Accelerated Dynamical Approaches for a Class of Distributed Optimization with Set Constraints

In this work, strongly convex and generally convex distributed optimization problems with set constraints are studied. In order to search for the optimal solution with faster decay rate, two accelerated dynamical systems are proposed, which can directly express the convergence rate under a unified framework. The two systems use control protocol and projection operator with auxiliary variable to figure out uniform constraints and set constraints, respectively. Firstly, a scale-time accelerated dynamical system with accelerated decay rate O(1/t
²
) is put forward to work out distributed optimizations with generally convex objective functions on the basis of the asymptotic vanishing damping (AVD) model. Secondly, an accelerated dynamical system based on heavy ball (HB) flow is proposed to resolve the same optimizations with strongly convex objective functions, and the exponential decay rate is obtained. Using the concept of the strong Lyapunov condition, the accelerated distributed optimizations with constraints for generally and strongly convex functions is analyzed under a unified form. In addition, the existence of a unique solution for the proposed systems is proved by means of the Cauchy-Lipschitz-Picard theorem. Finally, numerical examples are simulated, which verify that the systems are effective.