[Paper Review] A variational approach to the mean field planning problem

Carlo Orrieri, Alessio Porretta+1 more|arXiv (Cornell University)|Jul 25, 2018

Geometric Analysis and Curvature Flows12 references4 citations

TL;DR

This paper introduces a variational framework for the first-order mean field planning problem using convex analysis and optimal transport theory. By formulating the system as a duality between a primal entropy-transport problem and a dual Hamilton-Jacobi optimization, the authors establish existence and partial uniqueness of weak solutions via contact-defect measures and renormalized solutions, with a Lagrangian characterization through dynamic plans on path space.

ABSTRACT

We investigate a first-order mean field planning problem of the form \begin{equation} \left\lbrace\begin{aligned} -\partial_t u + H(x,Du) &= f(x,m) && ext{in } (0,T) imes \mathbb{R}^d, \\ \partial_t m - abla\cdot (m\,H_p(x,Du)) &= 0 && ext{in }(0,T) imes \mathbb{R}^d,\\ m(0,\cdot) = m_0, \; m(T,\cdot) &= m_T && ext{in } \mathbb{R}^d, \end{aligned} ight. \end{equation} associated to a convex Hamiltonian $H$ with quadratic growth and a monotone interaction term $f$ with polynomial growth. We exploit the variational structure of the system, which encodes the first order optimality condition of a convex dynamic optimal entropy-transport problem with respect to the unknown density $m$ and of its dual, involving the maximization of an integral functional among all the subsolutions $u$ of an Hamilton-Jacobi equation. Combining ideas from optimal transport, convex analysis and renormalized solutions to the continuity equation, we will prove existence and (at least partial) uniqueness of a weak solution $(m,u)$. A crucial step of our approach relies on a careful analysis of distributional subsolutions to Hamilton-Jacobi equations of the form $-\partial_t u + H(x,Du) \leq α$, under minimal summability conditions on $α$, and to a measure-theoretic description of the optimality via a suitable contact-defect measure. Finally, using the superposition principle, we are able to describe the solution to the system by means of a measure on the path space encoding the local behavior of the players.

Motivation & Objective

• To establish a variational structure for the first-order mean field planning problem with quadratic Hamiltonian and monotone interaction.
• To prove existence and partial uniqueness of weak solutions using duality and convex analysis in the context of optimal transport.
• To characterize solutions via a Lagrangian viewpoint using measures on the space of continuous curves.
• To analyze weak subsolutions to Hamilton-Jacobi equations under minimal summability assumptions on the source term.
• To introduce a contact-defect measure to describe optimality conditions in the variational framework.

Proposed method

• Formulate the mean field planning system as the first-order optimality condition of a convex dynamic optimal entropy-transport problem.
• Use the duality between the primal problem (minimizing entropy-transport cost) and the dual problem (maximizing an integral functional over subsolutions of the HJ equation).
• Employ renormalized solutions to the continuity equation to handle low-regularity densities and fluxes.
• Introduce a contact-defect measure to capture the discrepancy between the subsolution and the Hamilton-Jacobi equation at optimality.
• Apply the superposition principle to lift solutions from the Eulerian to the Lagrangian framework, describing the solution as a measure on the path space.
• Use anisotropic convolution and weighted $L^p$ spaces to handle convergence in measure and stability of subsolutions.

Experimental results

Research questions

• RQ1 Can the mean field planning problem with a first-order Hamiltonian and terminal constraints be reformulated as a convex variational problem?
• RQ2 How can weak subsolutions to Hamilton-Jacobi equations be analyzed under minimal integrability assumptions on the source term?
• RQ3 What is the role of the contact-defect measure in characterizing optimality in the variational framework?
• RQ4 How can the solution to the mean field planning system be represented in a Lagrangian setting using dynamic plans?
• RQ5 Under what conditions does the duality between the primal and dual problems yield existence and partial uniqueness of solutions?

Key findings

• Existence of a weak solution $(m, u)$ to the mean field planning system is established under convex Hamiltonian with quadratic growth and monotone $f$ with polynomial growth.
• The solution arises as the minimizer of a convex dynamic optimal transport problem with entropy regularization.
• The dual problem maximizes an integral functional over subsolutions of the Hamilton-Jacobi equation, with optimality characterized by a contact-defect measure.
• The contact-defect measure captures the failure of the subsolution to satisfy the HJ equation as a measure, enabling a precise duality description.
• A Lagrangian representation of the solution is achieved via the superposition principle, linking the solution to a probability measure on the space of continuous curves.
• Stability and convergence results for weak subsolutions are proven under minimal summability conditions on the right-hand side of the HJ inequality.