Year: 2023
Type: preprint
Abstract: We formulate and investigate a mean field optimization (MFO) problem over a set of probability distributions $μ$ with a prescribed marginal $m$. The cost function depends on an aggregate term, which is 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 (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.
Source: arXiv (Cornell University)
Authors: Liu, Kang, Pfeiffer, Laurent
Institutions: Centre National de la Recherche Scientifique, Institut national de recherche en sciences et technologies du numérique, École Polytechnique, Université Paris-Saclay, Laboratoire des signaux et systèmes, Centre de Mathématiques Appliquées de l'École polytechnique, CentraleSupélec, Institut Polytechnique de Paris, DISCO: Systèmes dynamiques interconnectés dans des environnements complexes
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4388184371