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The level set method for the two-sided max-plus eigenproblem

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Abstract

We consider the max-plus analogue of the eigenproblem for matrix pencils, A ⊗ x = λ ⊗ B ⊗ x. We show that the spectrum of (A,B) (i.e., the set of possible values of λ), which is a finite union of intervals, can be computed in pseudo-polynomial number of operations, by a (pseudo-polynomial) number of calls to an oracle that computes the value of a mean payoff game. The proof relies on the introduction of a spectral function, which we interpret in terms of the least Chebyshev distance between A ⊗ x and λ ⊗ B ⊗ x. The spectrum is obtained as the zero level set of this function.

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Acknowledgements

We thank Peter Butkovič and Hans Schneider for many useful discussions which have been at the origin of this work. We are also grateful to the referees for their careful reading and many useful remarks.

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Authors and Affiliations

  1. INRIA and Centre de Mathématiques Appliquées, École Polytechnique, CMAP, École Polytechnique, 91128, Palaiseau Cédex, France

    Stéphane Gaubert

  2. School of Mathematics, University of Birmingham, Edgbaston, B15 2TT, UK

    Sergeĭ Sergeev

Authors
  1. Stéphane Gaubert
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  2. Sergeĭ Sergeev
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Corresponding author

Correspondence to Sergeĭ Sergeev.

Additional information

The first author was partially supported by the Arpege programme of the French National Agency of Research (ANR), project “ASOPT”, number ANR-08-SEGI-005, by the Digiteo project DIM08 “PASO” number 3389, and by a LEA “Math Mode” grant for 2009–2010. The second author was supported by the EPSRC grant RRAH12809 and the RFBR grant 08-01-00601. He was with INRIA and Centre de Mathématiques Appliquées, École Polytechnique when the present (revised) version of this paper was prepared.

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Gaubert, S., Sergeev, S. The level set method for the two-sided max-plus eigenproblem. Discrete Event Dyn Syst 23, 105–134 (2013). https://doi.org/10.1007/s10626-012-0137-z

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