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multi-agent network; distributed optimization; linear algebraic equation; least squares solution; solvability verification
In this paper, we consider the problem of solving a linear algebraic equation $Ax=b$ in a distributed way by a multi-agent system with a solvability verification requirement. In the problem formulation, each agent knows a few columns of $A$, different from the previous results with assuming that each agent knows a few rows of $A$ and $b$. Then, a distributed continuous-time algorithm is proposed for solving the linear algebraic equation from a distributed constrained optimization viewpoint. The algorithm is proved to have two properties: firstly, the algorithm converges to a least squares solution of the linear algebraic equation with any initial condition; secondly, each agent in the algorithm knows the solvability property of the linear algebraic equation, that is, each agent knows whether the obtained least squares solution is an exact solution or not.
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