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Duality and Separation Theorems in Idempotent Semimodules

Abstract : We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce nonlinear projection on subsemimod- ules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to separate a point from a convex set. We also show that the projection minimizes the analogue of Hilbert's projective metric. We develop more generally a theory of dual pairs for idempotent semimodules. We obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings. We illustrate the results by showing polyhedra and half-spaces over the max-plus semiring.
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https://hal.inria.fr/inria-00071917
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Submitted on : Tuesday, May 23, 2006 - 7:14:32 PM
Last modification on : Friday, February 4, 2022 - 3:10:13 AM
Long-term archiving on: : Sunday, April 4, 2010 - 10:44:26 PM

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  • HAL Id : inria-00071917, version 1

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Guy Cohen, Stéphane Gaubert, Jean-Pierre Quadrat. Duality and Separation Theorems in Idempotent Semimodules. [Research Report] RR-4668, INRIA. 2002. ⟨inria-00071917⟩

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