Multi-agent Network Flow Design to Solve Matrix Equation Based on Nonsmooth Convex Optimization

This paper studies the distributed computation of a linear matrix equation in the form of $\sum_{i=1}^{r}A_{i}XB_{\dot{i}}=\sum_{i=1}^{r}C_{i}$, over multi-agent networks from an optimization perspective, with some nonsmooth requirements of the optimization variable $X$ at the same time. In this multi-agent network, agent $i$ can only get access to $A_{i}, B_{i}, C_{i}$ and communicate with its neighbors. Then, a distributed continuous-time algorithm, from a distributed constrained optimization viewpoint, is proposed to obtain the solution with balance between its least squares bias and requirements of the nonsmooth convex function, where the saddle point method and derivative feedback technique are employed to deal with complicated problem. With help of the Lyapunov stability and semi-stability analysis, we prove the convergence of the algorithm for any initial condition.