A Finite-Time Protocol for Distributed Time-Varying Optimization Over a Graph

In this paper we address a time-varying quadratic optimization problem over a graph under the assumption that the problem shares the same sparsity pattern as the static graph encoding the undirected network topology over which the multi-agent systems interacts. Notably, this framework allows to effectively model scenarios in which the optimization problem is inherently embedded within the network topology, e.g., flow balancing, electrical power system managements or packet routing problems. In this regard, we propose a finite-time distributed algorithm which allows the multi-agent system to track the time-varying optimal solution over time. Specifically, we first solve the frozen-time optimization problem, providing a necessary and sufficient condition for a solution to be globally optimal. Then, based on such condition, a continuous-time distributed nonsmooth algorithm is developed. Numerical simulations are provided to corroborate the theoretical findings.