This paper investigates the resource allocation problem for a group of agents communicating over a strongly connected directed graph, where the total objective function of the problem is composted of the sum of the local objective functions incurred by the agents. With local convex sets, we first design a continuous-time projection algorithm over a strongly connected and weight-balanced directed graph. Our convergence analysis indicates that when the local objective functions are strongly convex, the output state of the projection algorithm could asymptotically converge to the optimal solution of the resource allocation problem. In particular, when the projection operation is not involved, we show the exponential convergence at the equilibrium point of the algorithm. Second, we propose an adaptive continuous-time gradient algorithm over a strongly connected and weight-unbalanced directed graph for the reduced case without local convex sets. In this case, we prove that the adaptive algorithm converges exponentially to the optimal solution of the considered problem, where the local objective functions and their gradients satisfy strong convexity and Lipachitz conditions, respectively. Numerical simulations illustrate the performance of our algorithms.

中文翻译：

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通过连续时间算法在有向图上进行分布式资源分配

本文研究了一组通过强连通有向图进行通信的代理的资源分配问题，其中问题的总目标函数由代理产生的局部目标函数的总和组成。使用局部凸集，我们首先在强连接和权重平衡的有向图上设计连续时间投影算法。我们的收敛分析表明，当局部目标函数是强凸的时，投影算法的输出状态可以渐近收敛到资源分配问题的最优解。特别是，当不涉及投影操作时，我们在算法的平衡点处表现出指数收敛。第二，对于没有局部凸集的简化情况，我们在强连接和权重不平衡的有向图上提出了一种自适应连续时间梯度算法。在这种情况下，我们证明自适应算法以指数方式收敛到所考虑问题的最优解，其中局部目标函数及其梯度分别满足强凸性和 Lipachitz 条件。数值模拟说明了我们算法的性能。