Convergence Analysis of a Distributed Optimization Algorithm with a General Unbalanced Directed Communication Network

• Published in: IEEE transactions on network science and engineering Vol. 6; no. 3; pp. 237 - 248
• Main Authors: Li, Huaqing, Lu, Qingguo, Huang, Tingwen
• Format: Journal Article
• Language: English
• Published: Piscataway IEEE 01.07.2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Communication; Communication networks; Communications networks; Constraints; Convergence; Convexity; Coupling; Couplings; Distributed constrained optimization; Economic models; Linear programming; Load management; Mathematical analysis; Optimization; Power systems; primal-dual (sub)gradient algorithm; Smoothness; unbalanced directed network; uncoordinated step-sizes; Upper bounds
• ISSN: 2327-4697; 2334-329X
• DOI: 10.1109/TNSE.2018.2848288

In this paper, we discuss a class of distributed constrained optimization problems in power systems where the target is to optimize the sum of all agents’ local convex objective functions over a general unbalanced directed communication network. Each local convex objective function is known exclusively to a single agent, and the agents’ variables are constrained to global coupling linear constraint and individual box constraints. To collaboratively solve the optimization problems, existing distributed methods mostly require the communication network to be balanced or have the knowledge of in-neighbors’ out-degree for all agents, which are quite restrictive and hardly inevitable in practical applications. In contrast, we investigate a novel distributed primal-dual augmented (sub)gradient algorithm which utilizes a row-stochastic matrix (does not need each agent to know its in-neighbors out-degree) and employs uncoordinated step-sizes, and yet exactly converges to the optimal solution over a general unbalanced directed communication network. Under the assumptions of the strong convexity and smoothness on the aggregate objective functions, it is proved that the algorithm geometrically converges to the optimal solution if the uncoordinated step-sizes do not exceed the upper bound. An explicit analysis for the convergence rate of the proposed algorithm is also characterized. To manifest effectiveness and applicability of the proposed algorithm, three case studies are presented to solve two practical problems in power systems.