This paper investigates a distributed optimization problem based on the framework of a multi-agent system over a directed communication network, where the global cost function is the sum of the local cost functions of agents. The communication network is abstracted as a weight-unbalanced directed graph. First, a surplus-based accelerated algorithm with a fixed stepsize (SAAFS) is proposed by integrating the gradient tracking strategy into the surplus-based consensus protocol to address the problem considered. The matrix norm argument and matrix perturbation theory are employed to prove the linear convergence of SAAFS under the assumption that each local cost function is strongly convex with the Lipschitz continuous gradient. Second, the limitation of the stepsize, which is common to all agents, is relaxed in the cases of different stepsizes for each agent, such that the surplus-based accelerated algorithm with an uncoordinated stepsize (SAAUS) is proposed. It is proven that SAAUS also has a linear convergence rate if the upper bound of the uncoordinated stepsize at each agent is restricted by a sufficiently small positive number. Finally, two simulation examples are provided to evaluate the proposed algorithms and illustrate that both SAAFS and SAAUS achieve acceleration, particularly for ill-conditioned optimization problems.

本文研究了基于定向通信网络上的多智能体系统框架的分布式优化问题，其中全局成本函数是智能体局部成本函数的总和。通信网络被抽象为一个权重不平衡的有向图。首先，通过将梯度跟踪策略集成到基于盈余的共识协议中来解决所考虑的问题，提出了一种固定步长的基于盈余的加速算法（SAAFS）。矩阵范数假设每个局部成本函数在 Lipschitz 连续梯度上是强凸的，则采用参数和矩阵微扰理论来证明 SAAFS 的线性收敛性。其次，放宽了所有智能体共有的步长限制，在每个智能体的步长不同的情况下，提出了基于剩余的非协调步长加速算法（SAAUS）。已经证明，如果每个代理的不协调步长的上限受足够小的正数限制，SAAUS 也具有线性收敛速度。最后，提供了两个仿真示例来评估所提出的算法，并说明 SAAFS 和 SAAUS 都实现了加速，特别是对于病态优化问题。