Convex sets can be defined over ordered fields with a non-archimedean valuation. Then, tropical convex sets arise as images by the valuation of non-archimedean convex sets. The tropicalization of polyhedra and spectrahedra are of special interest, since they can be described in terms of deterministic and stochastic games with mean payoff. In that way, one gets a correspondence between classes of zero- sum games, with an unsettled complexity, and classes of semilagebraic convex optimization problems over non-archimedean fields. We shall discuss applications of this correspondence, including a counter example concerning the complexity of interior point methods, and the fact that non-archimedean spectrahedra have precisely the same images by the valuation as convex semi-algebraic sets. This is based on works with Allamigeon, Benchimol, Joswig and Skomra.
De Paul-André Melliès
De Noam Zeilberger
De Vincent Rivasseau
De Joseph Ben Geloun
De Darij Grinberg
De Natalia Iyudu
De Dominique Manchon
De Richard Kerner
De Stéphane Gaubert
De Albert Schwarz
De Clément Delcamp
De Clément Delcamp
De Thibaut Le Gouic
De François-Xavier Vialard
De Elsa Cazelles
De Olga Mula Hernandez
