The paper addresses large-scale, convex optimization problems that need to be solved in a distributed way by agents communicating according to a random time-varying graph. Specifically, the goal of the network is to minimize the sum of local costs, while satisfying local and coupling constraints. Agents communicate according to a time-varying model in which edges of an underlying connected graph are active at each iteration with certain non-uniform probabilities. By relying on a primal decomposition scheme applied to an equivalent problem reformulation, we propose a novel distributed algorithm in which agents negotiate a local allocation of the total resource only with neighbors with active communication links. The algorithm is studied as a subgradient method with block-wise updates, in which blocks correspond to the graph edges that are active at each iteration. Thanks to this analysis approach, we show almost sure convergence to the optimal cost of the original problem and almost sure asymptotic primal recovery without resorting to averaging mechanisms typically employed in dual decomposition schemes. Explicit sublinear convergence rates are provided under the assumption of diminishing and constant step-sizes. Finally, an extensive numerical study on a plug-in electric vehicle charging problem corroborates the theoretical results.

该论文解决了大规模的凸优化问题需要通过代理根据随机时变图进行通信以分布式方式解决。具体来说，网络的目标是最小化本地成本的总和，同时满足本地和耦合约束。代理根据时变模型进行通信，其中底层连接图的边在每次迭代时都处于活动状态，具有某些非均匀概率。通过依赖应用于等效问题重构的原始分解方案，我们提出了一种新颖的分布式算法，其中代理仅与具有活动通信链接的邻居协商总资源的本地分配。该算法被研究为具有逐块更新的次梯度方法，其中块对应于在每次迭代中处于活动状态的图边。由于这种分析方法，我们几乎可以肯定地收敛到原始问题的最优成本，并且几乎可以肯定地进行渐近原始恢复，而无需求助于通常在对偶分解方案中使用的平均机制。在递减和恒定步长的假设下提供显式次线性收敛速率。最后，对插电式电动汽车充电问题的广泛数值研究证实了理论结果。