HP Labs Technical Reports
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The Duality Theorem for Min-Max Functions
Gaubert, Stephanie; Gunawardena, Jeremy
Keyword(s): cycle time; generalized eigenvector; nonexpansive map; policy improvement
Abstract: The set of min-max functions F : Rn Rn is the least set containing coordinate substitutions and translations and closed under pointwise max, min, and function composition. The Duality Conjecture asserts that the trajectories of a min-max function, considered as a dynamical system, have a linear growth rate (cycle time) and shows how this can be calculated through a representation of F as an infimum of max- plus linear functions. We prove the conjecture using an analogue of Howard's policy improvement scheme, carried out in a lattice ordered group of germs of affine functions at infinity. The methods yield an efficient algorithm for computing cycle times.
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