  Considering the constrained stochastic optimization problem over a
time-varying random network, where the agents are to collectively minimize a
sum of objective functions subject to a common constraint set, we investigate
asymptotic properties of a distributed algorithm based on dual averaging of
gradients. Different from most existing works on distributed dual averaging
algorithms that mainly concentrating on their non-asymptotic properties, we not
only prove almost sure convergence and the rate of almost sure convergence, but
also asymptotic normality and asymptotic efficiency of the algorithm. Firstly,
for general constrained convex optimization problem distributed over a random
network, we prove that almost sure consensus can be archived and the estimates
of agents converge to the same optimal point. For the case of linear
constrained convex optimization, we show that the mirror map of the averaged
dual sequence identifies the active constraints of the optimal solution with
probability 1, which helps us to prove the almost sure convergence rate and
then establish asymptotic normality of the algorithm. Furthermore, we also
verify that the algorithm is asymptotically optimal. To the best of our
knowledge, it seems to be the first asymptotic normality result for constrained
distributed optimization algorithms. Finally, a numerical example is provided
to justify the theoretical analysis.
