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Exploiting Sparsity in Complex Polynomial Optimization

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Abstract

In this paper, we study the sparsity-adapted complex moment-Hermitian sum of squares (moment-HSOS) hierarchy for complex polynomial optimization problems, where the sparsity includes correlative sparsity and term sparsity. We compare the strengths of the sparsity-adapted complex moment-HSOS hierarchy with the sparsity-adapted real moment-SOS hierarchy on either randomly generated complex polynomial optimization problems or the AC optimal power flow problem. The results of numerical experiments show that the sparsity-adapted complex moment-HSOS hierarchy provides a trade-off between the computational cost and the quality of obtained bounds for large-scale complex polynomial optimization problems.

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Notes

  1. By a chord, we means an edge that joins two nonconsecutive nodes in a cycle.

  2. Even if CPOP (1.1) is not a QCQP, this operation could also strengthen the relaxation.

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Acknowledgements

Both authors were supported by the Tremplin ERC Stg Grant ANR-18-ERC2-0004-01 (T-COPS project). The second author was supported by the FMJH Program PGMO (EPICS project), as well as the PEPS2 Program (FastOPF project) funded by AMIES and RTE. This work has benefited from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Actions, Grant Agreement 813211 (POEMA) as well as from the AI Interdisciplinary Institute ANITI funding, through the French “Investing for the Future PIA3” program under the Grant agreement n\(^{\circ }\)ANR-19-PI3A-0004.

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Authors and Affiliations

  1. Academy of Mathematics and Systems Science, CAS, Beijing, China

    Jie Wang

  2. Laboratory for Analysis and Architecture of Systems, CNRS, Toulouse, France

    Victor Magron

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  1. Jie Wang
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  2. Victor Magron
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Correspondence to Victor Magron.

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Communicated by Vaithilingam Jeyakumar.

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Wang, J., Magron, V. Exploiting Sparsity in Complex Polynomial Optimization. J Optim Theory Appl 192, 335–359 (2022). https://doi.org/10.1007/s10957-021-01975-z

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  • DOI: https://doi.org/10.1007/s10957-021-01975-z

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