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Conic formulation of QPCCs applied to truly sparse QPs

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  • Immanuel M. Bomze

    (University of Vienna)

  • Bo Peng

    (University of Vienna)

Abstract

We study (nonconvex) quadratic optimization problems with complementarity constraints, establishing an exact completely positive reformulation under—apparently new—mild conditions involving only the constraints, not the objective. Moreover, we also give the conditions for strong conic duality between the obtained completely positive problem and its dual. Our approach is based on purely continuous models which avoid any branching or use of large constants in implementation. An application to pursuing interpretable sparse solutions of quadratic optimization problems is shown to satisfy our settings, and therefore we link quadratic problems with an exact sparsity term $$\Vert {{\mathsf {x}}}\Vert _0$$ ‖ x ‖ 0 to copositive optimization. The covered problem class includes sparse least-squares regression under linear constraints, for instance. Numerical comparisons between our method and other approximations are reported from the perspective of the objective function value.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:84:y:2023:i:3:d:10.1007_s10589-022-00440-5
    DOI: 10.1007/s10589-022-00440-5
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    References listed on IDEAS

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    1. Pietro Belotti & Pierre Bonami & Matteo Fischetti & Andrea Lodi & Michele Monaci & Amaya Nogales-Gómez & Domenico Salvagnin, 2016. "On handling indicator constraints in mixed integer programming," Computational Optimization and Applications, Springer, vol. 65(3), pages 545-566, December.
    2. Tiago Andrade & Fabricio Oliveira & Silvio Hamacher & Andrew Eberhard, 2019. "Enhancing the normalized multiparametric disaggregation technique for mixed-integer quadratic programming," Journal of Global Optimization, Springer, vol. 73(4), pages 701-722, April.
    3. 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.
    4. Yue Xie & Uday V. Shanbhag, 2021. "Tractable ADMM schemes for computing KKT points and local minimizers for $$\ell _0$$ ℓ 0 -minimization problems," Computational Optimization and Applications, Springer, vol. 78(1), pages 43-85, January.
    5. Jiahan Li, 2015. "Sparse and Stable Portfolio Selection With Parameter Uncertainty," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 33(3), pages 381-392, July.
    6. Bomze, Immanuel M., 2012. "Copositive optimization – Recent developments and applications," European Journal of Operational Research, Elsevier, vol. 216(3), pages 509-520.
    7. 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.
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