  Two distributed algorithms are described that enable all users connected over
a network to cooperatively solve the problem of minimizing the sum of all
users' objective functions over the intersection of all users' constraint sets,
where each user has its own private nonsmooth convex objective function and
closed convex constraint set, which is the intersection of a number of simple,
closed convex sets. One algorithm enables each user to adjust its estimate by
using a proximity operator of its objective function and the metric projection
onto one set randomly selected from the simple, closed convex sets. The other
is a distributed random projection algorithm that determines each user's
estimate by using a subgradient of its objective function instead of the
proximity operator. Investigation of the two algorithms' convergence properties
for a diminishing step-size rule revealed that, under certain assumptions, the
sequences of all users generated by each of the two algorithms converge almost
surely to the same solution. Moreover, convergence rate analysis of the two
algorithms is provided, and desired choices of the step size sequences such
that the two algorithms have fast convergence are discussed. Numerical
comparisons for concrete nonsmooth convex optimization support the convergence
analysis and demonstrate the effectiveness of the two algorithms.
