Consensus-Based Algorithm for Distributed Continuous-Time Convex Optimization Over Undirected and Directed Networks

In this article, the distributed continuous-time convex Optimization Problem (OP) is investigated over undirected and balanced directed graphs. The cost function of the distributed convex OP is determined as the sum of local convex functions where each of them is known only for one agent. The proposed algorithm consists of two main steps. The first step is a consensus-based scheme which is in combination with the gradient descent method. Employing the Lyapunov theory and LaSalle’s invariance principle, the convergence to the Optimal Solution (OS) is analyzed. Moreover, inspired by the average consensus, in the second step the Optimal Value (OV) of the distributed convex OP is calculated. Using consensus concepts converges to the OV is substantiated in the second step. Therefore, the offered algorithm can calculate the OS and the OV of the distributed convex OP with no need for the strong convexity assumption. Beyond the theoretical findings, the results from simulations are also showcased to demonstrate the efficiency and accuracy of the proposed algorithm.