Section: New Results

Stochastic optimization

Stochastic control

Participant : Frédéric Bonnans.

With J. Gianatti (U. Rosario) and F. Silva (U. Limoges) we obtained an extension of the Sakawa-Shindo algorithm (for computing a solution of the optimality system of a deterministic optimal control problem) to stochastic control problems. The paper is in progress.

Stochastic programming

Participants : Frédéric Bonnans, Nicolas Grebille, Faisal Wahid.

In the framework of the thesis of Nicolas Grébille, we continued our study of decomposition algorithms for a stochastic model of optimal electricity energy production.The energy production is divided in a number of zones. The idea is to constrain the energy flows between these zones, by linear feedback to the demand (wich is a random variable). The coefficients of the feedback are to be optimized. Then the problem is decomposed for each zone (and can then be solved easily by a SDDP type algorithm). We obtained encouraging preliminary numerical results in a three zones problem.

Faisal Wahid developed a mixed integer program model for hydro-power producers participating in the future intra-day French Electricity Balancing Market. He has also formulated the mixed integer stochastic dynamic program model for the more general hydro- bidding under uncertainty. The objective of this model is to produce optimal offer policies in the form of supply curves under a time inhomogeneous Markov process of electricity market clearing prices.

Dynamic programming and error estimates for stochastic control problems with maximum cost

Participants : Athena Picarelli, Hasnaa Zidani.

The paper [14] is concerned with stochastic optimal control for a running maximum cost. A direct approach based on dynamic programming techniques is studied leading to the characterization of the value function as the unique viscosity solution of a second order Hamilton- Jacobi-Bellman (HJB) equation with an oblique derivative boundary condition. A general numerical scheme is proposed and a convergence result is provided. Error estimates are obtained for the semi-Lagrangian scheme. These results can apply to the case of lookback options in finance. Moreover, optimal control problems with maximum cost arise in the characterization of the reachable sets for a system of controlled stochastic differential equations. Some numerical simulations on examples of reachable analysis are included to illustrate our approach.