Naser Osmanpour, 2018. "On solving quadratically constrained quadratic programming problem with one non-convex constraint," OPSEARCH, Springer;Operational Research Society of India, vol. 55(2), pages 320-336, June.
    2. Ben-Tal, A. & den Hertog, D., 2011. "Immunizing Conic Quadratic Optimization Problems Against Implementation Errors," Discussion Paper 2011-060, Tilburg University, Center for Economic Research.
    3. Yaohua Hu & Carisa Kwok Wai Yu & Xiaoqi Yang, 2019. "Incremental quasi-subgradient methods for minimizing the sum of quasi-convex functions," Journal of Global Optimization, Springer, vol. 75(4), pages 1003-1028, December.
    4. Zhijun Xu & Jing Zhou, 2023. "A simultaneous diagonalization based SOCP relaxation for portfolio optimization with an orthogonality constraint," Computational Optimization and Applications, Springer, vol. 85(1), pages 247-261, May.
    5. Xinzhen Zhang & Chen Ling & Liqun Qi, 2011. "Semidefinite relaxation bounds for bi-quadratic optimization problems with quadratic constraints," Journal of Global Optimization, Springer, vol. 49(2), pages 293-311, February.
    6. Peiping Shen & Kaimin Wang & Ting Lu, 2020. "Outer space branch and bound algorithm for solving linear multiplicative programming problems," Journal of Global Optimization, Springer, vol. 78(3), pages 453-482, November.
    7. 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.
    8. Immanuel Bomze & Werner Schachinger & Gabriele Uchida, 2012. "Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization," Journal of Global Optimization, Springer, vol. 52(3), pages 423-445, March.
    9. Cheng Lu & Zhibin Deng & Jing Zhou & Xiaoling Guo, 2019. "A sensitive-eigenvector based global algorithm for quadratically constrained quadratic programming," Journal of Global Optimization, Springer, vol. 73(2), pages 371-388, February.
    10. Xiaodong Ding & Hezhi Luo & Huixian Wu & Jianzhen Liu, 2021. "An efficient global algorithm for worst-case linear optimization under uncertainties based on nonlinear semidefinite relaxation," Computational Optimization and Applications, Springer, vol. 80(1), pages 89-120, September.
    11. Jing Zhou & Shu-Cherng Fang & Wenxun Xing, 2017. "Conic approximation to quadratic optimization with linear complementarity constraints," Computational Optimization and Applications, Springer, vol. 66(1), pages 97-122, January.
    12. Li, Xiaobo & Natarajan, Karthik & Teo, Chung-Piaw & Zheng, Zhichao, 2014. "Distributionally robust mixed integer linear programs: Persistency models with applications," European Journal of Operational Research, Elsevier, vol. 233(3), pages 459-473.
    13. Immanuel Bomze & Markus Gabl, 2021. "Interplay of non-convex quadratically constrained problems with adjustable robust optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(1), pages 115-151, February.
    14. Jianzhe Zhen & Ahmadreza Marandi & Danique de Moor & Dick den Hertog & Lieven Vandenberghe, 2022. "Disjoint Bilinear Optimization: A Two-Stage Robust Optimization Perspective," INFORMS Journal on Computing, INFORMS, vol. 34(5), pages 2410-2427, September.
    15. M. Locatelli, 2023. "KKT-based primal-dual exactness conditions for the Shor relaxation," Journal of Global Optimization, Springer, vol. 86(2), pages 285-301, June.
    16. Amir Beck & Dror Pan, 2017. "A branch and bound algorithm for nonconvex quadratic optimization with ball and linear constraints," Journal of Global Optimization, Springer, vol. 69(2), pages 309-342, October.
    17. Marandi, Ahmadreza, 2017. "Aspects of quadratic optimization - nonconvexity, uncertainty, and applications," Other publications TiSEM d2b9c576-7128-4ee4-939a-7, Tilburg University, School of Economics and Management.
    18. Satoshi Suzuki, 2021. "Karush–Kuhn–Tucker type optimality condition for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential," Journal of Global Optimization, Springer, vol. 79(1), pages 191-202, January.
    19. Hu, Yaohua & Li, Gongnong & Yu, Carisa Kwok Wai & Yip, Tsz Leung, 2022. "Quasi-convex feasibility problems: Subgradient methods and convergence rates," European Journal of Operational Research, Elsevier, vol. 298(1), pages 45-58.
    20. Rujun Jiang & Duan Li, 2019. "Second order cone constrained convex relaxations for nonconvex quadratically constrained quadratic programming," Journal of Global Optimization, Springer, vol. 75(2), pages 461-494, October.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jglopt:v:71:y:2018:i:2:d:10.1007_s10898-018-0617-2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.