10 problems

Logarithmic ansatz conjecture. The optimal value function is conjectured to have the form

Logarithmic-factor removal conjecture. The logarithmic factor can be removed, namely there exists a constant $C$, depending only on $T$, $H$, $L_F$, and $L_G$, such that for every…

Let a potential mean field game be one for which the optimizers of an additional mean field control problem are also solutions to the mean field game, although some mean field game…

Optimal convergence-rate conjecture. The optimal rate in the $\bm{d}_1$-regular case should be

HJB–SMP connection conjecture. The two different primal-dual formulations are connected: the Lions derivative of the HJB dual optimizer, evaluated along the optimal state process,…

Let $\mathcal{V}(t,\mu)$ be the value function and let $\mathcal{V}_1(t,\mu)$ and $\mathcal{V}_2(t,\mu)$ be the expressions appearing in the two control regimes. Let $\Pi_3$ denote…

Let $N$ agents have exchangeable dynamics with weak coupling, and let $x^{(N)}$ and $u^{(N)}$ denote the relevant aggregate state and control quantities. Write $\mathbb{E}_0$ for e…

Conjectured $A_n$ relations. The entries of the matrix $A_n$ satisfy the same equations as those of $B_n$, namely

The discussion concerns gradient methods with feedback controls for mean-field control problems, including Algorithm $\mathrm{NAG}$, whose iterates are feedback controls and whose…

Let $(X(s),Y(s))$ be the state and adjoint components of the mean-field forward-backward stochastic differential equation associated with the optimal control problem. Let…