Tropical methods for solving stochastic control problems
Monotone discretizations of Hamilton-Jacobi-Bellman equations lead to the dynamic programming equation of a multi-stage stochastic optimization problem. We develop several lower complexity numerical algorithms for solving such equations by combining tropical or max-plus method, numerical probabilistic approach, and stochastic dual dynamic programming method. This talk is based on joint works with Jean-Philippe Chancelier, Benoit Tran and Eric Fodjo. See in particular arXiv:1709.09049, arXiv:1810.12870, arXiv:2010.10619.
Population protocols and probabilistic automata as linear dynamical systems
In this talk I will discuss two a priori different problems related to linear dynamical systems. The first is a well-known model for distributed systems and the second an important notion in program verification. I will show an unexpected connection between the two models, and present our current understanding for these problems, and conclude with a conjecture tying the two problems together.
On the complexity of semidefinite and hyperbolic programming
Semidefinite programming (SDP) can be seen as a special case of a more general conic optimization problem called hyperbolic programming (HP). Although by construction HP is less explicit than SDP, the question whether there is a difference between these two classes is still an open problem in the theory of hyperbolic polynomials (and it is conjectured that this is not the case). I will make an overview of some known results, mostly concerning complexity questions around SDP and HP and concerning the problem of computing determinantal representations.