Linear Convergence in Optimization Over Directed Graphs With Row-Stochastic Matrices

• Published in: IEEE transactions on automatic control Vol. 63; no. 10; pp. 3558 - 3565
• Main Authors: Xi, Chenguang, Mai, Van Sy, Xin, Ran, Abed, Eyad H., Khan, Usman A.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.10.2018; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Acceleration; Agents and autonomous systems; Algorithms; Convergence; Convex functions; cooperative control; Cost function; directed graphs; Graph theory; Graphs; Indexes; Linear programming; Mathematical analysis; Matrix methods; Multiagent systems; Optimization; optimization algorithms
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2018.2797164

This paper considers a distributed optimization problem over a multiagent network, in which the objective function is a sum of individual cost functions at the agents. We focus on the case when communication between the agents is described by a directed graph. Existing distributed optimization algorithms for directed graphs require at least the knowledge of the neighbors' out-degree at each agent (due to the requirement of column-stochastic matrices). In contrast, our algorithm requires no such knowledge. Moreover, the proposed algorithm achieves the best known rate of convergence for this class of problems, <inline-formula><tex-math notation="LaTeX"> O(\mu ^k)</tex-math></inline-formula> for <inline-formula><tex-math notation="LaTeX">0<\mu <1</tex-math> </inline-formula>, where <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula> is the number of iterations, given that the objective functions are strongly convex and have Lipschitz-continuous gradients. Numerical experiments are also provided to illustrate the theoretical findings.