Decentralized resolution of finite-state, non-convex, and aggregative optimal control problems

Kang Liu (1, 2, 3) , Nadia Oudjane (4) , Laurent Pfeiffer (2)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 DISCO - Dynamical Interconnected Systems in COmplex Environments
3 L2S - Laboratoire des signaux et systèmes
4 EDF R&D OSIRIS - Optimisation, Simulation, Risque et Statistiques pour les Marchés de l’Energie

A general class of large-scale, nonconvex, and non-smooth optimization problems is introduced. It has the form of a multi-agent problem, where the agents interact through an aggregative term. A convex relaxation of the problem is provided together with an estimate of the relaxation gap. A numerical method, called stochastic Frank-Wolfe algorithm, is presented. The method allows to find approximate solutions of the original problem in a decomposed fashion. The convergence of the method is guaranteed from a theoretical point of view. An aggregative deterministic optimal control problem is formulated, with discrete state-space and discrete time. It is shown that the stochastic Frank-Wolfe algorithm can be applied to the optimal control problem; in particular, it amounts to solve at each iteration a series of small-scale optimal control problems, corresponding to each agent. These sub-problems are solved by dynamic programming. Numerical results are presented, for a toy model of the charging management of a battery fleet.

Domaines: Optimisation et contrôle [math.OC]

Kang Liu, Nadia Oudjane, Laurent Pfeiffer. Decentralized resolution of finite-state, non-convex, and aggregative optimal control problems. 2022. ⟨hal-03642127v1⟩