Distributed optimization for multi-agent systems with constraints set and communication time-delay over a directed graph

This paper is concerned with the problem of distributed optimization for a multi-agent system with constraints set and communication time-delay over a directed graph. The considered cost function is a summation of all local cost functions associated with each agent. Firstly, a novel distributed algorithm is developed to solve such a problem, where auxiliary state variables are also exchanged to compensate the nonzero gradient of local cost function and accelerate the convergence of estimate states to the optimal point. Secondly, the minimizer of distributed optimization of a multi-agent network is determined by the variational inequality in spite of the existence of time delay. Furthermore, delay-dependent and delay-free sufficient conditions on the convergence of states of agents to the optimal point are derived by constructing a new Lyapunov-Krasovskii functional, respectively. Finally, a numerical example and a comparison are provided to validate the obtained results.