Distributed Adaptive Convex Optimization on Directed Graphs via Continuous-Time Algorithms

This note considers the distributed optimization problem on directed graphs with nonconvex local objective functions and the unknown network connectivity. A new adaptive algorithm is proposed to minimize a differentiable global objective function. By introducing dynamic coupling gains and updating the coupling gains using relative information of system states, the nonconvexity of local objective functions, unknown network connectivity, and the uncertain dynamics caused by locally Lipschitz gradients are tackled concurrently. Consequently, the global asymptotic convergence is established when the global objective function is strongly convex and the gradients of local objective functions are only locally Lipschitz. When the communication graph is strongly connected and weight-balanced, the algorithm is independent of any global information. Then, the algorithm is naturally extended to unbalanced directed graphs by using the left eigenvector of the Laplacian matrix associated with the zero eigenvalue. Several numerical simulations are presented to verify the results.