We present a distributed algorithm to solve a multi-agent optimization problem, where the global objective function is the sum $n$ convex objective functions. Our focus is on constrained problems where the agents' estimates are restricted to be in different convex sets. The interconnection topology among the $n$ agents has directed links and each agent $i$ can only communicate with agents in its neighborhood determined by a directed graph. In this article, we propose an algorithm called \underline{D}irected \underline{C}onstrained-\underline{Dist}ributed \underline{A}lternating \underline{D}irection \underline{M}ethod of \underline{M}ultipliers (DC-DistADMM) to solve the above multi-agent convex optimization problem. During every iteration of the DC-DistADMM algorithm, each agent solves a local convex optimization problem and utilizes a finite-time "approximate" consensus protocol to update its local estimate of the optimal solution. To the best of our knowledge the proposed algorithm is the first ADMM based algorithm to solve distributed multi-agent optimization problems in directed interconnection topologies with convergence guarantees. We show that in case of individual functions being convex and not-necessarily differentiable the proposed DC-DistADMM algorithm converges at a rate of $O(1/k)$, where $k$ is the iteration counter. We further establish a linear rate of convergence for the DC-DistADMM algorithm when the global objective function is strongly convex and smooth. We numerically evaluate our proposed algorithm by solving a constrained distributed $\ell_1$-regularized logistic regression problem. Additionally, we provide a numerical comparison of the proposed DC-DistADMM algorithm with the other state-of-the-art algorithms in solving a distributed least squares problem to show the efficacy of the DC-DistADMM algorithm over the existing methods in the literature.

中文翻译：

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DC-DistADMM：用于有向图上的约束分布式优化的 ADMM 算法

我们提出了一种分布式算法来解决多智能体优化问题，其中全局目标函数是和 $n$ 凸目标函数。我们的重点是受约束的问题，其中代理的估计被限制在不同的凸集中。$n$个代理之间的互连拓扑具有有向链接，每个代理$i$只能与由有向​​图确定的邻域内的代理通信。在这篇文章中，我们提出了一种称为\underline{D}irected \underline{C}onstrained-\underline{Dist}ribted \underline{A}lternating \underline{D}irected \underline{}方法的\underline{ M}ultipliers (DC-DistADMM) 来解决上述多智能体凸优化问题。在 DC-DistADMM 算法的每次迭代中，每个代理解决一个局部凸优化问题，并利用有限时间的“近似”共识协议来更新其对最优解的局部估计。据我们所知，所提出的算法是第一个基于 ADMM 的算法，用于解决具有收敛保证的定向互连拓扑中的分布式多代理优化问题。我们表明，在单个函数是凸的且不一定可微的情况下，所提出的 DC-DistADMM 算法以 $O(1/k)$ 的速率收敛，其中 $k$ 是迭代计数器。当全局目标函数是强凸和平滑时，我们进一步为 DC-DistADMM 算法建立了线性收敛速度。我们通过解决约束分布式 $\ell_1$ 正则化逻辑回归问题来对我们提出的算法进行数值评估。此外，我们提供了所提出的 DC-DistADMM 算法与解决分布式最小二乘问题的其他最先进算法的数值比较，以展示 DC-DistADMM 算法相对于文献中现有方法的功效。