This paper proposes scalable distributed least square algorithms for solving large-scale linear algebraic equations via multi-agent network. We consider two kinds of different decomposition structures of $Ax=b$, in which each agent only knows a sub-block of linear equation instead of the complete row or column information in the existing results. By introducing one extra variable, the least square problem of linear equation can be transformed into a distributed constrained optimization problem. Based on the augmented Lagrange method, two continuous-time scalable distributed primal–dual algorithms are developed, in which all the states of agents have fewer dimension and can be even scalars. Subsequently, the discrete-time distributed algorithms with constant step-sizes are proposed by using the Euler discretization method, and the explicit bounds of constant step-sizes are provided. Based on the KKT condition and Lyapunov stability, we show that the proposed continuous-time and discrete-time distributed algorithms collaboratively obtain a least square solution with a linear convergence speed, and also own an additional property that verifies whether the obtained solution is an exact solution. Finally, some simulation examples are carried out to verify the effectiveness of the proposed algorithms.

本文提出了可扩展的分布式最小二乘算法，用于通过多智能体网络求解大规模线性代数方程。我们考虑两种不同的分解结构$\mathrm{一个}X=b$，其中每个智能体只知道线性方程的一个子块，而不知道现有结果中的完整行或列信息。通过引入一个额外的变量，可以将线性方程的最小二乘问题转化为分布式约束优化问题。基于增广拉格朗日方法，开发了两种连续时间可扩展的分布式原始对偶算法，其中代理的所有状态都具有较少的维度，甚至可以是标量。随后，利用欧拉离散化方法提出了具有恒定步长的离散时间分布算法，并给出了恒定步长的显式界限。基于 KKT 条件和 Lyapunov 稳定性，我们证明了所提出的连续时间和离散时间分布式算法协同获得具有线性收敛速度的最小二乘解，并且还具有验证获得的解是否是精确的附加属性解决方案。最后，通过一些仿真实例验证了所提算法的有效性。