Balance of Communication and Convergence: Predefined-Time Distributed Optimization Based on Zero-Gradient-Sum

This article proposes a distributed optimization algorithm with a convergence time that can be assigned in advance according to task requirements. To this end, a sliding manifold is introduced to achieve the sum of local gradients approaching zero, based on which a distributed protocol is derived to reach a consensus minimizing the global cost. A novel approach for convergence analysis is derived in a unified settling time framework, resulting in an algorithm that can precisely converge to the optimal solution at the prescribed time. The method is interesting as it simply requires the primal states to be shared over the network, which implies less communication requirements. The result is extended to scenarios with time-varying objective function, by introducing local gradients prediction and nonsmooth consensus terms. Numerical simulations are provided to corroborate the effectiveness of the proposed algorithms.