Augmented Lagrangian Methods for Transport Optimization, Mean Field Games and Degenerate Elliptic Equations

Abstract

Many problems from mass transport can be reformulated as variational problems under a prescribed divergence constraint (static problems) or subject to a time-dependent continuity equation, which again can be formulated as a divergence constraint but in time and space. The variational class of mean field games, introduced by Lasry and Lions, may also be interpreted as a generalization of the time-dependent optimal transport problem. Following Benamou and Brenier, we show that augmented Lagrangian methods are well suited to treat such convex but non-smooth problems. They include in particular Monge historic optimal transport problem. A finite-element discretization and implementation of the method are used to provide numerical simulations and a convergence study.

Appendix : Details of Step 2 for MFGs

Let $\gamma >0$ and $\rho _1\ge 0$ be given, then define

$$\begin{aligned} N_2(\rho ):=:=\left\{ \begin{array}{ll} \frac{\gamma }{2} ( \rho -\rho _1)^2, &{}\quad \hbox {if } \rho \ge 0, \\ +\infty , &{}\quad \hbox {otherwise} \end{array}\right. , \, N_{1}(\rho ):=\left\{ \begin{array}{ll} \gamma \vert \rho -\rho _1\vert ,&{} \quad \hbox {if }\rho \ge 0, \\ +\infty , &{}\quad \hbox {otherwise.} \end{array}\right. \end{aligned}$$

For $N=N_1, N_2$, we have to compute: $N^*$ as well as the two proximal operators:

• Terminal prox, given $c_0$, solve:

  $$\begin{aligned} \inf _{c} \left\{ N^*(c)+\frac{r}{2} \vert c-c_0\vert ^2\right\} . \end{aligned}$$

  (45)

• Quadratic Hamiltonian prox (recall that we have taken for simplicity $H(p)=\frac{1}{2} \vert p\vert ^2$ in the MFG), given $(a_0,b_0)\in \mathbb {R}\times \mathbb {R}^d$, solve

  $$\begin{aligned} \inf _{(a,b)\in \mathbb {R}\times \mathbb {R}^d} \left\{ N^*\left( a+\frac{1}{2} \vert b\vert ^2 \right) +\frac{r}{2}\left( \vert a-a_0\vert ^2 + \vert b-b_0\vert ^2 \right) \right\} . \end{aligned}$$

  (46)

1.1 Proximal computations for $N_2$:

The Legendre transform of $N_2$ is explicitly given by

$$\begin{aligned} N_2^*(\lambda ):=\left\{ \begin{array}{ll} \frac{\lambda ^2}{2\gamma }+\lambda \rho _1, &{}\quad \hbox {if }\lambda \ge -\gamma \rho _1, \\ -\gamma \frac{\rho _1^2}{2}, &{}\quad \hbox {otherwise.} \end{array}\right. \end{aligned}$$

In this case, the solution of the terminal proximal problem (45) is:

$$\begin{aligned} c=\left\{ \begin{array}{ll} c_0, &{}\quad \hbox {if }c_0\le -\gamma \rho _1, \\ \frac{ r c_0-\rho _1}{ r+\gamma ^{-1}}, &{}\quad \hbox {otherwise} \end{array}\right. \end{aligned}$$

(47)

Let us consider now the Hamiltonian-prox problem (46). It is convenient to formulate the optimality condition for (46) by setting

$$\begin{aligned} \lambda :=\left( a+\frac{1}{2} \vert b\vert ^2\right) ,\; r\mu :=\left( N_2^*\right) '(\lambda )=\left( N_2^*\right) ' \left( a+\frac{1}{2} \vert b\vert ^2\right) \end{aligned}$$

we then have

$$\begin{aligned} a=a_0-\mu , \; b=\frac{b_0}{1+\mu }. \end{aligned}$$

Defining $\lambda _0=a_0+\frac{1}{2} \vert b_0\vert ^2$, the optimal $(a,b)$ is given by

• Case 1: $\lambda \ge -\gamma \rho _1$ then $\mu$ has to be a (non-negative) root of the (cubic) equation

  $$\begin{aligned} r \mu =\rho _1+\frac{\lambda }{\gamma }= \rho _1+\frac{1}{\gamma } \left( a_0-\mu +\frac{1}{2} \frac{\vert b_0\vert ^2}{(1+\mu )^2}\right) \end{aligned}$$

  (48)

  and the solvability of this equation on $\mathbb {R}_+$ is equivalent to $\lambda _0\ge -\gamma \rho _1$.

• Case 2: $\lambda _0<-\gamma \rho _1$ then $(a,b)=(a_0, b_0)$.

1.2 Prox computations for $N_1$

The Legendre transform of $N_1$ is:

$$\begin{aligned} N_1^*(\lambda ):=\left\{ \begin{array}{ll} -\gamma \rho _1, &{}\quad \hbox {if }\lambda \le -\gamma , \\ \rho _1 \lambda ,&{}\quad \hbox {if } \lambda \in [ -\gamma , \gamma ]\\ +\infty , &{}\quad \hbox {otherwise}. \end{array}\right. \end{aligned}$$

Rewriting the proximal problem (45) as the inclusion $0\in (c-c_0)+\frac{1}{r} \partial N_1^*(c)$ and distinguishing the different possible cases for $\partial N_1^*(c)$, one finds

$$\begin{aligned} c=\left\{ \begin{array}{ll} c_0, &{}\quad \hbox {if } c_0<-\gamma \\ -\gamma ,&{}\quad \hbox {if } c_0\in [-\gamma , -\gamma +\frac{\rho _1}{r}], \\ c_0-\frac{\rho _1}{r}, &{}\quad \hbox {if } c_0\in ]-\gamma +\frac{\rho _1}{r}, \gamma +\frac{\rho _1}{r}[,\\ \gamma , &{}\quad \hbox {if } c_0 \ge \gamma +\frac{\rho _1}{r} . \end{array}\right. \end{aligned}$$

For the second problem (46) which corresponds to the conditions:

$$\begin{aligned} 0\in (a-a_0, b-b_0) +\frac{1}{ r} \partial N_1^*(\lambda )(1, b), \text { with } \lambda =a+\frac{1}{2} \vert b \vert ^2, \end{aligned}$$

we find as optimal $(a,b)$:

$$\begin{aligned} (a,b)\!=\!\left\{ \begin{array}{ll} (a_0, b_0),&{}\quad \hbox {if } \lambda _0 <-\gamma ,\\ (a(\mu ), b(\mu )), &{}\quad \hbox {with } \mu \!\in \! [0, \frac{\rho _1}{r}] \hbox { solving } (49) \hbox { (with } - \hbox {sign) if } \lambda _0^* \!\le \!-\gamma \!\le \! \lambda _0, \\ (a_0^*, b_0^*), &{}\quad \hbox {if } \lambda _0^*\in ]-\gamma , \gamma [,\\ (a(\mu ), b(\mu )),&{}\quad \hbox {with } \mu \ge \frac{\rho _1}{r} \hbox { solving } (49) \hbox {(with +sign) if }\lambda _0^*\ge \gamma , \end{array}\right. \end{aligned}$$

where we have defined

$$\begin{aligned} a_0^*:= & {} a_0-\frac{\rho _1}{r},\; b_0^*:=\frac{r b_0}{\rho _1+r}, \; \lambda _0^*:=a_0^*+\frac{1}{2} \vert b_0^*\vert ^2\nonumber \\ a(\mu ):= & {} (a_0-\mu ), \; b(\mu ):=\frac{b_0}{1+\mu },\nonumber \\ \pm \gamma= & {} (a_0-\mu )+\frac{1}{2} \frac{\vert b_0 \vert ^2}{(1+\mu )^2}. \end{aligned}$$

(49)