Parallel Subgradient Method for Nonsmooth Convex Optimization with a Simple Constraint

In this paper, we consider the problem of minimizing the sum of nondifferentiable, convex functions over a closed convex set in a real Hilbert space, which is simple in the sense that the projection onto it can be easily calculated. We present a parallel subgradient method for solving it and the two convergence analyses of the method. One analysis shows that the parallel method with a small constant step size approximates a solution to the problem. The other analysis indicates that the parallel method with a diminishing step size converges to a solution to the problem in the sense of the weak topology of the Hilbert space. Finally, we numerically compare our method with the existing method and state future work on parallel subgradient methods.