A Fenchel dual gradient method enabling regularization for nonsmooth distributed optimization over time-varying networks

In this paper, we develop a regularized Fenchel dual gradient method (RFDGM), which allows nodes in a time-varying undirected network to find a common decision, in a fully distributed fashion, for minimizing the sum of their local objective functions subject to their local constraints. Different from most existing distributed optimization algorithms that also cope with time-varying networks, RFDGM is able to handle problems with general convex objective functions and distinct local constraints, and still has non-asymptotic conver-gence results. Specifically, under a standard network connectivity condition, we show that RFDGM is guaranteed to reach e-accuracy in both optimality and feasibility within O(1/e(2) ln 1/e) iterations. Such iteration complexity can be improved to O(1/e ln 1/e) if the local objective functions are strongly convex but not necessarily differentiable. Finally, simulation results demonstrate the competence of RFDGM in practice.