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Tropical polyhedra have been recently used to represent disjunctive invariants in static analysis. To handle larger instances, the tropical analogues of classical linear programming results need to be developed. This motivation leads us to study a general tropical linear 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 combinatorialy. In other words, strategies play in tropical linear programming the role of Lagrange multipliers in classical linear programming. We use this idea to design a Newton-like algorithm to solve tropical linear programming problems, by reduction to a sequence of auxiliary mean payoff game problems.

Tropical linear programming and parametric mean payoff games