Abstract
Semimodules over idempotent semirings like the max-plus or tropical semiring have much in common with convex cones. This analogy is particularly apparent in the case of subsemimodules of the n-fold Cartesian product of the max-plus semiring: It is known that one can separate a vector from a closed subsemimodule that does not contain it. Here we establish a more general separation theorem, which applies to any finite collection of closed subsemimodules with a trivial intersection. The proof of this theorem involves specific nonlinear operators, called here cyclic projectors on idempotent semimodules. These are analogues of the cyclic nearest-point projections known in convex analysis. We obtain a theorem that characterizes the spectrum of cyclic projectors on idempotent semimodules in terms of a suitable extension of Hilbert's projective metric. We also deduce as a corollary of our main results the idempotent analogue of Helly's theorem.
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References
F. L. Baccelli, G. Cohen, G. J. Olsder, and J. P. Quadrat, Synchronization and Linearity, Wiley, New York (1992).
H. H. Bauschke, J. M. Borwein, and A. S. Lewis, “The method of cyclic projections for closed convex sets in Hilbert space,” in: Y. Censor and S. Reich, eds., Recent Developments in Optimization Theory and Nonlinear Analysis, Contemp. Math., Vol. 204, Amer. Math. Soc., Providence (1997), pp. 1–42.
G. Birkhoff, Lattice Theory, Amer. Math. Soc., Providence (1967).
T. S. Blyth and M. F. Janowitz, Residuation Theory, Pergamon Press (1972).
P. Butkovič, H. Schneider, and S. Sergeev, “Generators, extremals and bases of max cones,” Linear Algebra Appl., 421, 394–406 (2007), arXiv:math.RA/0604454.
G. Cohen, S. Gaubert, and J. P. Quadrat, “Duality and separation theorems in idempotent semimodules,” Linear Algebra Appl., 379, 395–422 (2004), arXiv:math.FA/0212294.
G. Cohen, S. Gaubert, J. P. Quadrat, and I. Singer, “Max-plus convex sets and functions,” in: G. Litvinov and V. Maslov, eds., Idempotent Mathematics and Mathematical Physics, Contemp. Math., Vol. 377, Amer. Math. Soc., Providence (2005), pp. 105–129, arXiv:math.FA/0308166.
R. A. Cuninghame-Green, “Projections in minimax algebra,” Math. Programming, 10, No. 1, 111–123 (1976).
R. A. Cuninghame-Green, Minimax Algebra, Lect. Notes Economics Math. Systems, Vol. 166, Springer, Berlin (1979).
R. A. Cuninghame-Green and P. Butkovič, “The equation A ⊗ x = B ⊗ y over (max, +),” Theor. Comput. Sci., 293, 3–12 (2003).
M. Develin and B. Sturmfels, “Tropical convexity,” Documenta Math., 9, 1–27 (2004), arXiv:math.MG/0308254.
H. G. Eggleston, Convexity, Cambridge Univ. Press (1958).
S. Gaubert and R. Katz, “The Minkowski theorem for max-plus convex sets,” Linear Algebra Appl., 421, 356–369 (2007), arXiv:math.GM/0605078.
S. Gaubert and F. Meunier, private communication (2006).
J. Golan, Semirings and Their Applications, Kluwer, Dordrecht (2000).
V. N. Kolokoltsov and V. P. Maslov, Idempotent Analysis and Applications, Kluwer, Dordrecht (1997).
G. L. Litvinov, “Maslov dequantization, idempotent and tropical mathematics: A brief introduction,” J. Math. Sci., 140, No. 3, 426–444 (2007).
G. Litvinov and V. Maslov, eds., Idempotent Mathematics and Mathematical Physics, Contemp. Math., Vol. 377, Amer. Math. Soc., Providence (2005).
G. L. Litvinov, V. P. Maslov, and G. B. Shpiz, “Idempotent functional analysis. An algebraic approach,” Math. Notes, 69, No. (5), 696–729 (2001), arXiv:math.FA/0009128.
V. Nitica and I. Singer, “The structure of max-plus hyperplanes,” Linear Algebra Appl., 426, 382–414 (2007).
R. D. Nussbaum, “Convexity and log convexity for the spectral radius,” Linear Algebra Appl., 73, 59–122 (1986).
R. T. Rockafellar, Convex Analysis, Princeton Univ. Press (1970).
S. N. Samborskii and G. B. Shpiz, “Convex sets in the semimodule of bounded functions,” in: V. P. Maslov and S. N. Samborskii, eds., Idempotent Analysis, Adv. Sov. Math., Vol. 13, Amer. Math. Soc., Providence (1992), pp. 135–137.
K. Zimmermann, “A general separation theorem in extremal algebras,” Ekonomicko-Matematický Obzor, 13, No. 2, 179–201 (1977).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 4, pp. 31–52, 2007.
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Gaubert, S., Sergeev, S. Cyclic projectors and separation theorems in idempotent convex geometry. J Math Sci 155, 815–829 (2008). https://doi.org/10.1007/s10958-008-9243-8
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DOI: https://doi.org/10.1007/s10958-008-9243-8