Asynchronous Algorithms for Decentralized Resource Allocation Over Directed Networks

In this article, we consider a class of decentralized resource allocation problems over directed networks, where each node only communicates with its in-neighbors and attempts to minimize its own cost when network-wide resource constraints as well as local capacity limits are satisfied. Decentralized optimization to solve this problem has been a significant focus within engineering research due to its advantages in scalability, robustness, and flexibility. Most existing methods are synchronous while few works are devoted to asynchronously solving the problem. The problem becomes even more challenging when the networks are directed. To address the resource allocation problem when the above issues are considered, we propose a novel decentralized asynchronous algorithm based on the gossip-based communication protocol and epigraph strategy. An important feature of the algorithm is that it is implemented in a completely decentralized manner in the case of asynchronous communication and directed networks. We provide theoretical proof to guarantee the convergence of the proposed algorithm, which indicates that it can successfully allocate the optimal resource. When solving the resource allocation problem over time-varying directed networks, we further discuss a related decentralized asynchronous algorithm according to the random sleep protocol. Numerical examples are given to demonstrate the viability and performance of the algorithms.