Multiorder, Kleene stars and cyclic projectors in the geometry of max cones

Abstract

Max cones are the subsets of the nonnegative orthant R+ of the n-dimensional real space Rn closed under scalar multiplication and componentwise maximisation. Their study is motivated by some practical applications which arise in discrete event systems, optimal scheduling and modelling of synchronization problems in multiprocessor interactive systems. We investigate the geometry of max cones, concerning the role of the multiorder principle, the Kleene stars, and the cyclic projectors. The multiorder principle is closely related to the set covering conditions in max algebra, and gives rise to important analogues of some theorems of convex geometry. We show that, in particular, this principle leads to a convenient representation of certain nonlinear projectors onto max cones. The Kleene stars are fundamental in max algebra since they accumulate weights of optimal paths and yield generators for max-algebraic eigenspaces of matrices. We examine the role of their column spans called Kleene cones, as building blocks in the Develin-Sturmfels cellular decomposition. Further we show that the cellular decomposition gives rise to new max-algebraic objects which we call row and column Kleene stars. We relate these objects to the maxalgebraic pseudoinverses of matrices and to tropical versions of the colourful Caratheodory theorem. The cyclic projectors are specific nonlinear operators which lead to the so-called alternating method for finding a solution to homogeneous two-sided systems of max-linear equations. We generalize the alternating method to the case of homogeneous multi-sided systems, and we give a proof, which uses the cellular decomposition idea, that the alternating method converges in a finite number of iterations to a positive solution of a multi-sided system if a positive solution exists. We also present new bounds on the number of iterations of the alternating method, expressed in terms of the Hilbert projective distance between max cones. 2000 Mathematics Subject Classification. Primary: 15A48, 52B11; Secondary: 52A20.