Almost sure convergence of random projected proximal and subgradient algorithms for distributed nonsmooth convex optimization

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 using the proximity operator of its objective function and the metric projection onto one constraint set randomly selected from a number of simple, closed convex sets. The other determines each user's estimate using the subdifferential 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. It also showed that the rate of convergence depends on the step size and that a smaller step size results in quicker convergence. The results of numerical evaluation using a nonsmooth convex optimization problem support the convergence analysis and demonstrate the effectiveness of the two algorithms.