## Section: Research Program

### Tropical methods applied to optimization, perturbation theory and matrix analysis

Tropical algebraic objects appear as a deformation of classical objects thought various asymptotic procedures. A familiar example is the rule of asymptotic calculus,

${e}^{-a/\u03f5}+{e}^{-b/\u03f5}\asymp {e}^{-min(a,b)/\u03f5}\phantom{\rule{5.0pt}{0ex}},\phantom{\rule{2.em}{0ex}}{e}^{-a/\u03f5}\times {e}^{-b/\u03f5}={e}^{-(a+b)/\u03f5}\phantom{\rule{5.0pt}{0ex}},$ | (1) |

when $\u03f5\to {0}^{+}$. Deformations of this kind have been studied in different contexts: large deviations, zero-temperature limits, Maslov's “dequantization method” [97], non-archimedean valuations, log-limit sets and Viro's patchworking method [116], etc.

This entails a relation between classical algorithmic problems and tropical algorithmic problems, one may first solve the $\u03f5=0$ case (non-archimedean problem), which is sometimes easier, and then use the information gotten in this way to solve the $\u03f5=1$ (archimedean) case.

In particular, tropicalization establishes a connection between polynomial systems and piecewise affine systems that are somehow similar to the ones arising in game problems. It allows one to transfer results from the world of combinatorics to “classical” equations solving. We investigate the consequences of this correspondence on complexity and numerical issues. For instance, combinatorial problems can be solved in a robust way. Hence, situations in which the tropicalization is faithful lead to improved algorithms for classical problems. In particular, scalings for the polynomial eigenproblems based on tropical preprocessings have started to be used in matrix analysis [85], [88].

Moreover, the tropical approach has been recently applied to construct examples of linear programs in which the central path has an unexpectedly high total curvature [61], and it has also led to positive polynomial-time average case results concerning the complexity of mean payoff games. Similarly, we are studying semidefinite programming over non-archimedean fields [49], [29], with the goal to better understand complexity issues in classical semidefinite and semi-algebraic programming.