Distributed Continuous-Time Convex Optimization on Weight-Balanced Digraphs

• Published in: IEEE transactions on automatic control Vol. 59; no. 3; pp. 781 - 786
• Main Authors: Gharesifard, Bahman, Cortes, Jorge
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.03.2014; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Automatic control; Convergence; Convex functions; Directed graphs; distributed optimization; Dynamic tests; Dynamics; Eigenvalues and eigenfunctions; Graph theory; Graphs; Invariance; Laplace equations; Linear programming; networked control systems; Optimization; Trajectory
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2013.2278132

This technical note studies the continuous-time distributed optimization of a sum of convex functions over directed graphs. Contrary to what is known in the consensus literature, where the same dynamics works for both undirected and directed scenarios, we show that the consensus-based dynamics that solves the continuous-time distributed optimization problem for undirected graphs fails to converge when transcribed to the directed setting. This study sets the basis for the design of an alternative distributed dynamics which we show is guaranteed to converge, on any strongly connected weight-balanced digraph, to the set of minimizers of a sum of convex differentiable functions with globally Lipschitz gradients. Our technical approach combines notions of invariance and cocoercivity with the positive definiteness properties of graph matrices to establish the results.