A Fast Proximal Gradient Method and Convergence Analysis for Dynamic Mean Field Planning

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In this paper, we propose an efficient and flexible algorithm to solve dynamic mean-field planning problems based on an accelerated proximal gradient method.

• Besides an easy-to-implement gradient descent step in this algorithm, a crucial projection step becomes solving an elliptic equation whose solution can be obtained by conventional methods efficiently.
• By induction on iterations used in the algorithm, we theoretically show that the proposed discrete solution converges to the underlying continuous solution as the grid size increases.
• Furthermore, we generalize our algorithm to mean-field game problems and accelerate it using multilevel and multigrid strategies.