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Robust linear algebra

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  • Bertsimas, Dimitris
  • Koukouvinos, Thodoris

Abstract

We propose a robust optimization (RO) framework that immunizes some of the central linear algebra problems in the presence of data uncertainty. Namely, we formulate linear systems, matrix inversion, eigenvalues–eigenvectors and matrix factorization under uncertainty, as robust optimization problems using appropriate descriptions of uncertainty. The resulting optimization problems are computationally tractable and scalable. We show in theory that RO improves the relative error of the linear system by reducing the condition number of the underlying matrix. Moreover, we provide empirical evidence showing that the proposed approach outperforms state of the art methods for linear systems and matrix inversion, when applied on ill-conditioned matrices. We show that computing eigenvalues–eigenvectors under RO, corresponds to solving linear systems that are better conditioned than the nominal and illustrate with numerical experiments that the proposed approach is more accurate than the nominal, when perturbing ill-conditioned matrices. Finally, we demonstrate empirically the benefit of the robust Cholesky factorization over the nominal.

Suggested Citation

  • Bertsimas, Dimitris & Koukouvinos, Thodoris, 2024. "Robust linear algebra," European Journal of Operational Research, Elsevier, vol. 314(3), pages 1174-1184.
  • Handle: RePEc:eee:ejores:v:314:y:2024:i:3:p:1174-1184
    DOI: 10.1016/j.ejor.2023.11.036
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    References listed on IDEAS

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    1. Bertsimas, Dimitris & Copenhaver, Martin S., 2018. "Characterization of the equivalence of robustification and regularization in linear and matrix regression," European Journal of Operational Research, Elsevier, vol. 270(3), pages 931-942.
    2. Zhen, Jianzhe & den Hertog, Dick, 2017. "Centered solutions for uncertain linear equations," Other publications TiSEM e625bf4d-40d2-4a94-be66-6, Tilburg University, School of Economics and Management.
    3. Stefania Corsaro & Marina Marino, 2006. "Interval linear systems: the state of the art," Computational Statistics, Springer, vol. 21(2), pages 365-384, June.
    4. Jianzhe Zhen & Dick Hertog, 2017. "Centered solutions for uncertain linear equations," Computational Management Science, Springer, vol. 14(4), pages 585-610, October.
    5. Ming Yuan & Yi Lin, 2007. "Model selection and estimation in the Gaussian graphical model," Biometrika, Biometrika Trust, vol. 94(1), pages 19-35.
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