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New semidefinite relaxations for a class of complex quadratic programming problems

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Abstract

In this paper, we propose some new semidefinite relaxations for a class of nonconvex complex quadratic programming problems, which widely appear in the areas of signal processing and power system. By deriving new valid constraints to the matrix variables in the lifted space, we derive some enhanced semidefinite relaxations of the complex quadratic programming problems. Then, we compare the proposed semidefinite relaxations with existing ones, and show that the newly proposed semidefinite relaxations could be strictly tighter than the previous ones. Moreover, the proposed semidefinite relaxations can be applied to more general cases of complex quadratic programming problems, whereas the previous ones are only designed for special cases. Numerical results indicate that the proposed semidefinite relaxations not only provide tighter relaxation bounds, but also improve some existing approximation algorithms by finding better sub-optimal solutions.

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Notes

  1. In the literature, the term “gap" means the difference between the optimal value of an optimization problem and its lower/upper bound. Here we borrow the term “gap" for convenience, but the meaning is different.

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Authors and Affiliations

  1. School of Economics and Management, North China Electric Power University, Beijing, 102206, China

    Yingzhe Xu & Cheng Lu

  2. School of Economics and Management, University of Chinese Academy of Sciences, Beijing, 100190, China

    Zhibin Deng

  3. MOE Social Science Laboratory of Digital Economic Forecasts and Policy Simulation at UCAS, Beijing, 100190, China

    Zhibin Deng

  4. State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Ya-Feng Liu

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  1. Yingzhe Xu
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  2. Cheng Lu
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  3. Zhibin Deng
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  4. Ya-Feng Liu
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Correspondence to Cheng Lu.

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Lu’s research has been supported by National Natural Science Foundation of China Grant Nos. 12171151 and 11771243. Deng’s research has been supported by the National Natural Science Foundation of China Grant No. T2293774, by the Fundamental Research Funds for the Central Universities E2ET0808X2, and by the grant from MOE Social Science Laboratory of Digital Economic Forecast and Policy Simulation at UCAS. The work of Y.-F. Liu was supported in part by the National Natural Science Foundation of China Grant Nos. 11688101, 12021001 and 12022116.

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Xu, Y., Lu, C., Deng, Z. et al. New semidefinite relaxations for a class of complex quadratic programming problems. J Glob Optim 87, 255–275 (2023). https://doi.org/10.1007/s10898-023-01290-z

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  • DOI: https://doi.org/10.1007/s10898-023-01290-z

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