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Interplay of non-convex quadratically constrained problems with adjustable robust optimization

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  • Immanuel Bomze

    (University of Vienna)

  • Markus Gabl

    (University of Vienna)

Abstract

In this paper we explore convex reformulation strategies for non-convex quadratically constrained optimization problems (QCQPs). First we investigate such reformulations using Pataki’s rank theorem iteratively. We show that the result can be used in conjunction with conic optimization duality in order to obtain a geometric condition for the S-procedure to be exact. Based upon known results on the S-procedure, this approach allows for some insight into the geometry of the joint numerical range of the quadratic forms. Then we investigate a reformulation strategy introduced in recent literature for bilinear optimization problems which is based on adjustable robust optimization theory. We show that, via a similar strategy, one can leverage exact reformulation results of QCQPs in order to derive lower bounds for more complicated quadratic optimization problems. Finally, we investigate the use of reformulation strategies in order to derive characterizations of set-copositive matrix cones. Empirical evidence based upon first numerical experiments shows encouraging results.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:mathme:v:93:y:2021:i:1:d:10.1007_s00186-020-00726-6
    DOI: 10.1007/s00186-020-00726-6
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    References listed on IDEAS

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    1. Yanıkoğlu, İhsan & Gorissen, Bram L. & den Hertog, Dick, 2019. "A survey of adjustable robust optimization," European Journal of Operational Research, Elsevier, vol. 277(3), pages 799-813.
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    Cited by:

    1. Markus Gabl, 2023. "Sparse conic reformulation of structured QCQPs based on copositive optimization with applications in stochastic optimization," Journal of Global Optimization, Springer, vol. 87(1), pages 221-254, September.
    2. 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.

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