A Fast Iterative PDE-Based Algorithm for Feedback Controls of Nonsmooth Mean-Field Control Problems

SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2737-A2773, August 2024.
Abstract. We propose a PDE-based accelerated gradient algorithm for optimal feedback controls of McKean–Vlasov dynamics that involve mean-field interactions both in the state and action. The method exploits a forward-backward splitting approach and iteratively refines the approximate controls based on the gradients of smooth costs, the proximal maps of nonsmooth costs, and dynamically updated momentum parameters. At each step, the state dynamics is approximated via a particle system, and the required gradient is evaluated through a coupled system of nonlocal linear PDEs. The latter is solved by finite difference approximation or neural network-based residual approximation, depending on the state dimension. We present exhaustive numerical experiments for low- and high-dimensional mean-field control problems, including sparse stabilization of stochastic Cucker–Smale models, which reveal that our algorithm captures important structures of the optimal feedback control and achieves a robust performance with respect to parameter perturbation.