Mean field optimization problems: stability results and Lagrangian discretization

TL;DR

This paper studies a mean field optimization problem involving probability distributions, establishing optimality conditions, duality, and stability, and proposes a method for approximating solutions using discretization and stochastic algorithms.

Contribution

It introduces a first-order optimality condition, proves strong duality, and develops a stability analysis and solution recovery method for mean field optimization problems, especially in the context of mean field games.

Findings

Established first-order optimality conditions for MFO.

Proved strong duality in the MFO framework.

Developed a stability analysis and solution recovery method using approximate marginals.

Abstract

We formulate and investigate a mean field optimization (MFO) problem over a set of probability distributions $\mu$ with a prescribed marginal $m$. The cost function depends on an aggregate term, which is the expectation of $\mu$ with respect to a contribution function. This problem is of particular interest in the context of Lagrangian potential mean field games (MFGs) 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 a previous publication of ours to derive a complete resolution method.