A Discontinuous Projection-Based Algorithm for Solving Distributed Optimization with Linear Equation Constraints

Within the context of distributed optimization, the convergence rate and the communication cost are two important issues to be considered when designing the algorithms. To address these problems, we proposed a discontinuous projection-based dynamics for solving distributed optimization problem with general objective functions and separated linear equation constraints. To save the communication resource, the proposed protocol is skillfully designed without introducing any auxiliary variable where only the communication of local estimations on the system decision variables are involved. Based on Lyapunov stability theory and nonsmooth analysis, it is shown that the states of all the agents can achieve finite-time consensus and then converge to the optimal solution of the considered optimization problem asymptotically. It is further revealed that, with the condition that the objective function has invertible Hessian matrix, the exponential convergence to the optimal solution will be guaranteed. Finally, a numerical example is performed to verify the theoretical results.