Accelerated Projection Algorithm Based on Smoothing Approximation for Distributed Nonsmooth Optimization

You Zhao, Xiaofeng Liao, Xing He

Abstract:

In this article, a distributed smoothing accelerated projection algorithm (DSAPA) is proposed to address constrained nonsmooth convex optimization problems over undirected multiagent networks in a distributed
manner, where the objective function is free of the assumption of the Lipschitz gradient or strong convexity. First,
based on a distributed exact penalty method, the original
optimization problem is translated to a problem of standard
assignment without consensus constraints. Then, a novel
DSAPA by combining the smoothing approximation with
Nesterov’s accelerated schemes, is proposed. In addition,
we provide a systematic analysis to derive an upper bound
on the convergence rate in terms of the objective function based on penalty function and to choose the optimal
step size accordingly. Our results demonstrate that the
proposed DSAPA can reach O(log(k)/k) when the optimal
step size is chosen. Finally, the effectiveness and correctness of the proposed algorithm are verified by numerical
and practical application examples.