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Duality and separation theorems in idempotent semimodules

Guy Cohen·Stephane Gaubert·Jean-Pierre Quadrat
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Abstract

We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: 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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Guy Cohen, Stephane Gaubert, Jean-Pierre Quadrat. 2003-09-29. Duality and separation theorems in idempotent semimodules. https://doi.org/10.1016/j.laa.2003.08.010

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