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Exploiting term sparsity in noncommutative polynomial optimization

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Abstract

We provide a new hierarchy of semidefinite programming relaxations, called NCTSSOS, to solve large-scale sparse noncommutative polynomial optimization problems. This hierarchy features the exploitation of term sparsity hidden in the input data for eigenvalue and trace optimization problems. NCTSSOS complements the recent work that exploits correlative sparsity for noncommutative optimization problems by Klep et al. (MP, 2021), and is the noncommutative analogue of the TSSOS framework by Wang et al. (SIAMJO 31: 114–141, 2021, SIAMJO 31: 30–58, 2021). We also propose an extension exploiting simultaneously correlative and term sparsity, as done previously in the commutative case (Wang in CS-TSSOS: Correlative and term sparsity for large-scale polynomial optimization, 2020). Under certain conditions, we prove that the optima of the NCTSSOS hierarchy converge to the optimum of the corresponding dense semidefinite programming relaxation. We illustrate the efficiency and scalability of NCTSSOS by solving eigenvalue/trace optimization problems from the literature as well as randomly generated examples involving up to several thousand variables.

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Notes

  1. https://github.com/wangjie212/NCTSSOS.

  2. The polynomials are available at https://wangjie212.github.io/jiewang/code.html.

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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) and EDF, Thales, Orange et Criteo, as well as the PHC Proteus Program (QUANTPOP project \(\hbox {n}^{\circ }\)46195TA). This work has benefited from the European Union’s Horizon 2020 research and innovation programme 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 \(\hbox {n}^{\circ }\)ANR-19-PI3A-0004.

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  1. Laboratoire d’Analyse et d’Architecture des Systèmes (LAAS), Toulouse, France

    Jie Wang & Victor Magron

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  1. Jie Wang
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Correspondence to Victor Magron.

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Wang, J., Magron, V. Exploiting term sparsity in noncommutative polynomial optimization. Comput Optim Appl 80, 483–521 (2021). https://doi.org/10.1007/s10589-021-00301-7

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