Exponential convergence of a distributed algorithm for solving linear algebraic equations

• Autores: Ji Liu, A. Stephen  Morse, Angelina Nedic, Tamer Başar
• Localización: Automatica: A journal of IFAC the International Federation of Automatic Control, ISSN 0005-1098, Vol. 83, 2017, págs. 37-46
• Idioma: inglés
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• Resumen

  • Abstract In a recent paper, a distributed algorithm was proposed for solving linear algebraic equations of the form A x = b assuming that the equation has at least one solution. The equation is presumed to be solved by m agents assuming that each agent knows a subset of the rows of the matrix A b , the current estimates of the equation’s solution generated by each of its neighbors, and nothing more. Neighbor relationships are represented by a time-dependent directed graph N ( t ) whose vertices correspond to agents and whose arcs characterize neighbor relationships. Sufficient conditions on N ( t ) were derived under which the algorithm can cause all agents’ estimates to converge exponentially fast to the same solution to A x = b . These conditions were also shown to be necessary for exponential convergence, provided the data about A b available to the agents is “non-redundant”. The aim of this paper is to relax this “non-redundant” assumption. This is accomplished by establishing exponential convergence under conditions which are the weakest possible for the problem at hand; the conditions are based on a new notion of graph connectivity. An improved bound on the convergence rate is also derived.