The tropicalization of non-archimedean convex semialgebraic sets and its relation with mean payoff games

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 tropicalizations 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 semialgebraic 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, especially:

 

[1] X. Allamigeon, P. Benchimol, S. Gaubert, and M. Joswig,

What Tropical Geometry Tells Us about the Complexity of Linear Programming, SIAM review, 63(1), 123--164 (2021).

[2] X. Allamigeon, S. Gaubert, and M. Skomra. Tropical spectrahedra, Discrete Comput. Geom., 63, 507--548 (2020)

 

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