The duality theorem for min-max functions

The set of min-max functions F : ℝ″ → ℝ″ 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 maxplus 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.