Exponential convergence rate of distributed optimisation for multi-agent systems with constraints set over a directed graph

In this study, the authors propose a novel distributed continuous-time algorithm based on projection and gradient to solve the optimisation problem of a multi-agent system under any initialisation manner over a directed graph. The considered cost function is a summation of all local cost functions with local constraints set. The point of the proposed protocol is that a new scheme is proposed to offset the non-zero local gradients of local cost functions at the minimiser. The optimal solution of the proposed algorithm is shown using the variational inequality under some conditions. Moreover, exponential convergence rate of the designed algorithm is verified with the help of the identity transformation and Lypaunov stability theory. Finally, a numerical example and a comparison are provided to demonstrate the effectiveness of the theoretical results obtained.