  Convergence of a projected stochastic gradient algorithm is demonstrated for
convex objective functionals with convex constraint sets in Hilbert spaces. In
the convex case, the sequence of iterates ${u_n}$ converges weakly to a point
in the set of minimizers with probability one. In the strongly convex case, the
sequence converges strongly to the unique optimum with probability one. An
application to a class of PDE constrained problems with a convex objective,
convex constraint and random elliptic PDE constraints is shown. Theoretical
results are demonstrated numerically.
