  In this paper, we present a stochastic augmented Lagrangian approach on
(possibly infinite-dimensional) Riemannian manifolds to solve stochastic
optimization problems with a finite number of deterministic constraints.We
investigate the convergence of the method, which is based on a stochastic
approximation approach with random stopping combined with an iterative
procedure for updating Lagrange multipliers. The algorithm is applied to a
multi-shape optimization problem with geometric constraints and demonstrated
numerically.
