arXiv - CS - Systems and Control Pub Date : 2020-03-30 , DOI: arxiv-2003.13742
Vivek Khatana; Murti V. Salapaka

We focus on the problem of minimizing a finite sum $f(x) = \sum_{i=1}^n f_i(x)$ of functions of $n$ functions $f_i$, where $f_i$ are convex and available only locally to an agent $i$. The $n$ agents are connected in a directed network $\mathcal{G}(\mathbf{V},\mathbf{E})$, where each agent $i$ can only communicate with agents in its neighborhood determined by $\mathcal{G}(V,E)$. In this article, we present the Directed-Distributed Alternating Direction Method of Multiplier (D-DistADMM) Algorithm, which is an Alternating Direction Method of Multiplier (ADMM) based scheme and utilizes a finite-time approximate'' consensus method to solve the above optimization problem distributively. At each iteration of the proposed scheme the agents solve their local optimization problem and utilize an approximate consensus protocol to update a local estimate of the global optimization variable. We show that for convex and not-necessarily differentiable objective functions $f_i$'s the proposed \textit{D-DistADMM} method converges at a rate $O(1/k)$. We further demonstrate the applicability of our algorithm by solving a distributed least-squares problem.

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