Scalable distributed least square algorithms for large-scale linear equations via an optimization approach

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.