Distributed Fenchel Dual Gradient Methods Enabling a Smoothing Technique for Nonsmooth Optimization

In this paper, we develop a class of distributed Fenchel dual gradient methods that enable a smoothing technique in order to solve nonsmooth convex optimization over networks with time-varying topologies, where the nodes are required to find a global optimal decision that minimizes the sum of their own objectives subject to their individual constraints. Specifically, we first apply a smoothing technique to the Fenchel dual of the problem, so that a strongly convex and smooth approximation of the Fenchel dual function can be obtained. We then adopt a family of weighted gradient methods to solve such a smoothed Fenchel dual problem, which can be implemented over time-varying networks in a decentralized fashion. Under a standard network connectivity condition, we derive a linear rate of convergence to the optimal value of the smoothed Fenchel dual problem for the proposed algorithms. Based on this result, we further show that an approximate primal solution reaches ε-accuracy in optimality and feasibility of the original problem within O([1/(ε
²
)] ln[1/ε]) iterations.