  We consider the control of McKean-Vlasov dynamics (or mean-field control)
with probabilistic state constraints. We rely on a level-set approach which
provides a representation of the constrained problem in terms of an
unconstrained one with exact penalization and running maximum or integral cost.
The method is then extended to the common noise setting. Our work extends
(Bokanowski, Picarelli, and Zidani, SIAM J. Control Optim. 54.5 (2016), pp.
2568--2593) and (Bokanowski, Picarelli, and Zidani, Appl. Math. Optim. 71
(2015), pp. 125--163) to a mean-field setting. The reformulation as an
unconstrained problem is particularly suitable for the numerical resolution of
the problem, that is achieved from an extension of a machine learning algorithm
from (Carmona, Lauri{\`e}re, arXiv:1908.01613 to appear in Ann. Appl. Prob.,
2019). A first application concerns the storage of renewable electricity in the
presence of mean-field price impact and another one focuses on a mean-variance
portfolio selection problem with probabilistic constraints on the wealth. We
also illustrate our approach for a direct numerical resolution of the primal
Markowitz continuous-time problem without relying on duality.
