分布式连续时间近似投影协议用于最短距离优化问题
Distributed Continuous-time Approximate Projection Protocols for Shortest Distance Optimization Problems
AI总结：
本文研究了多智能体网络中分布式最短距离优化问题，提出基于近似投影的连续时间动态协议以解决投影不确定性问题，并分别给出了非空交集和空交集情况下保证最优收敛的步长和近似角条件。
摘要：
本文研究了多智能体网络中分布式最短距离优化问题，即多智能体协作最小化与某些凸集的二次距离之和，其中每个集仅与一个智能体相关联。为解决具有投影不确定性的优化问题，我们提出了一种基于新近似投影概念的分布式连续时间动态协议。每个智能体只能获得其凸集边界上的近似投影点，并在时间变化的通信图上与邻居通信。首先，我们证明无论近似角有多大，系统状态始终有界，且如果步长的下限大于零，则相对于所有初始条件统一有界。然后，在非空交集和空交集的两种情况下，分别给出了保证最优收敛的步长和近似角条件。此外，我们还给出了空交集情况下的最优解特性，并在恒定步长和近似角情况下给出了智能体估计与最优点之间的收敛误差。
英文摘要：
In this paper, we investigate the distributed shortest distance optimization problem for a multi-agent network to cooperatively minimize the sum of the quadratic distances from some convex sets, where each set is only associated with one agent. To deal with the optimization problem with projection uncertainties, we propose a distributed continuous-time dynamical protocol based on a new concept of approximate projection. Here each agent can only obtain an approximate projection point on the boundary of its convex set, and communicate with its neighbors over a time-varying communication graph. First, we show that no matter how large the approximate angle is, the system states are always bounded for any initial condition, and uniformly bounded with respect to all initial conditions if the inferior limit of the stepsize is greater than zero. Then, in the two cases, nonempty intersection and empty intersection of convex sets, we provide stepsize and approximate angle conditions to ensure the optimal convergence, respectively. Moreover, we give some characterizations about the optimal solutions for the empty intersection case and also present the convergence error between agents' estimates and the optimal point in the case of constant stepsizes and approximate angles.
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