## Section: New Results

### Stochastic programming and stochastic control

#### Convergence results on Howard's algorithm

Participants : O. Bokanowski, S. Maroso, H. Zidani.

In [17] , we have studied some convergence results of Howard's algorithm
for the resolution of , where
B^{a} is a matrix, c^{a} is a vector, and is a compact set.
We show a global superlinear convergence result, under a
monotonicity assumption on the matrices B^{a} . Extensions of
Howard's algorithm for a max-min problem of the form
are also
proposed. In the particular case of an obstacle problem of the
form min(Ax-b, x-g) = 0 , where A is an N×N matrix
satisfying a monotonicity assumption, we show the convergence of
Howard's algorithm in no more than N iterations, instead of the
usual 2^{N} bound. Still in the case of an obstacle problem, we
establish the equivalence between Howard's algorithm and a
primal-dual active set algorithm [M. Hintermüller, K. Ito, and
K. Kunisch, *SIAM J. Optim.* , 13 (2002), pp. 865–888]. The
algorithms are illustrated on the discretization of nonlinear PDEs
arising in the context of mathematical finance (American option
and Merton's portfolio problem), of front propagation problems,
and for the double-obstacle problem.

#### Numerical methods for solving swing options

Participants : F. Bonnans, H. Zidani, N. Touzi [ CMAP ] , M. Mnif [ LAMSIN,ENIT ] , I. Ben Latifa [ LAMSIN,ENIT ] .

We started this year a projet on the numerical methods for solving the variational inequalities for the second order HJB equations, and the application to swing options in a model with jumps.

#### Natural Liquefied Gas trading

Participants : F. Bonnans, Z. Cen, T. Christel [ Total Gaz ] .

We study a multi-stage stochastic optimization problem where randomness is only present in the objective function. After building a Markov chain by using vectorial quantization tree method, we rely on the dual dynamic programming method (DDP) to solve the optimization problem. The combination of these 2 methods presents the capacity of dealing with high-dimension state variables problem. Finally, some numerical tests applied to energy markets have been performed, which show that the method provides good convergence rate. The report is published in [32] .

#### Hybrid vehicles

We started a project with Renault aiming at developing mathematical tools to model and simulate scenarios allowing to improve the performances of electric car engines. Let us recall that the hard point of the electric technology lies on the restricted kilometric autonomy related to a weak density energy of the battery and a weak speed of its refill in energy (8 hours for the slow load). The goal of this project consists in:

* Maximizing the absolute autonomy (hybrid technology)

* Minimizing the autonomy variability: Optimal management of the energy by taking into account the available informations of navigation

These problems can be described by stochastic dynamic optimization problems. The objective of our project is twofold:

* Theoretical and numerical study of related stochastic optimization problems

* Validation on the model associated to the problem of hybrid car engine.