Proximal Methods for Stationary Mean Field Games with Local Couplings

Abstract

We address the numerical approximation of Mean Field Games with local couplings. For powerlike Hamiltonians, we consider both the stationary system introduced in [51,53] and also a similar system involving density constraints in order to model hard congestion effects [65,57]. For finite difference discretizations of the Mean Field Game system as in [3], we follow a variational approach. We prove that the aforementioned schemes can be obtained as the optimality system of suitably defined optimization problems. In order to prove the existence of solutions of the scheme with a variational argument, the monotonicity of the coupling term is not used, which allow us to recover general existence results proved in [3]. Next, assuming next that the coupling term is monotone, the variational problem is cast as a convex optimization problem for which we study and compare several proximal type methods. These algorithms have several interesting features, such as global convergence and stability with respect to the viscosity parameter, which can eventually be zero. We assess the performance of the methods via numerical experiments.

A Projected Primal–Dual Method for Solving Constrained Monotone Inclusions

“…(v) In the context of primal-dual problem (P 0 )-(D 0 ), (3.2) reduces to (3.42) and our conditions on the parameters coincide with [15,16]. Without strong convexity of f and g * , we deduce from Theorem 3.2((ii)) the weak convergence of the sequences generated by (3.42), generalizing results in [1,7,16]. When f or g * is strongly convex, Theorem 3.2((iii)) yields an accelerated and projected version of [1].…”

“…However, this is not always possible since, in several applications, the matrices involved are singular or very bad conditioned (see discussion in [7,17]). If it is difficult to compute P L −1 b but we can project onto R −1 c, we can rewrite (4.1) as the problem of findinĝ…”

“…In this context, the primal-dual methods proposed in [1,2,3,13,14,15,16] impose feasibility through Lagrange multiplier updates. A disadvantage of this approach is that such algorithms are usually slow and their primal iterates do not necessarily satisfy any of the constraints (see, e.g., [7]), leading to unfeasible approximate primal solutions. By projecting onto the affine subspace generated by the constraints, previous problem is solved.…”

Some remarks on mean field games

“…Uniqueness (under suitable monotonicity conditions) and existence have been proved for this system in [24,25]. Numerical methods are being developed, let us cite [1,5] for examples of this growing literature. Let us also mention the questions of long time average [9,8] or learning [6].…”

On the implementation of a primal-dual algorithm for second order time-dependent Mean Field Games with local couplings

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We study a numerical approximation of a time-dependent Mean Field Game (MFG) system with local couplings. The discretization we consider stems from a variational approach described in [14] for the stationary problem and leads to the finite difference scheme introduced by Achdou and Capuzzo-Dolcetta in [3]. In order to solve the finite dimensional variational problems, in [14] the authors implement the primal-dual algorithm introduced by Chambolle and Pock in [20], whose core consists in iteratively solving linear systems and applying a proximity operator.

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