Privacy-preserving distributed optimization algorithm for directed networks via state decomposition and external input

In this paper, we study the privacy-preserving distributed optimization problem on directed graphs, aiming to minimize the sum of all agents' cost functions and protect the sensitive information. In the distributed optimization problem of directed graphs, agents need to exchange information with their neighbors to obtain the optimal solution, and this situation may lead to the leakage of privacy information. By using the state decomposition method, the algorithm ensures that the sensitive information of the agent will not be obtained by attackers. Before each iteration, each agent decomposes their initial state into two sub-states, one sub-state for normal information exchange with other agents, and the other sub-state is only known to itself and invisible to the outside world. Unlike traditional optimization algorithms applied to directed graphs, instead of using the push-sum algorithm, we introduce the external input, which can reduce the number of communications between agents and save communication resources. We prove that in this case, the algorithm can converge to the optimal solution of the distributed optimization problem. Finally, a numerical simulation is conducted to illustrate the effectiveness of the proposed method.