  Tropical polyhedra have been recently used to represent disjunctive
invariants in static analysis. To handle larger instances, tropical analogues
of classical linear programming results need to be developed. This motivation
leads us to study the tropical analogue of the classical linear-fractional
programming problem. We construct an associated parametric mean payoff game
problem, and show that the optimality of a given point, or the unboundedness of
the problem, can be certified by exhibiting a strategy for one of the players
having certain infinitesimal properties (involving the value of the game and
its derivative) that we characterize combinatorially. We use this idea to
design a Newton-like algorithm to solve tropical linear-fractional programming
problems, by reduction to a sequence of auxiliary mean payoff game problems.
