Submitted to “ Dynamics and Stability of Systems ” , appears as HP Laboratories Technical Report HPL - BRIMS - 97 - 13 , August 1997

HPL-BRIMS-97-13, August 1997 Dynamics of min-max functions Jean Cochet-Terrasson BPRE, Etat Major de l'Arm ee de l'Air, 24 Bd. Victor, 75015 Paris, France St ephane Gaubert INRIA, Domaine de Voluceau, 78153 Le Chesnay Cedex, France [email protected] Jeremy Gunawardena (corresponding author) BRIMS, Hewlett-Packard Labs, Filton Road, Stoke Gi ord, Bristol BS12 6QZ, UK Tel : + 44 117 922 8216 Fax : +44 117 922 9190 Email : jhcg@hplb:hpl:hp:com June 25, 1997 Abstract Functions F : Rn ! Rn which are nonexpansive in the `1 norm and homogeneous, Fi(x1+ h; ; xn+ h) = Fi(x1; ; xn)+ h, (so-called topical functions) have appeared recently in the work of several authors. They include (after suitable transformation) nonnegative matrices, Leontie substitution systems, Bellman operators of games and of Markov decisions processes, examples arising from discrete event systems (digital circuits, computer networks, etc) and the min-max functions studied in this paper. Any topical function F can be approximated by min-max functions in a way which preserves some of the dynamics of F . We attempt, therefore, to clarify the dynamics of min-max functions, with a view to developing a generalised Perron-Frobenius theory for topical functions. Our main concern is with the existence of generalised xed points, where F (x1; ; xn) = (x1+h; ; xn+h), which correspond to nonlinear eigenvectors, and with the cycle time vector, (F ) = limk!1 F k(~x)=k 2 Rn, which generalises the spectral radius. The Duality Conjecture for min-max functions asserts that this limit always exists, shows how it can be calculated and implies the strong result that F has a xed point if, and only if, (F ) = (h; ; h). We use an analogue of Howard's policy improvement scheme from optimal control to prove a xed point theorem of equivalent strength, which is independent of the Duality Conjecture, and we give a proof of the Conjecture itself for n = 2.