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Extended trust-region problems with one or two balls: exact copositive and Lagrangian relaxations

Author

Listed:
  • Immanuel M. Bomze

    (University of Vienna)

  • Vaithilingam Jeyakumar

    (University of New South Wales)

  • Guoyin Li

    (University of New South Wales)

Abstract

We establish a geometric condition guaranteeing exact copositive relaxation for the nonconvex quadratic optimization problem under two quadratic and several linear constraints, and present sufficient conditions for global optimality in terms of generalized Karush–Kuhn–Tucker multipliers. The copositive relaxation is tighter than the usual Lagrangian relaxation. We illustrate this by providing a whole class of quadratic optimization problems that enjoys exactness of copositive relaxation while the usual Lagrangian duality gap is infinite. Finally, we also provide verifiable conditions under which both the usual Lagrangian relaxation and the copositive relaxation are exact for an extended CDT (two-ball trust-region) problem. Importantly, the sufficient conditions can be verified by solving linear optimization problems.

Suggested Citation

  • Immanuel M. Bomze & Vaithilingam Jeyakumar & Guoyin Li, 2018. "Extended trust-region problems with one or two balls: exact copositive and Lagrangian relaxations," Journal of Global Optimization, Springer, vol. 71(3), pages 551-569, July.
    • Handle: RePEc:spr:jglopt:v:71:y:2018:i:3:d:10.1007_s10898-018-0607-4
    DOI: 10.1007/s10898-018-0607-4
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    References listed on IDEAS

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    1. Karthik Natarajan & Chung Piaw Teo & Zhichao Zheng, 2011. "Mixed 0-1 Linear Programs Under Objective Uncertainty: A Completely Positive Representation," Operations Research, INFORMS, vol. 59(3), pages 713-728, June.
    2. 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.
    3. Jean B. Lasserre & Kim-Chuan Toh & Shouguang Yang, 2017. "A bounded degree SOS hierarchy for polynomial optimization," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 5(1), pages 87-117, March.
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    Cited by:

    1. 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.
    2. Ana Maria A. C. Rocha & M. Fernanda P. Costa & Edite M. G. P. Fernandes, 2018. "Preface to the Special Issue “GOW’16”," Journal of Global Optimization, Springer, vol. 71(3), pages 441-442, July.
    3. Fabián Flores-Bazán & Giandomenico Mastroeni, 2022. "First- and Second-Order Optimality Conditions for Quadratically Constrained Quadratic Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 118-138, June.
    4. Nadav Hallak & Marc Teboulle, 2020. "Finding Second-Order Stationary Points in Constrained Minimization: A Feasible Direction Approach," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 480-503, August.

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