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Centered solutions for uncertain linear equations

Author

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  • Jianzhe Zhen

    (Tilburg University)

  • Dick Hertog

    (Tilburg University)

Abstract

Our contribution is twofold. Firstly, for a system of uncertain linear equations where the uncertainties are column-wise and reside in general convex sets, we derive convex representations for united and tolerable solution sets. Secondly, to obtain centered solutions for uncertain linear equations, we develop a new method based on adjustable robust optimization (ARO) techniques to compute the maximum size inscribed convex body (MCB) of the set of the solutions. In general, the obtained MCB is an inner approximation of the solution set, and its center is a potential solution to the system. We use recent results from ARO to characterize for which convex bodies the obtained MCB is optimal. We compare our method both theoretically and numerically with an existing method that minimizes the worst-case violation. Applications to the input–output model, Colley’s Matrix Rankings and Article Influence Scores demonstrate the advantages of the new method.

Suggested Citation

  • Jianzhe Zhen & Dick Hertog, 2017. "Centered solutions for uncertain linear equations," Computational Management Science, Springer, vol. 14(4), pages 585-610, October.
    • Handle: RePEc:spr:comgts:v:14:y:2017:i:4:d:10.1007_s10287-017-0290-9
    DOI: 10.1007/s10287-017-0290-9
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    References listed on IDEAS

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    1. Robert L. Smith, 1984. "Efficient Monte Carlo Procedures for Generating Points Uniformly Distributed over Bounded Regions," Operations Research, INFORMS, vol. 32(6), pages 1296-1308, December.
    2. Hladík, Milan & Popova, Evgenija D., 2015. "Maximal inner boxes in parametric AE-solution sets with linear shape," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 606-619.
    3. Zhen, J. & den Hertog, D., 2015. "Computing the Maximum Volume Inscribed Ellipsoid of a Polytopic Projection," Discussion Paper 2015-004, Tilburg University, Center for Economic Research.
    4. Zhen, J. & den Hertog, D., 2015. "Computing the Maximum Volume Inscribed Ellipsoid of a Polytopic Projection," Other publications TiSEM 4a51526e-c7f0-436f-aeaa-f, Tilburg University, School of Economics and Management.
    5. Bram L. Gorissen & Hans Blanc & Dick den Hertog & Aharon Ben-Tal, 2014. "Technical Note---Deriving Robust and Globalized Robust Solutions of Uncertain Linear Programs with General Convex Uncertainty Sets," Operations Research, INFORMS, vol. 62(3), pages 672-679, June.
    6. Blanc, J.P.C. & den Hertog, D., 2008. "On Markov Chains with Uncertain Data," Discussion Paper 2008-50, Tilburg University, Center for Economic Research.
    7. Burer Samuel, 2012. "Robust Rankings for College Football," Journal of Quantitative Analysis in Sports, De Gruyter, vol. 8(2), pages 1-22, June.
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    Cited by:

    1. Bertsimas, Dimitris & Koukouvinos, Thodoris, 2024. "Robust linear algebra," European Journal of Operational Research, Elsevier, vol. 314(3), pages 1174-1184.

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