This work introduces the StoMADS-PB algorithm for constrained stochastic
blackbox optimization, which is an extension of the mesh adaptive direct-search
(MADS) method originally developed for deterministic blackbox optimization
under general constraints. The values of the objective and constraint functions
are provided by a noisy blackbox, i.e., they can only be computed with random
noise whose distribution is unknown. As in MADS, constraint violations are
aggregated into a single constraint violation function. Since all functions
values are numerically unavailable, StoMADS-PB uses estimates and introduces
so-called probabilistic bounds for the violation. Such estimates and bounds
obtained from stochastic observations are required to be accurate and reliable
with high but fixed probabilities. The proposed method, which allows
intermediate infeasible iterates, accepts new points using sufficient decrease
conditions and imposing a threshold on the probabilistic bounds. Using Clarke
nonsmooth calculus and martingale theory, Clarke stationarity convergence
results for the objective and the violation function are derived with
probability one