Mean field optimization problems: stability results and Lagrangian discretization

¹ Université Bourgogne Europe, CNRS, Institut de Mathématiques de Bourgogne, 21000, Dijon, France
² Université Paris-Saclay, CNRS, CentraleSupélec, Inria, Laboratoire des signaux et systémes, 91190, Gif-sur-Yvette, France

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 10 January 2025
Accepted: 1 April 2026

Abstract

We formulate and investigate a convex optimization problem defined on a set of probability measures μ with prescribed marginal m, which we call Mean Field Optimization (MFO) problem. The cost function depends on an aggregate term, defined as the expectation of μ with respect to a contribution function. This problem is of particular interest in the context of Lagrangian potential mean field games and their discretization. We provide a first-order optimality condition and prove strong duality. We investigate stability properties of the MFO problem with respect to the prescribed marginal, from both primal and dual perspectives. In our stability analysis, we propose a method for recovering an approximate solution to an MFO problem with the help of an approximate solution to an MFO with a different marginal m, typically an empirical distribution. We combine this method with the stochastic Frank–Wolfe algorithm of [Bonnans et al. SIAM J. Optim. 33 (2023) 3083–3113] to derive a complete resolution method.