Distributed Algorithm for Achieving Minimum l1 Norm Solutions of Linear Equation

Jingqiu Zhou, Xuan Wang, Shaoshuai Mou, Brian D. O. Anderson

Abstract: Ahstract- This paper proposes a distributed algorithm for multi-agent networks to achieve a minimum <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$l_{1}$</tex> -norm solution to a linear equation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$Ax=b$</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$A$</tex> has full row rank. When the underlying network is undirected and fixed, it is proved that the proposed algorithm drive all agents' individual states to converge in finite-time to the same minimum <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$l_{1}$</tex> -norm solution. Numerical simulations are also provided as validation of the proposed algorithms.