Notice
Tropical convexity: application to linear programming and mean payoff games
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Descriptif
Linear programming, and more generally convex semialgebraic programming, makes sense over any ordered nonarchimedean field, like a field of real Puiseux series. Nonarchimedean instances encode classical parametric instances of convex programs with a suitably large parameter. Tropical geometry allows one to study such instances by combinatorial means. In particular, it reveals that, under a genericity condition, solving a nonarchimedean feasibility problem is equivalent to deciding who the winner is in a mean payoff game. Indeed, nonarchimedean linear programs correspond to deterministic mean payoff games, whereas nonarchimedean semidefinite programs correspond to perfect information stochastic mean payoff games. In this way, one can apply methods from convex programming to mean payoff games, and vice versa. This approach has led to several results: a counter example, with a family of linear programs, with large coefficients, for which interior point methods have a non strongly polynomial behavior (they make a number of iterations exponential in the number of constraints and variables); a theorem transferring complexity results concerning pivoting methods in linear programming to mean payoff games;
We will present here a general overview of these results, discussing tools from tropical geometry used in this approach.
This is based on a series of work, with M. Akian et A. Guterman, and then with X. Allamigeon, P. Benchimol, M. Joswig, M. Skomra, and N. Vandame
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