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Exploiting term sparsity in noncommutative polynomial optimization

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  • Jie Wang

    (Laboratoire d’Analyse et d’Architecture des Systèmes (LAAS))

  • Victor Magron

    (Laboratoire d’Analyse et d’Architecture des Systèmes (LAAS))

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.

Suggested Citation

  • Jie Wang & Victor Magron, 2021. "Exploiting term sparsity in noncommutative polynomial optimization," Computational Optimization and Applications, Springer, vol. 80(2), pages 483-521, November.
    • Handle: RePEc:spr:coopap:v:80:y:2021:i:2:d:10.1007_s10589-021-00301-7
    DOI: 10.1007/s10589-021-00301-7
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    References listed on IDEAS

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    1. Laurent, M., 2009. "Sums of squares, moment matrices and optimization over polynomials," Other publications TiSEM 9fef820b-69d2-43f2-a501-e, Tilburg University, School of Economics and Management.
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    Cited by:

    1. Jie Wang & Victor Magron, 2022. "Exploiting Sparsity in Complex Polynomial Optimization," Journal of Optimization Theory and Applications, Springer, vol. 192(1), pages 335-359, January.
    2. Tong Chen & Jean-Bernard Lasserre & Victor Magron & Edouard Pauwels, 2022. "A sublevel moment-SOS hierarchy for polynomial optimization," Computational Optimization and Applications, Springer, vol. 81(1), pages 31-66, January.

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