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.

This paper mainly studied the stochastic stability and the design of state feedback controllers for nonlinear singular continuous semi-Markov jump systems under false data injection attacks. Based on the Lyapunov function and the implicit function theorem, a basic stochastic stability condition of the system was given to ensure that the nonlinear singular semi-Markov jump system under attack was regular, impulse-free, unique, and stochastically stable. On this basis, the stochastic admissible linear matrix inequality conditions of the system were obtained by using the singular value decomposition of the matrix and Schur's complement lemma. To design the state feedback controller, based on the upper and lower bounds of the time-varying transition probability of the semi-Markov jump system and the singular value decomposition method, the stochastic stable linear matrix inequality condition of the closed-loop system under the false data injection attack was established. Finally, the validity and feasibility of the results were verified by numerical examples.

The issue of privacy leakage in distributed consensus has garnered significant attention over the years, but existing studies often overlook the challenges posed by limited communication in algorithm design. This paper addresses the issue of privacy preservation in distributed weighted average consensus under limited communication scenarios. Specifically targeting directed and unbalanced topologies, we propose a privacy-preserving implementation protocol that incorporates the Paillier homomorphic encryption scheme. The protocol encrypts only the 1-bit quantized messages exchanged between agents, thus ensuring both the correctness of the consensus result and the confidentiality of each agent’s initial state. To demonstrate the practicality of the proposed method, we carry out numerical simulations that illustrate its ability to reach consensus effectively while ensuring the protection of private information.