Discrete potential mean field games: duality and numerical resolution

• Published in: Mathematical programming Vol. 202; no. 1-2; pp. 241 - 278
• Main Authors: Bonnans, J. Frédéric, Lavigne, Pierre, Pfeiffer, Laurent
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-11-2023; Springer; Springer Nature B.V; Springer Verlag
• Subjects: Calculus of Variations and Optimal Control; Optimization; Combinatorics; Full Length Paper; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Numerical methods; Optimal control; Optimization and Control; Theoretical; ADM-G; 91A16; 90C25; 49N80; Duality theory; 49N15; Dynamic programming; Mean field games; Kolmogorov equation; Primal-dual optimization; ADMM; 91A50; duality theory; dynamic programming; primal-dual optimization; primal-dual proximal methods
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-023-01934-8

We propose and investigate a general class of discrete time and finite state space mean field game (MFG) problems with potential structure. Our model incorporates interactions through a congestion term and a price variable. It also allows hard constraints on the distribution of the agents. We analyze the connection between the MFG problem and two optimal control problems in duality. We present two families of numerical methods and detail their implementation: (i) primal-dual proximal methods (and their extension with nonlinear proximity operators), (ii) the alternating direction method of multipliers (ADMM) and a variant called ADM-G. We give some convergence results. Numerical results are provided for two examples with hard constraints.