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Polynomial-time computation of the joint spectral radius for some sets of nonnegative matrices

Author

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  • BLONDEL, Vincent

    (UNIVERSITE CATHOLIQUE DE LOUVAIN, Division of Applied Mathematics)

  • NESTEROV, Yu

    (Université catholique de Louvain (UCL). Center for Operations Research and Econometrics (CORE))

Abstract

We propose two simple upper bounds for the joint spectral radius of sets of nonnegative matrices. These bounds, the joint column radius and the joint row radius, can be computed in polynomial time as solutions of convex optimization problems. We show that for general matrices these bounds are within a factor 1/n of the exact value, where n is the size of the matrices. Moreover, for sets of matrices with independent column uncertainties of with independent row uncertainties, the corresponding bounds coincide with the joint spectral radius. In these cases, the joint spectral radius is also given by the largest spectral radius of the matrices in the set. As a byproduct of these results, we propose a polynomial-time technique for solving Boolean optimization problems related to the spectral radius. We also consider economics and engineering applications of our results which were never considered practice due to their intrinsic computational complexity.

Suggested Citation

  • BLONDEL, Vincent & NESTEROV, Yu, 2008. "Polynomial-time computation of the joint spectral radius for some sets of nonnegative matrices," CORE Discussion Papers 2008034, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  • Handle: RePEc:cor:louvco:2008034
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    File URL: http://www.uclouvain.be/cps/ucl/doc/core/documents/dp2008_34.pdf
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    More about this item

    Keywords

    joint spectral radius; joint column radius; joint row radius; nonnegative matrices; asynchronous systems; convex optimization; Leontief model.;

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