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On Tractable Convex Relaxations of Standard Quadratic Optimization Problems under Sparsity Constraints

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  • Immanuel Bomze

    (University of Vienna)

  • Bo Peng

    (University of Vienna)

  • Yuzhou Qiu

    (The University of Edinburgh)

  • E. Alper Yıldırım

    (The University of Edinburgh)

Abstract

Standard quadratic optimization problems (StQPs) provide a versatile modelling tool in various applications. In this paper, we consider StQPs with a hard sparsity constraint, referred to as sparse StQPs. We focus on various tractable convex relaxations of sparse StQPs arising from a mixed-binary quadratic formulation, namely, the linear optimization relaxation given by the reformulation–linearization technique, the Shor relaxation, and the relaxation resulting from their combination. We establish several structural properties of these relaxations in relation to the corresponding relaxations of StQPs without any sparsity constraints, and pay particular attention to the rank-one feasible solutions retained by these relaxations. We then utilize these relations to establish several results about the quality of the lower bounds arising from different relaxations. We also present several conditions that ensure the exactness of each relaxation.

Suggested Citation

  • Immanuel Bomze & Bo Peng & Yuzhou Qiu & E. Alper Yıldırım, 2025. "On Tractable Convex Relaxations of Standard Quadratic Optimization Problems under Sparsity Constraints," Journal of Optimization Theory and Applications, Springer, vol. 204(3), pages 1-36, March.
    • Handle: RePEc:spr:joptap:v:204:y:2025:i:3:d:10.1007_s10957-024-02593-1
    DOI: 10.1007/s10957-024-02593-1
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    References listed on IDEAS

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    1. Xin Chen & Jiming Peng, 2015. "New Analysis on Sparse Solutions to Random Standard Quadratic Optimization Problems and Extensions," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 725-738, March.
    2. Immanuel M. Bomze & Werner Schachinger & Reinhard Ullrich, 2018. "The Complexity of Simple Models—A Study of Worst and Typical Hard Cases for the Standard Quadratic Optimization Problem," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 651-674, May.
    3. Immanuel M. Bomze & Bo Peng, 2023. "Conic formulation of QPCCs applied to truly sparse QPs," Computational Optimization and Applications, Springer, vol. 84(3), pages 703-735, April.
    4. Jianjun Gao & Duan Li, 2013. "Optimal Cardinality Constrained Portfolio Selection," Operations Research, INFORMS, vol. 61(3), pages 745-761, June.
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