Random Sleep Scheme-Based Distributed Optimization Algorithm Over Unbalanced Time-Varying Networks

This article considers a category of constrained convex optimization problems over multiagent networks. The networked agents aim at collaboratively minimizing the sum of all locally known objective functions over a common convex set. Each agent possesses only its local convex function and its state is constrained to a privately known convex set. A novel distributed algorithm is proposed over time-varying unbalanced directed networks based on epigraph form of the original optimization problem and consensus theory. By incorporating the random sleep scheme, the proposed algorithm allows each agent to independently and randomly decide whether to calculate subgradient and take projection at each iteration, which alleviates the cost of subgradient observation. Besides, it neither resorts to doubly stochastic weight matrices (but only row-stochastic) nor the information of the graph sequence to execute. The convergence of the algorithm is explicitly analyzed under conditions that the sequence of time-varying directed graphs is uniformly jointly strongly connected and the subgradients of all local objective functions are bounded over a convex set. The optimization algorithm ensures zero-gap on the expected distance between the estimated value of each agent and the exact optimal solution. The two simulation cases are presented to demonstrate the practicability of the algorithm and correctness of the obtained theoretical results.