IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v204y2025i3d10.1007_s10957-024-02593-1.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-024-02593-1
    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/s10957-024-02593-1?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

    for a different version of it.

    References listed on IDEAS

    as
    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.
    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. Bomze, Immanuel M. & Gabl, Markus & Maggioni, Francesca & Pflug, Georg Ch., 2022. "Two-stage stochastic standard quadratic optimization," European Journal of Operational Research, Elsevier, vol. 299(1), pages 21-34.
    2. Lili Pan & Ziyan Luo & Naihua Xiu, 2017. "Restricted Robinson Constraint Qualification and Optimality for Cardinality-Constrained Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 175(1), pages 104-118, October.
    3. Dimitris Bertsimas & Ryan Cory-Wright, 2022. "A Scalable Algorithm for Sparse Portfolio Selection," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1489-1511, May.
    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. Adrian Gepp & Geoff Harris & Bruce Vanstone, 2020. "Financial applications of semidefinite programming: a review and call for interdisciplinary research," Accounting and Finance, Accounting and Finance Association of Australia and New Zealand, vol. 60(4), pages 3527-3555, December.
    6. Immanuel M. Bomze & Werner Schachinger, 2020. "Constructing Patterns of (Many) ESSs Under Support Size Control," Dynamic Games and Applications, Springer, vol. 10(3), pages 618-640, September.
    7. Carrizosa, Emilio & Ramírez-Ayerbe, Jasone & Romero Morales, Dolores, 2024. "Mathematical optimization modelling for group counterfactual explanations," European Journal of Operational Research, Elsevier, vol. 319(2), pages 399-412.
    8. Martin Branda & Max Bucher & Michal Červinka & Alexandra Schwartz, 2018. "Convergence of a Scholtes-type regularization method for cardinality-constrained optimization problems with an application in sparse robust portfolio optimization," Computational Optimization and Applications, Springer, vol. 70(2), pages 503-530, June.
    9. 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.
    10. Papp, Dávid & Regős, Krisztina & Domokos, Gábor & Bozóki, Sándor, 2023. "The smallest mono-unstable convex polyhedron with point masses has 8 faces and 11 vertices," European Journal of Operational Research, Elsevier, vol. 310(2), pages 511-517.
    11. Justin A. Sirignano & Gerry Tsoukalas & Kay Giesecke, 2016. "Large-Scale Loan Portfolio Selection," Operations Research, INFORMS, vol. 64(6), pages 1239-1255, December.
    12. Xiaojin Zheng & Xiaoling Sun & Duan Li & Jie Sun, 2014. "Successive convex approximations to cardinality-constrained convex programs: a piecewise-linear DC approach," Computational Optimization and Applications, Springer, vol. 59(1), pages 379-397, October.
    13. Samim Ghamami & Paul Glasserman, 2019. "Submodular Risk Allocation," Management Science, INFORMS, vol. 65(10), pages 4656-4675, October.
    14. Jiawei Chen & Huasheng Su & Xiaoqing Ou & Yibing Lv, 2024. "First- and second-order optimality conditions of nonsmooth sparsity multiobjective optimization via variational analysis," Journal of Global Optimization, Springer, vol. 89(2), pages 303-325, June.
    15. Janusz Miroforidis, 2021. "Bounds on efficient outcomes for large-scale cardinality-constrained Markowitz problems," Journal of Global Optimization, Springer, vol. 80(3), pages 617-634, July.
    16. Tahereh Khodamoradi & Maziar Salahi & Ali Reza Najafi, 2021. "Cardinality-constrained portfolio optimization with short selling and risk-neutral interest rate," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 44(1), pages 197-214, June.
    17. Sarat Moka & Matias Quiroz & Vali Asimit & Samuel Muller, 2025. "A Scalable Gradient-Based Optimization Framework for Sparse Minimum-Variance Portfolio Selection," Papers 2505.10099, arXiv.org.
    18. Mei Choi Chiu & Chi Seng Pun & Hoi Ying Wong, 2017. "Big Data Challenges of High‐Dimensional Continuous‐Time Mean‐Variance Portfolio Selection and a Remedy," Risk Analysis, John Wiley & Sons, vol. 37(8), pages 1532-1549, August.
    19. Hyunglip Bae & Haeun Jeon & Minsu Park & Yongjae Lee & Woo Chang Kim, 2025. "A Cholesky decomposition-based asset selection heuristic for sparse tangent portfolio optimization," Papers 2502.11701, arXiv.org.
    20. Caihua Chen & Xindan Li & Caleb Tolman & Suyang Wang & Yinyu Ye, 2013. "Sparse Portfolio Selection via Quasi-Norm Regularization," Papers 1312.6350, arXiv.org.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    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:joptap:v:204:y:2025:i:3:d:10.1007_s10957-024-02593-1. 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.