Duality and separation theorems in idempotent semimodules. (English) Zbl 1042.46004

Summary: 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.


46A20 Duality theory for topological vector spaces
06F07 Quantales
46A55 Convex sets in topological linear spaces; Choquet theory
Full Text: DOI arXiv


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