I am a PhD student in optimization since september 2017 at the CERMICS, École des Ponts ParisTech and the CMAP, École Polytechnique under the supervision of Jean-Philippe Chancelier and Marianne Akian.
The dynamic programming approach applied to stochastic control problems allows one to find optimal feedback policies but it requires solving a
nonlinear equation or a fully nonlinear partial differential equation of Hamilton-Jacobi type over the state space.
Hence, this method suffers from the curse of dimensionality, for instance grid-based methods like finite-difference or finite element methods
have a complexity exponential in the dimension of the state space.
Several methods were created to bypass this difficulty by assuming some structure on the problem. Examples are the max-plus based method of McEneaney and the Stochastic Dual Dynamic Programming (SDDP) algorithm of Pereira and Pinto.
We aim to associate and compare these methods in order to solve more general structures, namely problems involving a finite set-valued (or switching) control and a continuum-valued control, knowing that the value function associated to a fixed switching strategy is convex.
With Marianne Akian and Jean-Philippe Chancelier, 31 pages, 5 figures. In this article, we build a common framework for both the SDDP algorithm and a discrete time version of Zheng Qu's min-plus algorithm to solve deterministic multistage optimization problems. We prove its convergence under mild technical assumptions.