A Distributed Algorithm for Computing a Common Fixed Point of a Finite Family of Paracontractions

• Published in: IEEE transactions on automatic control Vol. 63; no. 9; pp. 2833 - 2843
• Main Authors: Fullmer, Daniel, Morse, A. Stephen
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-09-2018; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Autonomous agents; Convergence; Distributed algorithms; Estimates; Estimating techniques; Graph theory; Linear matrix inequalities; multi-agent systems; nonlinear dynamical systems; Nonlinear equations; optimization; Radio frequency
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2018.2800644

A distributed algorithm is described for finding a common fixed point of a family of m > 1 nonlinear maps Mi : ℝ n → ℝ n assuming that each map is a paracontraction and that at least one such common fixed point exists. The common fixed point is simultaneously computed by m agents assuming each agent i knows only Mi, the current estimates of the fixed point generated by its neighbors, and nothing more. Each agent recursively updates its estimate of a fixed point by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a time-varying directed graph N(t). It is shown under suitably general conditions on N(t), that the algorithm causes all agent estimates to converge to the same common fixed point of the m nonlinear maps.